Theories of Everything with Curt Jaimungal - Scott Aaronson: The Greatest Unsolved Problem in Math

Episode Date: December 11, 2023

YouTube link https://youtu.be/1ZpGCQoL2Rk Scott Aaronson joins us to explore quantum computing, complexity theory, Ai, superdeterminism, consciousness, and free will. TIMESTAMPS:- 00:00:00 Introduction- 00:02:27 Turing universality & computational efficiency- 00:12:35 Does prediction undermine free will?- 00:15:16 Newcomb's paradox- 00:23:05 Quantum information & no-cloning- 00:33:42 Chaos & computational irreducibility- 00:38:33 Brain duplication, Ai, & identity- 00:46:43 Many-worlds, Copenhagen, & Bohm's interpretation  - 01:03:14 Penrose's view on quantum gravity and consciousness- 01:14:46 Superposition explained: misconceptions of quantum computing  - 01:21:33 Wolfram's physics project critique- 01:31:37 P vs NP explained (complexity classes demystified)- 01:53:40 Classical vs quantum computation- 02:03:25 The "pretty hard" problem of consciousness (critiques of IIT) NOTE: The perspectives expressed by guests don't necessarily mirror my own. There's a versicolored arrangement of people on TOE, each harboring distinct viewpoints, as part of my endeavor to understand the perspectives that exist. THANK YOU: To Mike Duffy, of https://dailymystic.org for your insight, help, and recommendations on this channel.   - Patreon: https://patreon.com/curtjaimungal (early access to ad-free audio episodes!)- Crypto: https://tinyurl.com/cryptoTOE- PayPal: https://tinyurl.com/paypalTOE- Twitter: https://twitter.com/TOEwithCurt- Discord Invite: https://discord.com/invite/kBcnfNVwqs- iTunes: https://podcasts.apple.com/ca/podcast/better-left-unsaid-with-curt-jaimungal/id1521758802- Pandora: https://pdora.co/33b9lfP- Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e- Subreddit r/TheoriesOfEverything: https://reddit.com/r/theoriesofeverything- TOE Merch: https://tinyurl.com/TOEmerch LINKS MENTIONED:- Scott's Blog: https://scottaaronson.blog/- Newcomb's Paradox (Scott's Blog Post): https://scottaaronson.blog/?p=30- A New Kind of Science (Stephen Wolfram): https://amzn.to/47BTiaf- Jonathan Gorard's Papers: https://arxiv.org/search/gr-qc?searchtype=author&query=Gorard,+J- Boson Sampling (Alex Arkhipov and Scott Aaronson): https://arxiv.org/abs/1011.3245- Podcast w/ Tim Maudlin on TOE (Solo): https://youtu.be/fU1bs5o3nss- Podcast w/ Tim Palmer on TOE: https://youtu.be/883R3JlZHXE

Is there always a clever shortcut for every problem where we can efficiently recognize a correct answer? And I think it's now recognized as one of the central unsolved problems in all of mathematics. Scott Aronson is a professor of theoretical computer science at UT Austin, particularly known for his work on quantum computing and complexity theory. Today we talk about free will, we talk about consciousness, complexity classes, superdeterminism, and even quantum computing. That last one in particular, we talk about what quantum supremacy actually means, rather than how it's promulgated by people like Michio Kaku and other popularizers of science. Scott explores and teaches these ideas with extreme simplicity, as well as joy, which is a rare combination. Welcome to this

channel, my name is Kurt Jaimungal, and for those of you who are unfamiliar, this is Theories of Everything, where we delve into the topics of mathematics, physics, artificial intelligence, and consciousness with depth and rigor. This commitment stems from a recognition that popular science articles often peddle superficial falsehoods, leaving a discerning audience, like yourself, yearning for technical accuracy and substantive discourse. In other words, the audience of Toe is the audience that's willing to invest the extra time to understand the nature of reality, and not be stuck in the mysticism that characterizes, say, Neil deGrasse Tyson,

explaining that, whoa, quantum mechanics is both a wave and a particle. Cool, bro. Like, what does that mean? In order to understand how that's misleading, one needs to know what a complex linear combination is. And so we'd rather explain that than broadcast that a particle is both up and down at the same time. Enjoy this podcast with Scott Aronson. Welcome, Professor. It's an honor to speak with you. I've been following you for a few years. Thanks. Great to be here. I mean that in a non-creepy way. What got you interested in computational complexity? Well, I mean, I got into computer science as an adolescent because I

wanted to create my own video games, mostly. And so I learned, you know, what I could about programming, you know, and really it was a revelation to find out that engineering a video game just reduces to writing these lines of code. That was not what I had expected. I like to say that for me, it was like learning where babies come from. Why didn't I find that out before? But as I learned more about it, I realized that while I like programming, I was really not any good at software engineering at, you know, making making my code work with other people it was probably, you know, on the more theoretical side, right? You know, thinking about what kinds of programs could and QBASIC, and these ancient languages, I figured, okay, to make a really sophisticated program, clearly you would need a more powerful language. And it was a second revelation to me that, no, you just very rapidly hit this ceiling of Turing universality, where just a very simple programming language

becomes capable in principle of expressing anything that any programming language could express, right? You know, and so then, you know, the, you could say, one of the biggest remaining questions is about efficiency, right? Is about, well, you know, which among one of the biggest remaining questions is about efficiency, right? It's about, well, you know, which among all of the problems that computers can solve, you know, which ones can they solve in a reasonable amount of time, right? Or a reasonable amount of memory. And so, you know, when I was 15 or 16, I would have learned what is the P versus NP problem. And, you know, and that problem

is just so stunning, you know, that humans could ask something, you know, so basic, and yet, you know, so concrete, you know, that has a definite answer one way or the other. And, you know, I had fantasies for, you know, a few months that, well, you know, all of these experts, you know, must have gotten, you know, stuck in a rut and I'll come in as a 15-year-old and, you know, with no preconceptions and I'll polish this problem off. You know, I think it was good to have that experience. Yeah. Once in my life, you know, once was enough to get, you know, to get disabused of that. And then, yeah, at some point, I learned about quantum computing,

which we can talk about more, but that actually changes the rules of computational complexity based on our best current theory of physics. And that was then irresistible to understand because, you know, somehow these very basic questions in physics and in computer science were merging with each other, right? And, you know, it was all, you know, a story about computational complexity. I mean, you know, if you don't care about complexity, then there's

basically no reason to build a quantum computer, right? You know, anything it can do can be simulated by a classical computer, albeit exponentially slower. Okay, so you need complexity theory to even pose the questions about, you know, what are quantum computers good for? And, you know, but this was a field where there was a lot of low-hanging fruit, you know, in the late 1990s when I started really getting into it. And, you know, I was also extremely interested in AI. I thought maybe I would do that. But, you know, again, there was the difficulty that AI so often boils down to in practice to software engineering, which I wasn't so good at. in practice, to software engineering, which I wasn't so good at. Now, it was when I, so I was an undergrad at Cornell. When I applied for grad school, you know, I got into where I really wanted to go, which was UC Berkeley. But it was the AI people who recruited me there, not the theory people.

But I sort of, you know, secretly, I guess, wanted to do quantum complexity theory. So after a year of doing AI, you know, I switched. And so, you know, and then I ended up, you know, computational complexity and quantum computing ended up being interesting enough that I've, you know, spent more than 20 years of my life on them. And only now, finally, in the last couple of years, I'm circling back to AI with this stuff that I'm doing for OpenAI. Yeah, I'd love to speak with you about your work at OpenAI. First, is computational complexity, algorithmic complexity, and quantum complexity distinct? Well, I would say that computational complexity is the whole field that studies sort of the inherent computational resources that are needed to solve problems, right?

And that includes time, memory, it could include energy, randomness, parallelism, and quantum computing is a part of that, right? You know, or you could say quantumness is another computational resource that you can throw into the mix and then see how it changes things. So, yeah, but it is the field that studies sort of the inherent capabilities and limitations of algorithms. You know, sometimes, you know, people who are only interested in just, you know, solving a practical problem with the best algorithm that they can find for that problem, they might not call themselves complexity theorists, right? They might just be algorithms people. But as soon as you start asking the question, what is the best algorithm for this task in terms of the scaling of resources as the input gets larger and larger? How do I know that it's the best algorithm, right? What else would it imply if there were a faster algorithm? As soon as you start asking things like that, then you're doing complexity theory. It sounds like it's easy to show that something is a more efficient algorithm than another,

but to show that something is the best, how do you go about doing that? Yes, well, good question. The field has been struggling with that for half a century. question. You know, the field has been struggling with that for half a century. So yeah, in order to give a faster, you know, in order to show that there is a faster algorithm to solve a given problem, typically, the way you do it is you just give that algorithm, you know, you, you give it, you know, and that could already be very non trivial, because you have to analyze the algorithm, you have to prove that it works. And you have to prove that it actually terminates after this reasonable amount of time. Okay. So that can, that can already be, be non-trivial. Okay. But, but now, you know,

if you ask, is this the best algorithm, you know, how do we know it's the best? Okay. Now you're trying to prove a negative, right? And that is inherently, you know, a vastly harder undertaking. And, and, you know, in some sense, you know, the, you know, the main reason why we, we view it as a, as a, as a goal to strive for at all, is that, you know, computer science was sort of born, you know, with knowledge about its own limitations, right? When Alan, with knowledge about its own limitations. When Alan Turing introduced the Turing machine, which is sort of the mathematical model for what a computer is in the 1930s,

he also, as sort of the key application of his new theory, he proved that certain problems are not solvable by any Turing machine, right, in any amount of time. Okay, this was the famous unsolvability of the halting problem. Okay, and it built on Gödel's incompleteness theorem, you know, which had been proved just five years prior. But, you know, it's now a statement that, you know, for certain problems, and famous example is the halting problem. I give you a program, and you have to determine whether it ever stops running, say, when run on a blank input.

And Turing showed that there is no program to solve this problem in any amount of time. And the argument is basically self-referential. And Turing showed that there is no program to solve this problem in any amount of time. And the argument is basically self-referential. You say, well, suppose that there were such an algorithm, then we could contrive things in such a way that that algorithm would be fed its own code as input. And then it would have to do the opposite of whatever it does. It's like, if it halts, then it would have to, you know, when run on itself as input, then it would have to run forever. And if it runs forever, when run on itself as input, then it would have to halt. And since, you know, that's a contradiction, the only conclusion is

that the program can't have existed. Okay. And so, you know, we've known since the beginning of computer science that you can use these sort of self-referential methods to understand something about the limitations, you know, of any algorithm, right? In a kind of magical way, you know, without having to roll up your sleeves and delve into the details of what the Turing machine is doing, right? And then, you know, in the 1960s, you know, some of the first complexity theorists like Yoris Hartmanis, who passed away recently, and Richard Stearns, managed to go further. And they used similar self-referential arguments to show, for example, that there are problems involving n-bit inputs that are solvable in n-cubed steps but are not solvable in n-squared steps okay there are other problems that are solvable in n to the fourth steps but not in

n-cubed steps and so on can you give an example yeah the simplest example would just be i give you a program and now you have to decide whether it halts in n to the fourth steps or not And now you have to decide whether it halts in n to the fourth steps or not. That is solvable in slightly more than n to the fourth steps. But, you know, a sort of scaled down version of Turing's argument shows that it is not solvable in n cubed steps. And basically, basically, because if it were, then the program could predict what it itself is going to do faster than it can do it. Okay. And it's kind of like, you know, the, like, like, you know, uh, this is like a paradox that a five-year-old could understand. It's like, you know, if I could, you know, if I, if I knew for certain, you know, whether I'm going to raise one finger or two fingers, you know, 10 seconds from now, then I could just resolve to do the opposite of whatever I predicted I would do. And so that's not possible.

To you, does this touch on free will? Some people think it does. I mean, I tend to think that if there were a computer in another room, and it ran faster than my brain does, and it perfectly predicted what I was going to do before I do it, and maybe it just leaves its prediction in a sealed envelope, you know, but then after I take the action, then we can open the envelope and we can see that it perfectly predicted what I would do. I would say, you know, that, that would, that would really profoundly shake my, my sense of free will, you know, just speaking, speaking personally. Right. And, and I would say that based on the known laws of physics, we don't actually know whether that prediction machine can exist or not, right? It comes down to questions about how accurately would you have to scan someone's brain? Would you have to go all the way down to the quantum mechanical level? Would that not be necessary, right? And I would say that, you know, the thing that most people don't realize

is that this is an empirical question, you know, who's, you know, maybe whose answer we'll someday know, but we don't know it yet. And that's the, that's sort of, you know, what I would advocate as the best sort of empirical replacement for the free will question, right? And if you accepted that, then it's, you know, the fact that I myself can't predict, you know, my future actions is not really the core of the matter. You know, the question is just whether any machine could do it. Yeah, why is the sealed envelope important? Well, just because, well, because if I saw the prediction, then I could resolve to do the opposite. Yeah, so if this machine existed, does it still say something about your free will if you

were able to look at it and you could go against the wishes of the machine or the predictions of it? Well, yeah. I mean, you could say if that machine cannot be reliably built, you know, if any attempt to build it consistent with the laws of physics, you know, fails, then that seems to me like about as far as science, you know, could possibly go in saying that, well, you know, there seems to be something that, you know, that corresponds to part of what we mean by free will, right? There is this inherent unpredictability to our actions. And, you know, and conversely,

if the machine did exist, then that seems to be, to me, like about as far as science could possibly go towards saying, you know, actually, you know, free will is an illusion, right? Not, you know, not just in some abstract metaphysical way, but because, you know, here is the machine that predicts what you will do, you know, look at it, try it out. Yeah, you had a blog post on Newcomb's paradox. Yes. Can you please outline it and then what your proposed resolution is, if it exists? Sure. A Newcomb's paradox is the thing where, you know, we imagine this super intelligent

predictor, you know, just like I was talking about before, you know, that this sort of machine or being that, you know, knows what, you know, you're going to do before you do it. And it puts two boxes on a table, okay? And inside of the first box, you know, there might be nothing and there might be a million dollars, okay? And inside of the second box, there is definitely a thousand dollars. And now you have a choice. You can either take the first box only, or you can take both of the boxes. Okay, but now the predictor, the catch is that the predictor, you know, has told you in advance that if it predicts that you're going to take both boxes, then it will leave the first box empty. So it punishes greed. Yes.

Right. If, if it predicts that you're going to take only the first box, then it puts a million dollars in it. Okay. So, and, and let's say that the predictor has played this game with, you know, a thousand people before you, and it's never been wrong. Right. So then, then, you know, what do you do? Do you, do you, you know, as, as, you know, the people have actually made it into verbs, you know, do you one box or do you two box in the, in the, in the new comb paradox? And, you know, and, and there seemed to be like basic principles of rationality that, you know, that you could use to, to, to prove either answer is correct. Right. either answer is correct, right? On the one hand, you know, everyone who takes only the first box ends up, you know, about a million dollars richer than the people who try to take both, right? And, you know, by the whole setup of the problem is that, you know, that's because the predictor knew and, you know, it's over. But on the other hand, you know, by the time you're contemplating your decision, the million dollars is either in the box or not. Right. And so how could your decision possibly affect, you know,

what is in the box, it would seem like it would have to be a backwards in time causation. Right. And therefore, you know, you know, whatever is in the first box, you're going to have $1,000 more than that, if you take both boxes boxes and therefore you should take both right so so you know so we can prove two contradictory answers you know that is the basic setup of a paradox and uh uh you know and people have argued about this for half a century there is an enormous literature on this problem uh and you know many different points of view um you know i had a blog post back in 2006 where um you know i i suggested like like well what what what seemed to me like the natural resolution of this and and since then you know i've learned that you know that that other people have had

broadly similar ideas so you know some of them do cite that blog post of mine. But, you know, my resolution of the paradox was, okay, I think that, you know, in this scenario, you should take one box, right? You should one box. Okay, but the question is why, right? The question is, how can we possibly explain how your decision to one box could affect the predictor, could affect whether the predictor puts the money in the box. And now the key is, we have to think harder about what the world would be like with this predictor in it. The predictor contains within it a perfect simulation of you. I mean, whatever you're going to base your decision on, whatever childhood memory, whatever detail of your brain function, the predictor knows all of it, right? By hypothesis, right? But the way that I would describe that is that the predictor has effectively brought into being a second copy of you, a second instantiation of you, right? And now, you know, the key is that as you're

contemplating your decision, whether to one box or two box, you know, you have to think of yourself as somehow, you know, being both versions of you at once, right? Or, you know, or perhaps, you know, you don't know which one you are, right? If you are the simulation being run by the predictor, well, then, of course, your decision can affect what the predictor does. So, you know, you don't even, you know, in the scenario, you know, that was hypothesized, like, you have to be radically uncertain about where you physically are, about what time it is, right? Like, these are the kinds of things that you have to worry about in a world where there really could be perfect predictors of yourself, right? I mean, a different, you know, a much more boring resolution would be to say, well,

you know, I'm not going to worry about Newcomb's paradox because I believe that this predictor cannot exist at all. Right? And as I said, I regard that as an empirical question to which we don't yet know the answer. Two concepts that I see as related are the no-cloning theorem and computational irreducibility. So this is something popularized by Wolfram, which I know you know, so I'll get you to explain it to the audience. But the no-cloning may have implications that such a machine can't exist because it can't be a perfect copy of you. Yeah. Okay. So, so, so yeah. So, so by the way, no cloning and computationally reducibility are two totally different things. You know, we can, we can, we, we, we can talk about both of them.

Okay. But the no cloning theorem is just a very, very basic fact about quantum mechanics. Okay. And it says that there is no physical operation that you can do that takes as input an arbitrary quantum state, you know, an unknown quantum state, like let's say a qubit, you know, a quantum bit, a superposition of a, you know, zero and a one, okay, and that produces as output, two identical copies of that state. Okay, so there is no quantum Xerox machine that is possible. The ability to make perfect copies of things, this is fundamentally a classical phenomenon. It is not true quantumly.

I'm sorry to interrupt. I'm sure you've heard of Leibniz's law of indiscernible. Yeah. Okay, so this isn't that. This isn't saying indiscernible. Yeah. Okay. So this isn't that. No. This isn't saying like, look. This is a fact about physics that could have been false. But it is true because for a century, every experiment has told us that quantum mechanics is true. And as long as quantum mechanics remains the basis of physics, then this is true.

But it's not something that is a priori knowable. And so the sense of copying that I mean is copying the information. So think about classical bits. We all know that classical bits can be copied. If you have, this has been the bane of the music industry and the software industry, you know, if you have a file, right, you can make a copy of that file, right? You know, it won't be, you know, it'll have a different physical representation, you know, somewhere else in your computer's memory or in a different computer's memory, you know, but it will encode

the same bits of information. Okay. So Napster exists because the no-cloning theorem is purely quantum? Precisely. Precisely. Yeah. So classical information can be copied, okay? But what we're saying is that in quantum mechanics, you know, even the information cannot be copied, okay? So quantum, we have to take a step back and say, you know, what is quantum information, right? It's basically, you know, the, the, the basic building block in quantum information is, is, you know, what's called this quantum bit or qubit. Okay. And this is a bit that, you know, it doesn't have to be definitely zero or definitely one, you know, and, and, and, and, you know, and, and once people hear that, then they say, Oh, well, then, you know, that just sounds like a fancy way of saying, well, it's either zero or it's one, and you don't know which it is. Right. But, you know, we, we, we know how to deal with that classically, right? We could, you know, we could have a, a random bit, you know, a bit that has like a 40% chance of being one and a 60% chance of being zero.

And until you look, you don't know which it is. So you kind of have to think about both possibilities. But then once you look, you know, right? Okay, now that is not a qubit, right? Qubit is more interesting than that. Okay, because the key thing that quantum mechanics says is that to every possible configuration that a system could be in, like zero or one in the case of a bit, you have to assign not just a probability, you have to assign a complex number. These complex numbers are called amplitudes. They're the basic quantities of quantum mechanics, right? And so,

for example, a qubit, a quantum bit, might have a square root of a half amplitude to be zero, and it might have a minus square root of a half amplitude to be one, okay? Now, the, or, you know, they could actually be complex numbers, you know, a real plus an imaginary number. Okay. And now, when I make a measurement, then these amplitudes convert into probabilities. And the way they do that is one of the most famous rules in all of physics. It's called the Born rule. It says that the probability that I see a particular outcome is equal to the square of the absolute value of the amplitude for that outcome. So if I had an equal superposition, zero plus one divided by the square root of two, the qubit zero plus the qubit one divided by the square root of two, okay, and then I measure it, then I'm going to see zero or one equally likely, okay? But there are other things that I can do besides just measuring the qubit to ask it whether it is zero or one, okay? I can, you know,

when the qubit is isolated, then these amplitudes can change in time by by rules that are that are not familiar to our experience um uh you know and figure i can take the the list of all the amplitudes of all the possible states and i can uh do something to to my my system you know my particles or whatever that has the effect of doing a linear transformation on that list of amplitudes right so so you know, my particles or whatever, that has the effect of doing a linear transformation on that list of amplitudes, right? So, you know, you could say in some sense what, you know, quantum mechanics tells us is that, you know, the operating system of the universe is linear algebra, you know, it's matrices and vectors, right? My states are these vectors of amplitudes, matrices and vectors, right? My states are these vectors of amplitudes, these lists of complex numbers. My time evolution, the way the state changes over time while it's isolated, is that I apply what's called a norm-preserving linear transformation, you know, or also called a

unitary transformation, okay? These are like linear transformations, matrices that always map unit vectors to other unit vectors, right? So they always preserve the length of the vector, okay? But an example would be a rotation, right? Like I could take a qubit, you know, that is some intermediate state between zero and one, right? Somewhere on like the the unit circle where you know here's zero here's one here's minus zero here's minus you know minus one and i could rotate by a certain fixed angle right or i could reflect about an axis hey these are unitary transformations that i can do you know and then i measure and then i i you know measurement is a destructive operation it sort of collapses me to a single outcome. But now the key phenomenon that told physicists that the world works this way in the first place a century ago,

and it is sort of the signature that something quantum is going on is called interference. So now if I want to know, let's say, how likely is a particle to hit a certain spot on a screen, then I have to calculate the amplitude for that thing to happen. And then take the squared absolute value, and that gives me the probability. But the amplitude is a sum of a whole bunch of contributions. Okay. One from every possible path that the photon could have taken or the particle could have taken in order to reach, you know, this, this end point. Right. And now if some of those paths that it could have taken have make a positive contribution

and others make a negative contribution then they can interfere destructively and cancel each other out meaning like the total amplitude will be zero and then the particle will never be found there at all right whereas you know and here's the the even crazier part if i close off one of the paths like you know like the you know the famous two slit experiment where there are two slits that this particle could go through if i off one of the paths, like the famous two-slit experiment, where there are two slits that this particle could go through, if I block one of the two slits, well, now I only have a positive contribution or only a negative contribution, depending on which slit I blocked.

So now the particle can appear at that end point. So to say it again, by decreasing the number of paths that a particle can take to get somewhere, I can increase the chance that it gets there. Okay, that is something that, you know, just forget about all the low-level details of physics, right? That could never happen if the world were described by conventional probability theory, right? That, you know, that is sort of the sign that we have, you know, that to actually describe what physics is doing, we need different rules of probability, okay?

Which is, you know, a much more fundamental thing than, you know, than you might have imagined the laws of physics even talking about at all. Yeah. But they do. And so now, you know, okay so now we can come back to the no cloning theorem since you asked about it uh now uh you know a a a qubit is going to have some state like a times you know uh the qubit zero plus b times the qubit one right where a and b are

amplitudes okay so so you know the state of one qubit is described by a two-dimensional vector, you know, a list of two complex numbers, A and B. Okay, now what would it mean to make a copy of the qubit, right? It would mean that, well, now, you know, at the other end, I should have two qubits that are both in the state, A0 plus B1, it have two qubits that are both in the state a zero plus b one right okay and now okay uh and and and the way that in quantum mechanics the way that we describe sort of two systems that are just sitting there next to each other and that are you know separate from each other that haven't interacted right it's a mathematical operation called the tensor product, okay? But it basically just means, you know, we take like component-wise multiplication. So if I have a0 plus b1, you know, for my first qubit times a0 plus b1 for my second

qubit, okay, then I can, you know, just like in middle school algebra, you know, I can expand it out and I can say that's an amplitude of a squared for the qubits to both be zero. That's an amplitude of a b for the qubits to be zero and then one. It's an amplitude of a b for the qubits to be one and then zero. And it's an amplitude of b squared for the qubits to both be one. Right? So now I have a new vector. I want, you know, a squared, ab, ab, and b squared, okay? But now that we know that, now we've proved the no-cloning theorem, okay? Why have we proved it? Well, because that transformation that we just asked for is a nonlinear transformation, okay? You know, the amplitudes, you know, a and b,

okay it you know the amplitudes you know a and b you know were not replaced by linear functions of a and b they were replaced by non-linear functions such as a squared or you know b squared and that is a thing that unitary evolution in quantum mechanics just cannot do okay and so you know so so you know the way to you know there there, there are other ways to prove the no-cloning theorem, okay? But, you know, one way to prove it is really as simple as that. I see. Now, computational irreducibility. And then also, all of this has to do with Newcomb's paradox. All right, all right, all right. So, I mean, computational irreducibility is just, you know, a term that Stephen Wolfram uses for, you know, I would say, you know, a basic phenomenon that was, you know, known to many people before Wolfram. You know, he likes to, you know, treat everything as his invention. know, the fact that, you know, for many, many systems that are computationally universal,

like, you know, we cannot figure out how to, you know, predict their behavior faster than just by simulating the whole thing, right? So, you know, there are, you know, in some sense, science has gotten all, you know, as much leverage as it has over the past 400 years, because often we can model a system by something that is simpler than the system itself. So the orbits of the planets around the sun, the orbit of the moon around the earth. Ke you know, these look to me like, like ellipses, right? And then Newton explained, you know, from from a single simple, you know, law of gravitation, and from laws of motion, right? He explained why they should look like ellipses, right? And, and, and, and, you know, and then, you know, you can you can predict, you know, in some cases,

what the planets are going to be doing millions of years from now, right? Because, you know, in some cases what the planets are going to be doing millions of years from now. Right. Because, you know, the system is simple enough. OK, but, you know, there are many other systems. You know, we could take a lava lamp, for example, or the weather. Right. Where, you know, there is just so much dependence on the fine details of the system state at any one time that, you know, if you try to run a prediction, you know, to a future time, then, you know, your prediction will, you know, before long diverge from reality, okay? This is, you know, the famous butterfly effect, right? That, you know, unless you know, you know, the exact state of every particle and can then feed that

into your computer, right? Then, you know, like a small error, you know, exact state of every particle and can then feed that into your computer, then whatever small error you make in knowing the current state is going to blow up exponentially over time. That's the basic phenomenon of chaos. And computational irreducibility, I think it's just the term that Wolfram uses for the analog of chaos in discrete systems like cellular automata. Great. And so what does that have to do with free why the Newcomb predictor cannot exist, then, you know, the only candidate that I can put forward, you know, based on, you know, the physics and neuroscience and so

forth that I know about, is to say, well, maybe, you know, in order to make, you know, a well calibrated prediction of what a person is going to do, you know, you would really have to know, you know, not just like a crude approximation of the state of their brain, you know, which could mean like, like knowing the connectivity pattern of the neurons, you know, knowing the strengths of each synaptic link, you know, and so forth, right? Maybe that's not good enough, okay? You know, maybe you need to know like, like, you know, is this individual neuron going to fire or not, right, at this time, right? Because, you know, I mean, a single neuron firing or not firing could certainly trigger a cascade, you know, of, you know, of chaotic effects, right? Maybe if this neuron fires, it causes 10 neurons to fire, which in turn cause 100 neurons to fire and so forth. And before long, you know, you're going into industry rather than, you know, becoming a professor.

You've made a different choice or you're marrying a different person. Yeah. So a cosmic ray is responsible for Scott not being a quant. I mean, this is the question, right? You know, like what is the smallest change that you could have made? You know, and this is a standard trope of like, you know, time travel stories in science fiction, right?

Like when you go back in time, you know, if you change even the tiniest thing, I mean, you know, like usually they're like, oh, we have to walk very carefully and not kick any of the rocks. It's like, okay, you know, that's kind of silly. If you believe this at all, then, you know, the very fact that you're there, you know, disturbing the air molecules, you know, you're, you know, it's like, forget it, you know, you've already completely changed the future. Right. But, but, but okay, you know, now, now, if, if we really need to know whether a single neuron, you know, fires or not. Well, you know, we know that the sodium ion channels, you know, that control that right are modeled in neuroscience by something called the Hodgkin-Huxley equation, which is a stochastic differential equation.

OK, so it has a noise component. Right. And, you know, the neuroscientists would probably say, well, you know, we just treat it as thermal noise, right? We just treat it as, you know, a bunch of molecules are bumping around randomly. And that, you know, somehow, you know, sometimes it makes the sodium ion channel open, and other times it makes it close, right? But if you really needed to, you know, make an exact prediction, well, then maybe you need to, you know, to know the states of all these particles so precisely that, you know, you would have to violate the no cloning theorem, right? So, you know, let's be clear, you know, I don't know if that is true. I regard this as, you know, at least partly an empirical question, right? You

know, you like someone could address it by trying to build a you know a brain duplication device you know trying to build a prediction machine right and you know if they succeed then you know we'll know that the answer was one thing okay but uh um you know i mean there is also a philosophical question here, I should admit, of how good or how accurate does the copy have to be before you will accept that copy as being a new version of you. Right. So this is the famous thought experiment here. Imagine that someone has built a teleportation machine, right? And, you know, that can teleport you to Mars, you can visit Mars, and, you know, only, you know, only 10 minutes transit time, right? But, you know, the way that it works is that you will get, you know, your brain

will get scanned in as pure information, that information will get sent to Mars, you know, on Mars, a machine will reconstitute you from that information. And then the original version of you on Earth, well, you know, that'll just be painlessly euthanized or something, right? And so now, you know, the question is, do you agree to go in that machine, right? Is that a means of travel that you are comfortable with, right? And, you know, I think, well, you know, it might, you know, depend on the details of, you know i think well you know it might you know depend on the details of you know just how accurate you know is this copy right is it uh uh you know is it is it really perfect is it you know is it just good enough you know and these are you know like like you know there

have been philosophical thought experiments about this kind of thing for generations, okay? But, you know, we can already see with, you know, GPT, for example, with the, you know, with the AIs that have come online within the last few years, that, you know, these questions are going to come up, like, you know, you, you know, take a person who has some giant corpus of work, you know, tens of thousands of postings on the internet, right? And then, you know, you can train a model, you can train a language model to emulate that person, you know, as well as it can, as you know, character.ai and companies like that are already doing in a kind of crude way. You know, they let you, you know, talk to Einstein,

kind of crude way. You know, they let you, you know, talk to Einstein, talk to Taylor Swift, talk to Socrates or whatever, right? But, you know, I didn't find it that engrossing. They all kind of sounded the same to me. They all kind of sounded just like different language models. But imagine that that gets better, right? Imagine that like, you know, you make this doppelganger of yourself and then, you know, you all day, and you get up, and you see what it's done, and you say, yeah, those are totally the things that I would have done. How good does it have to be before you accept it as a replacement for yourself? Do we need to go as far as a teleportation thought experiment to Mars? Because even when you move a sub-millimeter amount, it could technically be that you just got destroyed in an recreated? Or, you know, should we think about it that way? Now, you know, we certainly in ordinary life, we don't think about, you know, I get on a bus, I get on a plane, you know, I move around as something that is destroying and reconstituting me, right? But now, you know, if you really want to get confused about this, you can think about a quantum teleportation, right? So there is a protocol by which you can transfer, you know, a quantum state from one place to a different place, okay? If you have two resources, you know,

number one is just classical communication, you know, the ability to send conventional bits, like let's say over the internet, for example. And the second resource is you need pre-shared quantum entanglement. So you need the sender and the receiver location to have pre-shared entangled quantum states, you know, that are, that were sort of correlated with each other beforehand. But if you have both of those things, there is this amazing protocol that was discovered 30 years ago, okay, that where you measure one, you know, you measure your quantum state, let's say Alice, you know, over on the left side, measures her state together with her entangled particle. And then she gets two bits of information that she sends over the internet to Bob. And then Bob, using those bits, applies some correction operations to his entangled particle.

And now voila, he now has exactly the same quantum state that, that, that, that Allison had before. Okay. Now, now, you know, you might say like, like, how is this possible? You know, was information sent faster than light? Well, no, no, it wasn't because, you know, we had to send these classical bits, right. And those only moved at the speed of light. So, you know, this is consistent with relativity. And then the next thing you could ask was, well, did this violate the no-cloning theory, right? Because I had my quantum state, and now somehow a new copy of the quantum state has popped up over at Bob's end, okay? But the key is, in order for this to work, Alice had to make a measurement that destroyed her copy of the state.

And does Alice know her copy of the state before? No, she doesn't. So she doesn't even know what she's sending? Right, exactly. She doesn't have to know. She doesn't have to know. She doesn't have to know or she can't know? She can know, but she doesn't have to.

Oh, I see. I see. And then Bob just ends up with a new copy of the same state, whatever it was. up with a new copy of the same state, whatever it was. Okay. And so you could say, you know, would you agree to be quantumly teleported to Mars? Right. Well, in that case, you know, that sounds potentially better or safer than just being, you know, sent as classical information, because in that case, it really is the same quantum state that would be reconstituted on Mars, right? You know, just like it would have been if you would just gotten into a spaceship and traveled to Mars in the conventional way.

Yeah. All right, great. Do you have a preferred interpretation of quantum mechanics? Well, so, you know, there is... Actually, if your views on this have changed, then it would be great to outline what they were prior and what changed them. Okay. I mean, usually when I teach this, teach quantum mechanics to undergrads in my quantum computing, you know, I try to teach it like comparative religion. You know, I try to, you know, I try to not tip my hand about, you know, which interpretation I'm leaning toward, but I've discovered something interesting over, you know, in recent years, which is that it's really hard to not, you know, make the majority of the students into many-worlders, you know. Once they see the, you know, make the majority of the students into many worlders, you know, once they, once they see the issue, you know, the, the pros and cons laid out, then kind of the, the, the,

the, you know, and then, and then we, you know, we, we asked, you know, as an ungraded question on the final, we ask, you know, what's your favorite interpretation? And then, you know, a consistently a majority say that there are many worlders.. Just to back up, many-worlds interpretation is just the one that says that the wave function, which is this list of amplitudes for all the possibilities that you could get, that is the fundamental reality. That is what the universe is. It is this list of amplitudes, right? And, you know, it evolves in time just by this unitary evolution. And, you know, the many-worlders would say,

that measurement is not real, right? Measurement is sort of our local perception, you know, from our local point of view, right? But it's not really a fundamental law of physics. Okay. So, so, so if, you know, and, and, and there's a sense in which that is, you know, the, the, the mathematically simplest or nicest picture that you could have, right? Where it just all unitary evolution, which is continuous, it's reversible, it's deterministic, right? You know, you don't have these weird probabilistic, you know, irreversible jumps. You don't have any of that. But the cost for saying that is that now, if let's say, you know, there's some qubit that's in a superposition of zero and one, and then we make a measurement of it, okay, then the way you have to describe that,

you know, by unitary evolution is that the whole system consisting of, you know, the qubit and the measuring device and me, you know, are now going to evolve to a new quantum state, okay, and that state will have two components. And in one of the components, the qubit is zero, and the measuring device registered it as zero. And my brain, you know, I looked at it, and I saw the zero, right? And in the other branch, you know, the qubit was in the state one, and the measuring device registered it as one, in the state one and the measuring device registered it as one and and my brain you know saw that okay so so you're you're led to this prediction you know that the universe is sort of constantly splitting into branches you know as it were uh or at least you know what what you know

what what what what what what what we would regard as sort of different uh uh approximately classical universes okay and where where you know, our lives could turn out differently, right? And so, you know, in some sense, like, you know, if you just treat, you know, if you were to treat, like, the qubit plus, you know, all of the atoms in the measuring device and all of the atoms in your body as just all quantum mechanical systems, right? All just obeying the same Schrodinger equation, the same laws of, you know, of unitary physics and nothing ever gets singled out as being an observer or, you know, having this sort of special role, then like that, that is the prediction that you, you know,

sort of special role, then like that, that is the prediction that you, you know, that you would get, you know, and so now, like, in some sense, the whole interpretation problem of quantum mechanics is what do you do with that fact? Okay. And so now, you know, there are, you know, a few different approaches, the original approach of, you know, Niels Bohr and Werner Heisenberg, and most of the other founders of quantum mechanics was to say, well, well, then, then, you know, the, you know, the, this wave function, this list of amplitudes is not real, right? It's just a mental device in our heads that we are using in order to calculate the probability that we will see this outcome or that one, right? What is real is like, you know, what we see when we make a measurement, right? And so then, you know, they would tend to say, you know, there is the classical world that we live in,

and then there is the quantum world, you know, which is the subatomic world, and measurement is somehow an interface between the two worlds, right? But then the problem that, you know, that's called the Copenhagen interpretation, right? But the problem that Copenhagen has always had is where do you actually draw the boundary between the quantum world and the classical world, right? You know, like nowadays, we can take much bigger systems and we can put them in superposition states, like even molecules with, you know, thousands of atoms in them, we can put in a superposition of going one way and going another way. Nothing as big as a Schrodinger cat, you know, yet, but you know, but but like after a century, no one has discovered any fundamental obstruction

to, you know, scaling up superpositions arbitrarily, right? And, you know, and quantum computing, you know, feeds into this discussion as well, because if you can build a scalable, you know, error corrected quantum computer, then you could have like millions or billions of qubits that are all, you know, you know, in a superposition of two different states. If you even loaded an AI program onto that quantum computer, and if the AI were conscious, then it could even be in a superposition of thinking one thought and thinking a different thought, right?

So, which was the original thought experiment that sort of led David Deutsch to, you know, propose the ideas of quantum computing in the first place in the late 1970s. Right. So the question is, like, where does the buck stop? Right. And, you know, and the Copenhagen approach has basically been to say, well, you know, there are certain questions that you're not allowed to ask. questions that you're not allowed to ask. You know, you just, you know, you know, you know, to say that, you know, we know a priori what it means to measure something and get a classical outcome. This is just a precondition of doing science. And so we have to just assume this, right? And, you know, you could say like that was an answer that was bound to not satisfy everyone forever, right. But,

you know, you can say that, that, that sort of option one, I, I view it as kind of, kind of the, the,

the giving up option, right. You just say, you know, the theory, you know, you know, you know,

it works for experiments and, and we're not going to, to, to, to, uh, uh, treat the whole universe,

including ourselves quantum mechanically. We're not even going to try to, to understand that. And then a second approach would be to say, yes, there is this whole wave function, right? There is this whole, you know, list of amplitudes for every possible outcome. But also, you know, there is one particular branch that is the real one, that is the, you know, is one particular branch that is the real one you know that is the act you know the one of actual experience right and and and so like you have this giant ocean of of amplitudes but then there's also a cork in the ocean that just gets pushed around by the waves right in a

way that matches the predictions of quantum mechanics the the Born rule, right? That is what David Bohm tried to do and Louis de Broglie, right? This is called the pilot wave or the de Broglie-Bohm interpretation of quantum mechanics, okay? And there are many different versions of it because you can write down like thousands of different such rules for how the quirk is going to go that will all make, that will all end up with the same predictions for any experiment that we can do. Okay. And then, you know, a third point of view, you know, would, you know, would be, you know, many worlds where you just bite the bullet. Okay. And you say the wave function is real and I refuse to introduce any additional ingredient, like any quark in this ocean,

right? Which means I'm not going to regard the other branches as any less real than my branch, right? I regard all of them as existing, right? I can't talk to the other branches. The fact that quantum mechanics is linear is the thing that prevents me from communicating with the other branches, right? But, you know, you know, if they're there in the equations, you know, if they're there in the theory, then I'm going to say that they're just as real as whichever branch you and I happen to experience. And then, you know, the fourth outcome, or sorry, the fourth option would be to say, well, none of these ideas are any good. None of these interpretations, you know, is acceptable. And therefore, you know, there must be something wrong with quantum mechanics itself. Right. And hopefully in the future, we'll discover a better theory of physics that, you know, says, okay, here is when the quantum state collapses. It happens when it gets this big or this massive, and there will just be some objective testable law of physics that describes

the collapse process. Okay, now that would be something new, right? That's not an interpretation. That would be a new and different physical theory that would overturn quantum mechanics as we know it today, right? But you can say, well, you know, then that has to be the truth, you know, quantum mechanics has to just be an approximation to some better theory that hasn't been discovered yet. So, you know, you ask, you know, what I'm partial to, you know, I'm kind of partial to an idea by the physicists, Lenny Susskind and Raphael Bousseau from a decade ago, which is that cosmology might be an important part of the story, right? So like you could say that, you know, the big, you know, one of the main problems, you know, for anyone who is trying to interpret quantum mechanics is to specify when does a measurement happen?

When does the buck stop and when should I regard this superposition of outcomes as having resolved into one or the other definite outcome? as having resolved into one or the other definite outcome, right? You know, like the many-worlders might say, well, there's no one right answer to that question, right? It's kind of like asking, like, how many grains of sand do I have to put together until it's a heap of sand? Exactly. Right? You know, but even they might want a rough and ready rule for when we can treat

an outcome as definite, okay? And now here is one possibility for such a rule, okay? So when you make a measurement, right, it's not just your brain that becomes entangled with the qubits or the particles that you measure, right? It's also, it's the air in the room that you're in, right? And the radiation in the room, right? You know, like, you know, there's a butterfly effect that happens, right? Like each thing, you know, each particle starts knocking around the nearby particles. And so there's a whole, you know, bubble of effects of whatever the outcome of that measurement was that spreads outward, you know, from you, you know, no faster than the speed of light. But, you know, but, but, but, you know, as soon as the information gets encoded into photons, then possibly, you know, this sphere of effects is expanding around you at the speed of light, right? And once the information about which measurement outcome

you saw is encoded into photons that are leaving the earth, right, and that are flying away from the earth at the speed of light, right, which eventually they will be, then you could say, well, even in principle, we could never catch those again, you know, as we would need if we wanted to re-cohere the superposition. You know, if we wanted to, like, show that, you know, see any effect of the other outcome, of the one that we didn't measure, right? So, you know, in order to get interference, you have to collect all of the qubits that were affected, right? And, you know, if many of those qubits are flying away from us at the speed of light,

well, then you can say, you know, how could we ever catch them again? Well, if there were some Well, then you could say, you know, how could we ever catch them again? Well, if there were some extraterrestrials who had thought to, like, enclose the solar system in perfectly reflecting mirrors, right? Well, then, okay, then the photons are going to bounce back, and then maybe we could be here, and we could see that this is still a quantum superposition. But, you know, I will assume that aliens have not done that, right? That, you know, that does not seem to be the case in our universe. Okay, and then you get into questions about cosmology. So, like, you know, there's this cosmological constant that was discovered in 1998, right? Which is the, you know, also known as the dark energy, right? It's the thing that is

pushing the galaxies away from each other at an exponential rate, right? And yeah, this has been, you know, the, you know, one of the, you know, the most important discoveries in all of physics for decades, right? And certainly in cosmology, right? The fact that this dark energy exists, what Einstein a century ago called the cosmological constant. It's actually not zero. We now know that, right? But now, if that constant had been negative, right, which it could have been, okay, a universe with a negative cosmological constant is one that in that in some sense has, does have a reflecting boundary, right? It's called an anti-de Sitter universe.

Okay. And in that kind of universe, like we would be like sort of trapped in a bubble where everything would be unitary. Right. And we're like any photons that are flying away from the earth, you know, eventually they can come back.

Right. And so, so no loss of quantum coherence would truly be permanent in that kind of universe, okay? But, you know, since 1998, we know that we don't live in that kind of universe, right? We live in a universe with a positive cosmological constant, a de-sitter universe, okay? And in that universe, you know, things really, you know, at least as far as anyone knows, no one knows for sure, and in that universe you know things really you know as at least as far as anyone knows no one knows for sure but it's you know it seems possible that things can

can fly off to infinity and we could just take that as our criterion for when a measurement has happened so in some sense you know we could say like it doesn't you know, we could say, like, it doesn't, you know, we could sort of harmonize the many worlds in the Copenhagen points of view by saying, like, yeah, at some formal level, yes, you know, we, you know, there is this whole wave function of the universe, or, you know, there's also a criterion, you know, for loss of quantum coherence, right? You know, the photons flying away from me at the speed of light, where after that happens, then I might as well say that the other branches are gone, right? They are now not empirically accessible to me, even in principle. So in other words, when you're teaching this

in the comparative religion sense, you're the Baha'i faith. Each of them has some semblance of the truth. Now, what was the name of the Susskind interpretation or the Susskind theory? I think they called it the multiverse interpretation. And by the way, I just saw Lenny Susskind a couple weeks ago, and he doesn't seem to believe his own interpretation anymore. But, you know, I still like it, though. Great. So the multiverse interpretation is separate from the many worlds, because those sound

similar. That's right, yes. And the many worlds, when you said that you measured, forgive the pun, your students at the end, and then they said that they like the many worlds, 80% or so. I think it was probably more like 55% with the rest. Okay, so a majority still. Yeah, with the rest split among Copenhagen, Bohm. I allowed agnosticism or what does it even matter or shut up and calculate or that there has to be new physics.

I allowed all those things. that there has to be new physics. You know, I allowed all those things. Yeah, it's surprising to me that number four, the provisional one that, hey, we don't have the current fundamental law, and so who cares about it, isn't more popular given that gravity isn't integrated into quantum mechanics. I mean, it does have adherence, right?

I mean, Roger Penrose is one very famous adherent, right? But, you know, I think he sort of hurts his case by tacking onto it like a whole enormous chain of speculations right that uh you know he thinks that that uh quantum gravity causes an objective collapse of the wave function uh and this collapse is uncomputable it cannot be simulated by a machine and the microtubules in our brains are somehow sensitive to this quantum gravitational collapse. And this is implicated in consciousness. Right? So that's the Penrose view. And you can say, like, you know, even if you might go with him, like, you know, to the first stop of

that train, you know, most of us are going to get off by the, you know, before the later stops. Right? Yeah. Okay, we're going to explore uncomputability shortly. Okay. So, oh yeah, what I was getting at was, did you measure the students initially and say, hey, what is your preferred interpretation in order for you to establish a difference? Yeah, that's a good question. Like, to do a controlled experiment, you ought to do that. The trouble is that until I, like, expose the students to all these interpretations, they don't even know what they are, right? Yeah. Most of them have not even heard of them.

Or if they have heard of them, then, you know, I'm not sure if they could define them. The reason is because in the popular press, many worlds is prominent. And so it may just be an effect of, hey, I like Sean Carroll. I listen to his podcast. Yeah, yeah. Well, I mean, I mean, I mean, Sean makes very, you know, I've, you know, he's been a good friend of mine since, you know, 2006 or so. And I think he does make very strong arguments for many worlds. And I say that even though, you know, I am not nearly as hardcore of a many world or as Sean is. Isn't there still the issue of having a globally well-defined measure in order to even state what the probability distribution is of different branches of the wave function? Can you explain

what that is? Yeah. So, I mean, you could say that the basic problem for, you know, if you want to be a many-worlder is you have to explain, well, why do we only perceive one world, is you have to explain, well, why do we only perceive one world, right? And not only that, but why do we perceive each world with these particular probabilities, these Born-Rule probabilities, right? But now, I have to say, I don't regard that as a problem for only the many worlds interpretation, right? I think, you could ask the same question with any interpretation, like where, where did these probabilities come from? Right. It's just that question takes on a different character depending on which interpretation you like, right? Because, you know, if, if, if you believe in new physics, then you have to postulate some

new law of physics that will then give rise to these probabilities, right? And, you know, you can check, you know, whether it does or it doesn't. And, you know, ideally, you wouldn't just, you know, stick them in. But, you know, in physics, it's always better if you can derive something, right? Rather than, you know something rather than just assuming it from the outset. notion that happens to always give you agreement with this Born rule, right? But then you could say, you know, why should it have been a rule of that kind, right? And then, you know, will we, you know, if we started out in some other distribution, would we reach that Born distribution as an equilibrium, right? So that's what the question looks like to a Bohmian. To a many-worlder, you know, the issue is that a many-worlder is committed to the view that all of the outcomes are real. All of the outcomes are experienced by someone. And so then they have to say, then what does it even mean to say that this outcome has this probability and that one has that probability?

And that one has that probability, right? Like, you know, how do you even make sense of that statement? It's like, you know, you have to imagine that there's, you know, like all of these beings are real, but somehow, you know, one of them is going to be picked to be, you know, your experience, right? And so somehow there is some, you know, metal law that governs that, right? And so there is a long history, you know, since Everett himself in the 1950s, of many worlders trying to, or claiming to derive the Bern rule, right? Derive the probabilities, okay? They always have to make some auxiliary assumption, you know, in these derivations, right? Because it's like,

you're starting with a picture that has just the wave function, you know, that has no probabilities in it. And then in the end, you know, you get a statement about probability, right? And so, so, you know, there has to be some step where you're just postulating that, yes, something is random, right? And, you know and and for a many worlder what that kind of looks like is um it's this thing called indexical uncertainty or or self-locating belief so like by basically uh um you know imagine that uh um you know you didn't know your own blood type right like you know you just hadn just hadn't gotten tested yet, right? But then you say, well, look,

there's this many people in the world who are type O, there's this many people who are type A, right? So I'm going to just assume that I was, you know, a randomly chosen person, right? You know, and there's something fundamentally weird about that, right? As soon as you start thinking about yourself as chosen randomly from the set of all people. Yeah, there's also a reference class. Yeah, exactly, exactly. Then

you can start wondering about things like, you know, why was I born in the, you know, late 20th century, as opposed to, you know, in medieval Spain, or, you know, or some other time, right? Why am I on Earth? Why am I not an alien on a different planet? Right? And, you know, or at some other time, right? Why am I on earth? Why am I not an alien on a different planet? Right. And, you know, it's not, it's not obvious if these questions have well-defined answers at all. Right. But, you know, what the many worlders need to do is to say, you know, there are all of

these branches of the wave function that are, you know, are all real. They all have real copies of you, but now you have to think of yourself as a, you know, as a randomly selected member of that ensemble, right? And then once you decide to do that, then, you know, you actually can give many mathematical arguments that, like, the Born probabilities are pretty much the only probabilities that would make sense, right? Like, you can show that, like, any other choice for what the probabilities would be make sense, right? Like you can show that like any other choice for what the probabilities would be, like if instead of the absolute square of the amplitude, suppose it were the absolute cube of the amplitude, right? Or, you know, the absolute value to the 2.8

power or something like that, right? You can show that that would give you like nonsensical things. It would lead to faster than light communication. It would lead to sort of massive violations of the laws of physics that we understand. So you can kind of give arguments for once you've decided to put a probability distribution over all these branches, why it should be this particular one, right? The harder part is to say, why should there have been a probability distribution at all? Uh-huh. Now, it's been a while since I've studied this, but it's my understanding that the space of pure quantum states is a projective Hilbert

space, which is a Kahler manifold. The symplectic structure gives rise to the dynamics, and the complex structure gives rise to the superpositions, and the Riemannian metric gives rise to the probabilities. That's probably all true. Those are already much fancier words than the ones that I ever use to talk about these things. Right, well, fancy words with a specific mathematical meaning. Some say that computability or quaternions are fancy words. Right, exactly. Yeah, what I mean is that when one says, well, where does the Born rule come from?

In a symplectic, sorry, in a Kaler manifold, if you have two of those structures, you get the third. So why isn't the question then, well, why do we have a complex structure? Or why do we have a symplectic structure? That is a different superb question. You could say like,

why should quantum mechanics have been based on complex numbers? You know, and you actually can define a variant of quantum mechanics that would only use real ampl as our ordinary, you know, complex quantum mechanics, right? It would lead to, you know, basically all of the same, you know, information and communication protocols, you know, such as quantum teleportation, you know, you'd have the same no cloning theorem, the same, all of that stuff. It just that there, there are certain things that would be less elegant in the, in elegant in the universe with real quantum mechanics. And some of those arise just because the real numbers are not algebraically closed. You can't take square roots of them.

And so if I have a unitary transformation that operates over, let's say, one second of time, and now I want to know, okay, but now what was the piece of it that operated only over the first half second? Right? Well, then, you know, as long as I have complex matrices, then I can just take a square root, right? And, you know, I'll get an answer to that question. Okay, with real matrices, there might not be a square root, you know, in the same number of dimensions. So for example, there's no, there's no two by two real square root of the matrix one, zero, zero, negative one, right? That would be, that would be an example, right? And we, we can see that because it has a negative one determinant, right? So, so, so, so that, that, that breaks, you know, various things

about, about the way that physicists use quantum mechanics. And then there are other more subtle things that also break, like the number of parameters that you need to describe a composite state. In complex quantum mechanics, it's exactly just the number of parameters that you need to describe the first piece times the number of parameters that you need to describe the second piece. Okay. But in real quantum mechanics, that's no longer true. By the way, it's also not true in quantum mechanics based on quaternions, right? With real numbers, you get an undercount, with quaternions, you get an overcount, and only with complex numbers does it work out exactly right. Okay. I never heard about the overcount. Yeah. Yeah. So, so, so, so, so, so there are these subtle things that just work out perfectly when, when, when, when quantum mechanics is defined over the complex numbers. But, you know, I got, I've asked mathematicians this question, like, you know, if you were God,

you know, designing the universe on a blackboard, right? Like, you know, why would, you know, you know, do you know why you would have chosen the complex numbers for this? In some sense, the deepest laws of physics that we know. And the mathematicians were like, well, come on, they're algebraically closed. Why wouldn't you want them? Yeah, I think you said this, that we used to think there were two logical operations and or or but then we found out with quantum mechanics there's complex linear combinations uh well yeah okay so i was saying like like in terms of how you can combine multiple possibilities

right like like it's um like like uh you know when when someone says that an electron, for example, well, it's not in its ground state. It's not in its excited state. It's in some kind of superposition of the two. Often, the first thing that they think that you mean is, well, then you must be saying that it's in both simultaneously. both simultaneously, right? And, you know, and the trouble is, if you take that too literally, that it leads to like, for example, a vision of what a quantum computer is, where it would just try all of the different solutions in parallel, right? And that's wrong, right? That's just like, that leads you to like, leads people to importantly wrong expectations of how useful a quantum computer would be if they

really think that it could try all the answers in parallel you know in the in the in the naive way right so then you you correct that and then they say oh so then so then what you must be saying instead is that you know the electron is in one state or the other and we just don't know which one right it's either in the ground state or these east. But no, you know, we're not saying that either. In some sense, like, it's a different ontological category that, you know, there was no, you know, ordinary English word for, because no one needed it before the 1920s, right? And that new ontological category is you have a complex linear combination of the two things. Do you believe there to be a fourth ontological category that involves the quaternions?

I mean, you could define quantum mechanics over the quaternions or over the reals for that matter. And that would give a subtly different answer. So yeah. But what would that look like? So yeah, so quantum mechanics over the quaternions turns out to be sick in, in, in various ways. Like it's, it's sort of not as in cool. Let me tell you what

I mean. And then, and then you can decide, okay. Uh, you know, at least naively in quaternionic quantum mechanics, you could send information faster than light. Okay. And in fact, like if, if Alice and Bob are far away, right, then, you know, it could, you know, even if Alice is on earth and Bob is on Mars, like it could matter which one of them does an operation first, right? Does Alice act first or does Bob act first? You know, and this is because the quaternions are non-commutative. So it matters. Even for separated, then it can't matter in which order, you know, they're done. Because, you know, to some observers, you know, Alice will be first and to other observers, Bob will be first, right? And these two perspectives have to be consistent with each other, okay? So, quaternionic quantum mechanics breaks that, you know, because the quaternions are not commutative.

So if you want to believe in it, then you need some way, you know, of making that effect go away at large enough distance scales or something like that. There's a physicist named Steve Adler who spent decades trying to make that work. But, you know, I talked to him a few years, and he said that he doesn't really believe it anymore. But now, real quantum mechanics, like I said, that one does make a lot more sense. But a real superposition, philosophically, I would think of as almost the same sort of thing as a complex superposition. They're just kind of different in detail. Now, the octonions have been chopped liver in this conversation. Yeah, Octonians don't even get started, right? Yeah, they might get a tiny bit started. Just explain, because that's a popular subject.

Well, okay, the Octonians, so there are these four, what are they, complete division algebras? Norm division algebras uh the norm division norm division algebras excuse me the the uh the reals the complex numbers you know which have two parameters the quaternions which have four parameters and the octonions which have eight parameters and um you know and like naively you would expect that it would you know it must just keep going after that point but uh you know it's a very important theorem from the 19th century that says that these are the only four, right? So it's sort of the progression stops after that.

So there are these four norm division algebras that are kind of special. But, you know, as you go to the larger and larger ones, you, you know, you lose certain properties. So like, you know, the reals are ordered, right? You know, the complex numbers and beyond are not ordered. Okay, but with the complex numbers, you could say you lose something, but you gain something, right? You gain that they're algebraically closed, right? But now, when you go to the, and the complex numbers are also commutative, they're associative, they satisfy pretty much all

of the basic properties that you would want in algebra, right? But now, you know, with the quaternions, that already starts falling apart, because the quaternions are associative, but they're not commutative. Okay, so A times B can be different from B times A. Which doesn't sound terrible, because non-communitivity is a hallmark of quantum mechanics. No, no, no. I mean, look, and octonians still have, you know, they have applications in, you know, even in computer graphics, in math and physics, you know, they are actually used, right? As I said, you know, if you want to use them as amplitudes in quantum mechanics, then there

is stuff that goes wrong you have to then somehow make that consistent with with relativity okay but now now the the the octonions are not commutative and they're not even associative right so you know it's funny it's like um um you know in like fifth or sixth grade right kids learn all these these these these terms you know this is the commutative property. This is the associative property, right? But until you've seen any examples of things that are not commutative or not associative, and this is just a bunch of words to memorize, right? I think that, you know, the time when you want to teach these terms is the time when

you want to teach things that don't satisfy them. I mean, for my daughter, my 10-year-old daughter, just the other day, I gave the example of something noncommutative, getting dressed, putting your pants on and then your underwear is not the same as the reverse, right? So, you know, so there are, you know, there are examples for non-commutativity that even a small child can understand. You know, non-associativity is a little bit harder. Well, you're lucky this camera is only here. Otherwise, you'd see that I don't pay much attention to the non-commutativity of the pants and underwear. Before we get on to consciousness, I want to talk about IIT, and I want to talk about P versus NP, or sorry, P equals NP, which by the way, what I learned as a teenager, I just learned the equation P equals NP, and I'm like, when N equals one, it's obvious.

Right, or P equals zero, right? Yeah, yeah, right. Anyhow, I want to get to that, but we mentioned Wolfram before, and Wolfram has a physics project, and so I'm curious if you had a chance to go through it and what your thoughts are. as a student was a review of his new kind of science book. And, you know, what I spent half of that review doing is just explaining why, you know, you know, his kind of model cannot account for the known phenomena of quantum mechanics. Right. And this is, you know, this is, this, this, this is not a fuzzy statement. Like this is, you know, this is, this is,

you know, this is, this, this, this is a thing that one can prove, right? So, you know, he, you know, what he wants basically is to, you know, reduce everything in the universe to cellular automata, right? To some, you know, to, you know, a bunch of bits, you know, maybe at the Planck scale or something like that, that are, you know, undergoing some sort of simple computational rules, you know, to, you know, a bunch of bits, you know, maybe at the Planck scale or something like that, that are, you know, undergoing some sort of simple computational rules, you know, and, you know, like that far along, yeah, I think, you know, that's great. That's, you know, that's like, you know, the whole physics community would like that, right? They would like, you know,

especially a theory of quantum gravity that would explain why, you know, what's called the Bekenstein-Hawking entropy is finite, right? Why are there only finitely many degrees of freedom in a black hole, right? Or apparently in any physical system, right? Like, what is it, you know, when you get down to the Planck scale of 10 to the minus 33 centimeters or 10 to the minus 43 seconds, right? Something seems to break in our picture of a smooth, continuous space-time, okay? And we can see that from thought experiments where, for example, if you tried to, you know, build a clock that was more reliable, you know, that could measure time more finely than 10 to the minus 43 seconds at a time, then your clock would take so much energy that it would just

collapse to a black hole instead. Right. Right. So, okay. So, there are these thought experiments that tell us that sort of something breaks at the Planck scale. But, you know, can you actually explain that by giving a theory where space and time are discrete or sort of discrete at the Planck scale, right? That is, I would say, a central part of the problem of quantum gravity, right? But now Wolfram, you know, wants to do something else. He wants to say, no, it's not that, you know, we have quantum mechanics with a, you know, a finite dimensional Hilbert space and, you know, with a discrete space and time. He wants to, you know, in some sense, get rid of quantum mechanics, right, and have it be a classical cellular automata, right? And then the problem is, well, we have, you know, in a mountain of evidence that quantum mechanics is both true and unavoidable, right? And so, you know, what does he do with that evidence, right? And, you know, and he kind of hand waves it away, right? This is, you know, this is the key problem, right? He, you know, in A New Kind of Science, he said, well, it's true that there are these experiments that you can do on

entangled particles, you know, that could be very far away that lead to these phenomena like the violation of the Bell inequality, right? Which is, you know, famously a thing that sort of you could not explain in a classical universe, right? And that's sort of just sort of a very crisp statement, right? Like, it doesn't matter what additional assumptions you make, you know, as long as they're reasonable ones, or sort of sane ones. You say, you know, I'm, you know, Alice and Bob, who are far away from each other, they measure their halves of, you know, of this entangled pair, and they get, and the statistics of the outcomes could not have been explained by any theory where the particles just secretly agreed in advance on, you know, whether to be spin up or spin down or whatever, right? That, you know, it can only

be explained by saying, well, until Alice and Bob made the measurements, it was an entangled superposition state, right? So, you know, this is, you know, one of the strongest arguments that, yes, you know, the world is really quantum mechanical, and, you know, we can't just replace it by something else, by, you know, certainly not by a local hidden variable theory. But then what Wolfram says, and this is in chapter nine of A New Kind of Science, he just says, well, okay, you know, this is no problem. Just imagine that sometimes there are long range threads between the particles, right? So usually space is local, but when, you know, two particles become entangled, then there's kind of like a thread that allows instantaneous communication between them,

right? Problem solved, right? This is like, you know, the Wolfram method, right? You just, you know, if you can imagine a way that the problem could be solved, then it's solved, right? And so what I spent a lot of my review doing is just explaining, no, sorry, it still doesn't work. It still doesn't give you a picture that's compatible with special relativity, okay? And so what you can get is that, you know, it's not just a statement about non-locality. It's sort of a statement that, you know, the universe cannot have, you know, if the measurements that Alice and Bob are going to make on the entangled particles have not been pre-decided, or sort of were not known in advance, then the outcomes of the measurements on the particles

also cannot have been pre-decided, right? They, you know, they cannot be explained by hidden variables that go back to the beginning of time. They must be sort of genuine new randomness, right? Or, you know, otherwise you get a violation of locality. Okay. So this is, so that was kind of the argument. And like I said, I just made that in 2002, you know, buried in my review of Wolfram's book. Now that same conclusion a few years later became famous when Conway, John Conway and Simon Koshin said something very similar in 2006, and they called it the free will theorem, right? And, you know, the way that they phrased it, it was that, like, if humans have free will, then subatomic particles must also have free will. And then that, you know,

of course, that got all over the popular press, right? But, you know, I would never have used the term free will here, right? Because I would say, you know, we could equally well just talk about randomness, right? And that, you know, but it's sort of a, I might have called it the freshly generated randomness theory, okay? And it sort of tells you that, you know, no model like, like, like Wolfram's from a new kind of science can explain, you know, even, you know, existing experiments, like the Bell inequality violation, you know, let alone, you know, all the, you know, the, the experiments of the future. Now, Wolfram never, never really accepted that. I think, you know, most of the community did, you know, he, he, he did not. Right. But if you look at like what he's done more recently with the, the Wolfram, you know,

I think he called it the Wolfram model of physics. Like, you know, um, what, what, what he basically does is he just says, okay, well, you know, wherever we have a result that we can't explain classically, then we'll just graft on the relevant part of quantum mechanics. Right. You know, and it's like, okay, you know, we'll assume that we have a multi-way system. So it can, you know, evolve into superposition. You know, it can evolve multiple ways simultaneously. And that's kind of like quantum mechanics.

Right. can evolve multiple ways simultaneously and that's kind of like quantum mechanics right and uh um you know and and and uh you know and and also like like we will you know say oh we we can derive general relativity except the derivation will basically be we like crib from a general relativity textbook we just find out what the einstein's equations are and we say okay okay, well, whatever is the cellular automaton, it's presumably it's something that satisfies those equations, right? So it's like, you know, it's not sort of playing by the rules of physics, right? Where you have to, you have to actually, you know, mathematically derive these things, right, from some simple starting postulate, then ideally you know make a novel prediction right that's that that's the gold standard right but uh you know i i so so so one thing that you know when i i talked to uh uh uh well from in person a couple years ago uh when uh he was in

austin and uh uh you know one one question one one thing that I kept trying to get clarity from him about was, does he predict that scalable quantum computers can work or not? And, you know, he wouldn't, you know, he would not give a clear prediction. Basically, you know, that like, you know, this multi-way theory, you know, it suggests that maybe quantum computers can't work. But if it turns out that they do work, then the theory can accommodate that also. And so it's sort of parasitic on existing science. It's like anything, he basically says, just tell me the prediction from existing science, and then I'll find a way to graft it onto this picture. Have you read any of the papers by Jonathan Gerard? Yeah, these are exactly the ones that I'm talking about, actually. Yeah, so I mean, maybe there's something there that I'm missing, but I spent some time on it, and that was the conclusion I came to that caused me to not spend more time on it. All right, so before we get to P, no, let's get to P equals N P. Please explain what what that is why p equals zero is not a valid solution and n equals one isn't so um p and np are uh two of the most fundamental

complexity classes uh which are just classes of problems that are solvable with different kinds of computational resources so p stands for polynomial time And it's the class of all of the yes or no problems that a conventional computer, you know, a deterministic digital computer, like the one that we're using, could solve using a number of steps that grows like a polynomial function of the number of input bits, okay? So, yeah, and that's our sort of rough and ready criterion for when an algorithm is efficient, right? If it takes, you know, if its running time scales polynomially with the length of the input, okay? This is not a perfect criterion because, you know, an algorithm that took n to the 10,000th time, right, would be wildly impractical. But what justifies it is that usually when something is

solvable in polynomial time, the polynomial is something like n or n squared or maybe n cubed, right? Whereas usually when something is not polynomial, you know, you get scaling that looks more like two to the n power, right? That is like exponential at n. And that just, you know, when n is in the hundreds, let's say, then, you know, you can do n squared, you can do n cubed, but two to the n is just forget about it, right? So that's why, like, from the 1960s onward, the polynomial versus exponential distinction kind of became fundamental to complexity theory. And so P is just all of the yes or no problems that are solvable by some algorithm that has polynomial scaling. So what are some examples of problems in P you know if i give you a string of of text and i ask is it a palindrome or not right you know or any kind of basic arithmetic you know most of the things we do with our

computers to be honest are things that are in p right uh uh um you know you know more interesting examples like uh given a map you know is every city reachable from every other one you know given a graph is it connected right uh given uh yeah yeah that's that that's in p uh given a bunch of boys and girls and who is willing to date whom you know can you pair them off so that everyone is happy with their partner that's's called the matching problem. That's also in P. And graphics like ray tracing, are you able to say whether that's P? Yeah, I think the basic problems underlying ray tracing would be in P also, yes. Now, you know, there are some, you know, and these are all things that, you know, one can learn as an undergraduate in computer science, right? You know, there are, you know,

undergraduate in computer science, right? You know, there are, you know, also much more, you know, much more non-trivial examples. Given a number written in binary, is it prime or composite? Okay, that's in P. It was only discovered to be in P 20 years ago, okay? That's called the Agrawal-Kyle-Saxena or AKS theorem, okay? So, you know, they gave a breakthrough algorithm. I mean, before that, we had known probabilistic algorithms, but they gave the first deterministic polynomial time algorithm for primality testing. If the number is composite, then these fast algorithms do not tell you the factors. Okay. You know, finding the factors seems to be a much much harder problem right which we can come back to okay but determining primality is in pitch right uh you know given uh

like a bunch of linear constraints you know is there a way to satisfy all of them that's called the linear programming problem okay again it's in p you know, for very non-trivial reasons, okay? So a lot of interesting things are NP, okay? But now there's this potentially larger class, which is called NP. Now, that does not stand for not polynomial, which, you know, some people think it stands for non-deterministic polynomial time. So what does that mean? So NP is the class of all the problems where sort of if there is a solution, then you can check the solution in polynomial time.

So if the answer is yes, you know, there is a solution, then there is some witness, some proof that if it's given to you, then you can check in polynomial time that, yeah, I guess that works, right? So let's, you know, go through some examples, okay? Solving a jigsaw puzzle, right? Like imagine a jigsaw puzzle with no picture on it, right? So you're just trying to fit all the pieces together, okay? That might be, you know, trying to fit all the pieces together. Okay. That might be, you know, incredibly cumbersome to do, right? You might have, you know, exponentially many, you know, possible ways of fitting the pieces together to try. Okay. But if someone has solved the puzzle, then they just have to show it to you. Right. And, you know, you can easily see that, yes, they have solved it. Okay. The famous

traveling salesman problem where, you know, you, you're given a bunch of cities and you're asked, let's say, is there a route that, you know, visits every city, like with at most 5,000 miles of total travel distance, right? Again, you know, you know, if there are N cities, there might be N factorial different, you know, routes that you would have to try out, right? That, you know, roots that you would have to try out, right? That, you know, with hundreds of cities, that could be massively expensive. But if someone finds a root that works, then they just have to show it to you. And it's easy to check. Okay, so that's also an NP problem. A factor in, you know, I asked,

let's say, does this number have a prime factor that ends with a three or some, you know, question like that, right? Where, again, you know, the fastest algorithms that we know for factoring take some sort of exponential time, at least algorithms for classical computers, okay? They actually take time. That's like exponential in the cube root of the number of digits. Okay. That's called the number field sieve. Okay. And, and, and that's extremely important for cryptography, right? Cause, uh, you know, all of the encryption that sort of currently protects, uh, uh, the internet, you know, is based on the belief that factoring or a few closely related problems are hard.

Yeah, yeah. Even for the factor of three, isn't there some algorithm that if you add the- Yeah, yeah, yeah. I said a factor ending in a three. Oh, sorry. Okay. Yeah, yeah, yeah. I agree. If you just want to know if the number is divisible by three, that is in P. Yes. Yeah, okay. Yeah. In fact, if you want to know if it's divisible by any fixed number, that's in P, right. Yeah. Okay. Yeah. In fact, in fact, if you want to know if it's divisible by any fixed number, that that's in P, right? But if I don't tell you the prime factors,

and they're like, let's say they're all enormous. And now, you know, you have to find them, right? Then that we don't know how to do in P. Okay. But it still is an NP problem. Because if someone succeeds in factoring the number, then they just have to show you the prime factors, right? And at least with a computer, it's easy enough to multiply them. And as I said, it's even easy enough to check that those numbers are prime. So factoring is another NP problem. Okay, so now the P versus NP question is just the question, question well how do these classes relate to each other

so it's it's it's pretty clear that p is contained in np right so you know if you can solve a problem yourself in polynomial time then you don't even need this witness right you just you know you just have the answer to it right yeah but now the the the uh the profound question is, is NP contained in P? OK, so if I can efficiently recognize the solution to a problem, then does that imply that I can efficiently find the solution? OK, and, you know, and so that's the P versus NP question. And as soon as you see this question, some people, it takes time to convince them of why it's even a question at all. Because they would say, well, obviously not.

Obviously, there are cases where you're going to need brute force search. And brute force search is going to take exponential time. But the trouble is that, you know, as we discussed, like with the example of the matching problem, or the linear programming problem, or the primality problem, right? There are cases where like naively, it looks like you have to try exponentially many possibilities. But if you think about it more, then there is a clever shortcut that only takes polynomial time, right? And so what P versus NP is asking is, is there always a clever shortcut for every problem where we can efficiently recognize a correct answer? And so, you know, that is for half a century, that's been sort of

the central unsolved problem of theoretical computer science. And I think it's now recognized as one of the central unsolved problems in all of mathematics. So, two questions. One, why is this the largest unsolved problem to you? You think this is the greatest unsolved problem in math, and potentially even physics? And then number two, this sounds... It's not a physics question, right? I mean, we could find a related physics question by asking, for example, can the all NP problems be solved in polynomial time by any physical means, right? Which could include a quantum computer or could even include a hypothetical quantum gravity computer or whatever, then we'd be asking a physics question. But P versus NP is a question that can be purely stated mathematically, meaning it has some platonic answer. We just haven't proven what it is.

What I meant was that I heard you say something along the lines of, look, the Clay Institute has many of these problems. Yes. And one of the problems is a physics problem, the Yang-Mills problem. Yeah. Well, you could say it's a mathematical physics problem, or it's a math problem that came from physics. But I mean, the argument that I give is just that if P equal to NP, and furthermore, if the algorithm were really efficient in practice, so not like N to the 10,000, but, you know, N squared or N or something, then that would not only let you solve the P versus NP problem, that would let you solve all of the other clay problems, you know, the Riemann hypothesis, the Yang-Mills problem, right, and all the other ones. And why is that? Because it would mean that we could just ask our computer, right, say like, you know, is there a proof of this theorem

in this formal language, you know, like Zermelo-Fraenkel set theory, that is at most, you know, a million symbols long, or at most a billion symbols long, right? And what it would mean if P equal to NP would be that if such a proof existed, then you could find it using a number of steps that only scaled polynomially with the length of the proof, right? So like in, you know, only polynomially more time than it would take to write down the proof, you could actually find the proof, right? And so in some sense, you know, mathematical creativity would have been automated. Uh-huh. So a proof of P equals NP would automatically have... Well, okay. I mean, the caveat is that the algorithm would have to be efficient in practice, right? But, you know, the question, like, you know, is there a proof of this theorem in some purely formal language, like first-order logic, ZF set theory, with at most this number of symbols? That is an NP problem, literally. And so that means that if P equaled NP, then it would also be a P problem.

Can you explain what quantum supremacy is? if P equal to NP, then it would also be a P problem. Can you explain what quantum supremacy is? So quantum supremacy is a term that was coined by the physicist John Preskill in 2012. And it's just referring to sort of the first experiment that you can do with a quantum computer that solves some benchmark problem much faster than we believe that it could be solved with a classical computer. Okay. And notice that I did not say a useful problem. Okay. It doesn't have to be useful.

It can be a completely artificial benchmark, but it has to be something that is well-defined. Okay. So, like, it can't just be, like, simulate this physical system with all of its noise, right? You have to give a mathematical specification of what calculation you want so that that same calculation could be done either on a quantum computer or on a classical computer, okay? And what we want to see is that the quantum computer is faster, to see is that the quantum computer is faster, you know, not just in classical brute force, but the fastest classical algorithm that anyone can design. And we want to see that that is so for sort of inherent scaling reasons, you know, not just for sort of accidental reasons of hardware,

but, you know, the quantum running time is sort of scaling polynomially in a way that the classical runtime is scaling exponentially with the size of the problem. So that's quantum supremacy. And so Preskill coined this term in order to describe sort of the kind of thing that I and others had been talking about, like, in the year or so prior. that I and others had been talking about like in the year or so prior. Right. So my then student, Alex Arkhipov and I, in 2011, we had a proposal called boson sampling.

Okay. Which was a proposal for like a very rudimentary kind of quantum computer. For example, could be, you know, built using, using photonic components. So you just generate a bunch of single photons,

you send them through a network of beam splitters, and then you measure where they end up. And that's it. That's all you do here. So we don't think that this is universal for quantum computation, or even universal for classical computation, for that matter, right? It's like, in some ways, it's a very, very limited model of computation. And yet, if you ask, like, okay, what is it doing? Well, it's sampling from a probability

distribution, right? Each time you run the experiment, you feed in the photons, you know, they will typically end up in different places, right? Because, you know, they sort of move around randomly, you know, and, you know, I mean, they, or rather they evolve in a superposition state, but then when you make a measurement, you collapse the superposition, okay? And you just see one outcome. So you probably never even see the same outcome twice. You know, sometimes there's two photons here, one here, zero here. You know, sometimes there's zero here, zero here, you know, three here and so forth. All right. So you see these different distributions.

Sorry, you see these different lists of photon occupation numbers, you know, numbers of photons in each sort of output port. But then you ask the question, okay, you know, you could ask, well, what is this useful for? We don't really have a good answer to that. Okay. But then you can ask a different question, which is how hard would it be for a classical computer to sample from the same probability distribution over photon locations, right? And what we did in 2011 was that we gave pretty strong evidence using complexity theory that that problem should be hard for a classical computer. Okay, you know, I could go through, you know, through what the evidence looked like. But basically, if there were a classical algorithm that could sample from exactly the same probability distribution as the sort of ideal version of this photonic experiment, then we'd say like that would have sort of staggering implications for complexity classes. It wouldn't quite mean that P equal to NP,

city classes. It wouldn't quite mean that P equal to NP, but it would mean something that's sort of morally almost as bad as that, which is called the collapse of the polynomial hierarchy. Okay, it's sort of like a more abstract version or higher up version of P equaling NP. Okay, we showed that that would follow if you had a fast, exact classical simulation of boson sampling. Okay. And then if your classical simulation is only approximate, which is sort of the more physically relevant case, because after all, the experiment itself isn't perfect either, right? You know, in the approximate case, we believe that that would collapse the polynomial hierarchy, but there we had to make a, you, you know, a yet another conjecture. Okay. Uh, you know, and, and that, that, that, and,

and I would say the, the, the, the status of that remains unresolved to this day. Okay. But, but, you know, it's, it's at least, it's at least seems very plausible that, you know, and we showed that like, you know, here is some well-stated problem about a function of matrices called, called the permanent. And if this problem is sufficiently hard, then boson sampling is hard to simulate using a classical computer, even a proximate. Okay, so, yeah, so we did that. And then independently from us, there were others like Bremner, Josa, and Shepard who were having, you know, related ideas, like about different kinds of rudimentary quantum computers that, you know, were not, you know, again, you know, were not obviously useful for anything, but that at least, you know, seem to give rise to these probability distributions that are hard to sample using a classical computer for, you know, and for broadly similar reasons. And, you know, so one reason why we cared about this was just pure complexity theory, right?

It was just pure, you know, can you encode these hard problems into the amplitudes of a quantum computer, right? But then, you know, we realized more and more as we worked on it that maybe, you know, the experimentalists will care about this because, you know, what we hard for a classical computer, you know, hard for the biggest supercomputers in the world to simulate. Okay, and so then, you know, we explicitly talked about that. And then, you know, the quantum optics experimentalists got very excited about it. And, you know, they decided to start doing it. You know, initially, like in 2013, it was like with three photons, four photons. This is all, you know, of course, trivial for a classical computer to simulate, right? The difficulty, ideally, you know, you would expect it to go maybe like two to the power of the number of photons, right? You know, less than that if there's noise or other things that, you know, you can take advantage of. But, but, but then what happened was that in 2014,

I think Google hired John Martinez, who was maybe the top superconducting qubits experimentalist in the world. And he said, you know, we want to build a 50 to 70 qubit quantum computer, you know, that's programmable using superconducting qubits. And we want to do something cool with it, right? And what is there to do with 50 qubits that's cool, right? Well, you know, there's not a whole lot, unfortunately, right? Most of the actually useful things, you know, they might need, like, you know, hundreds to thousands of qubits, and then crucially, you know, of a much higher quality than Google google was you know able to make where you could do like thousands of layers of gates you know and they couldn't do that they can do maybe 20 layers of gates right uh but so so so so what you could do well they realized okay

we can do some version of boson sampling right except you know adapt except more except now more adapted to their hardware so we talked to them about that, and we said, yeah, that sounds reasonable, but we then had to adapt the theory from boson sampling to superconducting qubits, to the kind of thing that they were building. So we did that like 2016, 2017. And then in 2019, Google announced that they had actually done this. So they built a 53 qubit device called Sycamore. And they used it

to sample from some probability distribution over 53 bit strings, you know strings that you get by just applying a random sequence of quantum operations to these qubits. And then at first they were saying, well, with the best classical algorithm that we can think of, it would take 10,000 years to do the same thing on a supercomputer. And the press loved right? And they ran with that number. And that turned out to be wildly over-optimistic. Right. So- I could imagine Michio Kaku would love that.

Yes. Oh, I'm sure he would, right? But the hard part in quantum computing, if you want to be intellectually honest about it, is that you always have to compare against what is the best thing that anyone can do with a classical computer, right? And you're only winning if you do better than that, okay? And

what's the best thing that you can do with a classical computer is often far from obvious. I mean, after all, that's what the P versus NP question is about, right? Right. So, and indeed what happened over the next couple of years is that people got better and better at simulating the Google type of experiment with classical computers by, you know, taking advantage of the noise and taking advantage of the fact that each qubit can only sort of talk to its nearest neighbors, you know, so you can, you can sort of take advantage of all the the current technological limitations of Google's experiment in order to simulate it faster with a classical computer. And I would say that the current situation is that the Google chip, you know, like, is still somewhat better than any classical solution that we have for the task of simulating itself. Like if you measure, let's say, by the total energy cost, right, or by, you know, the money

that it takes to run it, or the CO2 that's emitted. By steps? Well, the trouble with steps is that, you know, like with a classical, you can always roughly have the number of steps by just using twice as many cores. Okay. So these problems are very, very parallelizable. Right. So, you know, you can always do it faster if you're willing to like go to AWS or whatever and just say, you know, I want, you know, this massive number of cores. Right. But then, you know, for the comparison to be fair,

you should maybe be talking about money or you should be talking about the energy, the electricity that gets spent. And if you look at metrics like that, then I think that the quantum computer still wins on some tasks, but only by a couple orders of magnitude at this point. That sounds a bit human though, because if we're measuring it by money, firstly, that's human. But if we're measuring it by energy, isn't it then not based on the algorithm,

but based on our current cores? And maybe in the future, Apple comes out with M5. Of course it is. Of course it is. So you could say that you are fighting against a moving target, Of course it is. Of course it is. So you could say that, you know, you are fighting against a moving target, right? Like quantum supremacy could be achieved and then unachieved because, you know, the classical hardware and software will both get better. You know, and so if you want to claim that a quantum computer is better for something, right, then, you know, you may have to keep improving the quantum computer, you know, just for that statement to still be true. Okay. Now, the hope, let's be clear. Okay. The hope is that eventually you have an error-corrected, you know, programmable quantum computer. And at that point, you can scale up to as many operations as you want on as many qubits as you want okay and at that point you know you could use like millions of qubits to factor some enormous number right that like unless there's a breakthrough in classical algorithms that just cannot be done classically within the whole lifetime of the universe right so that's the eventual goal okay but we're not there yet okay and and you know with these you know what what what we can

do today are these sorts of sampling experiments right where you know we're solving problems that don't have a single right answer right there you know we're sampling from a distribution over possible answers and a key drawback of these problems is that even just to verify what the quantum computer is doing already appears to take exponential time with a classical computer. Which means that we're sort of inherently limited in how far we can scale this. Like, you know, if you scaled it to 300 qubits, even if it worked, how would you ever prove it? How would you ever convince a skeptic of what you had done? So we're sort of forced to stay in this regime, where the advantages that we can get over classical computers are relatively marginal ones. Okay. But you know, but but but but but but at least we can now do that. Right? Five years ago,

you know, we could not even do that. Okay. And, and now we can, right? Five years ago, you know, we could not even do that, okay? And now we can, right? And that's, you know, I think it's a step forward. I think it's, you know, it's taught the experimentalists a lot about how to actually integrate, you know, large numbers of qubits. And, you know, and one, maybe scientifically, the most important thing that they learned from these quantum supremacy experiments was that like the total amount of signal you get, what's called the circuit fidelity, is, well, it's falling off exponentially with the number of operations, you know, which sounds kind of bad, right? But the good news is that it's merely falling off exponentially and not faster than that. the good news is that it's merely falling off exponentially and not faster than that. Okay. So basically the total fidelity, you know, like let's say, you know, each individual two qubit operation has a fidelity in Google's experiment of about 99.5%. Right. And there's about a thousand such operations. Okay. And so, so then the total fidelity that you get for the circuit it looks like just 0.995 to

the power of a thousand right uh you know which is like the simplest prediction that you could possibly make and as long as that remains true then ultimately quantum error correction should work okay so you know the the people people who believe that scalable quantum computers are impossible, such as, you know, Gil Kalai is a famous one, right? You know, what they basically have to believe is that, you know, either quantum mechanics itself is going to break, which, you know, let's face it, that would be far more exciting than a mere success in building a quantum computer, right? That would be a revolution in physics. Or else they have to believe that there's some sort of conspiratorially correlated errors

that will violate all the assumptions of the theory of quantum error correction. But from the quantum supremacy experiments, we can now say we see no sign of those conspiratorially correlated errors. And you're referring to super determinism? No, I'm not. I'm not. I'm not. I mean, super determinism is so bizarre that it's not even clear that that would have any empirical consequences at all, right? Because that's just saying there's a giant cosmic conspiracy theory that just predetermined everything that would happen from the beginning of time, right? It was like, you can always believe that, but it's sort of explanatorily worthless, right? It doesn't, it has no power to explain anything, okay? So, the more interesting thing, if there were these

conspiracies in the errors affecting qubits, that would be something that would be empirically observable, right? And, you know, if the correlations were strong enough, then conceivably that could even kill quantum computation, okay? But, you know, there are some basic difficulties here. One is that it's very, very hard to design a model of correlated errors that only kills quantum computation and that wouldn't also kill classical computation, right? If you kill classical computation, then you've proved too much because, you know, we know that scalable classical computers can be built, right? So, okay, but then the other thing you can say is that, you know, from the quantum supremacy experiments, we see no sign of these correlated errors. The errors look pretty independent. And in that case, it's, you know,

as long as that remains true, then it's just a quantitative question. You just have to get the accuracy of each operation to be high enough, like instead of 0.995, maybe 0.9999, right? And then at that point, quantum error correction should work. Now, we both got to get going shortly, and I want to end on the topic of consciousness. So why don't you talk about your critiques for IIT, and then also what the pretty hard problem of consciousness is. But before you do that, I just wanted to ask you a yes or no question about if you saw Tim Palmer's response to your critiques on superdeterminism in his recent article.

Tim Palmer's article on superdeterminism without conspiracy is listed in the description, and you should know that Tim Palmer was on the Theories of Everything podcast, and that link is also in the description. He was on with Tim Modlin at the same time, talking about the interpretations of quantum mechanics. Yeah, I mean, I think it is a conspiracy. I mean, I think that he redefines terms in an utterly perverse way in order to make it look like it's not one. But, you know, I don't think that you actually explain anything new that you, you know,

couldn't have explained without this. And, you know, it's like, it's a problem because I only have so much time in life to spend, right? And like lot about Wolfram's ideas, about Tim Palmer's ideas, but there are so many ideas out there that seem like orders of magnitude more plausible to me than these. These people have succeeded at getting out to the public and putting their message before the public but like scientifically i regard these as worthless ideas okay so um uh so so okay um yeah uh but but but now uh uh consciousness yeah so so so so integrated information theory is another thing that i think has you know okay you know maybe like it was worth trying, it was worth thinking about, but that has also become, you know, a worthless idea, a pathological idea. Okay. So it's a, a proposed, you know, theory of, of consciousness, or at least of what things are

conscious and what things are not, uh, that was, uh, um, originated, originated by Giulio Tononi, okay, and, you know, also pursued by, you know, a bunch of other people such as Christophe Koch, I think, you know, Max Tegmark was into it. And basically what it says is that there was some, you know, you could take any complex system and then there is some quantitative measure that, you know, that you can calculate for that system that somehow measures how interconnected all of its components are. Right. And and the war, you know, they call that fee. Right. And the larger is that fee measure, the more conscious the system is. Right. Now, you know, you could say, what is this fee, right? Well, the actual definition of it keeps changing, right? So, you know, there's not actually the one fixed thing that you can critique, right?

They keep fiddling with the definition of it. And they said, you know, and if you read the papers about it, like what they say is, well, you know, we have these axioms of conscious experience, and then we derive phi from these axioms. But actually, there's no derivation at all. It's sort of like pseudo-mathematics. It's like they state the axioms, and then at some point, phi just appears. It just shows up, not having been derived in any way. So they show it's consistent with the axioms, but not derivable from them? just appears. It just shows up, you know, not having been derived in any way, right?

So they show it's consistent with the axioms, but not derivable from them? Yeah, I mean, the axioms are not even clear enough to me, for me to really understand what they are saying, right? But at least the fee measure, you know, at least like that seems reasonably clear. Like if you, once you've decided what are the subsystems, what does it mean for a subsystems to be connected to each other and so forth, then, you know, and you decided which version of phi you're working with, then, you know, you can actually calculate these numbers. Okay. And then, you know, their case, you know, they're like, they were neuroscientists and they said, well, there seems to be much higher connectivity in the cerebrum than in the cerebellum, for example, right? And the cerebrum is associated with consciousness and the cerebellum is not,

right? So that was the kind of evidence that they gave. Okay. But then I wrote a blog post about this like nine years ago where I pointed out that we could easily invent systems that have massively higher fee than the human cerebrum. Right. But that are just like a gigantic grid of XOR gates or something like that. Right. Without processing. Yeah. Well, OK. Yeah. Maybe, you know know they do process something like they compute the xor of a bunch of bits right but but it's all like completely regular it's like there's not even anything interesting or intelligent going on there let alone anything conscious right and yet you know if you calculate this number that they put forward then you'll get that the connectivity between

the different components, you know, can be arbitrarily large, right? Just depending on how big you make the thing, right? You could make it, you know, and I said, and to me, that seems like a reductio ad absurdum, right? That seems like, okay, you know, the theory is just making a, you know, like an utterly wrong or absurd prediction. And therefore, this connectivity, it might be correlated with consciousness in various cases, but it cannot be identical with consciousness, right? Because we can make it huge in systems that are obviously not conscious, okay? And so then, Tinoni's response to me was extremely

interesting. Okay. Because what he, I say like, the way I put it was like, he didn't just bite the bullet. He like basically devoured a bullet hoagie with mustard, right? He said, look, you know, the problem is that you're just, you know, using your intuition, but you have to rely on the theory. And what the theory tells you is that this grid is that this giant grid of exorgates is conscious. In fact, it's much more conscious than you are, right? And you have to be bold and just believe what the theory says. And my response at that point was like, okay, if a theory is sort of getting right the things that, the cases where we already thought we knew the answers, right? Then you might be interested in, you know, what does the theory say about the really hard cases? Like, let's say about a fetus or about a coma patient or about an earthworm or

about an AI program, right? But if the, you know, your measure of consciousness is already, you know, telling you that a potato chip bag is conscious, right? Or that, you know, any old, you know, sort of uninteresting physical system, you know, is conscious just, you know, because of this sort of, you know, this sort of, you know, mundane properties that it has, right? Then, okay, if it's gotten that wrong, then sort of what is there for it to get right, right? It's like, how else would we validate the theory in the first place,

except by seeing that it at least gives the right answers in the cases where we already thought that we understood what we meant by consciousness and so forth. So that was kind of the point where i got off the train and you know and this theory has just continued you know you know generating i don't know hundreds or thousands of publications over the last decade i saw that there was a letter recently by like a bunch of philosophers and neuroscientists who said that

you know integrated information theory is pseudoscience right and uh and then there was a response and there was you know i i didn't i didn't even wade into this because i felt like i had said my piece you know a decade ago yeah right well it's like it's like you know yeah these are these are ideas that are worth exploring but then once you get you know once you see that you you make a prediction that that's bad that that's that badly wrong then i think you have to you think you have to go back to the drawing board. You have to say, okay, well, then this is not a measure of consciousness. And instead, they just keep trying to rescue the theory. And at some point, that kind of thing does degenerate into pseudoscience.

kind of thing does degenerate into pseudoscience. So it sounds like what he's saying is that what you think of as a reductio ad absurdum is actually a productio ad correctum. You're proving something correct, and we should listen to the theory. So why can't we follow that same logic when it comes to the many worlds and say, look, on one hand, some people can say it's a reductio ad absurdum. You're predicting all of these other worlds. But then Sean Carroll may say, no, no, we should be listening. Well, I would say that the difference is that quantum mechanics gets, you know, every single case that we can check, it gets the right answer. Right. Right. You know, that's, that's been true for a hundred years. There were zero counterexamples, right. And that's now even with very, very complicated entangled quantum systems. And so then, you know, it is, it is natural to just want to do the extrapolation so then, you know, it is natural to just want

to do the extrapolation and say, you know, assume that this continues to be true up to arbitrary scales, then what does that mean? Because we've seen no sign otherwise, right? But like with consciousness, like I feel like if you want to decide what is or isn't conscious, like, you know, we have to start with, you know, we have to start with definitions, right? Like, what do we even mean by consciousness? Like it is, you know, whatever it is, like, it is some sort of thing that we regard ourselves as having, and that we regard this glasses cloth here as not having. Or at least not having anywhere near as much. Right, right. Yes, yes, exactly. It's not having nearly as much. If you

say that this glasses cloth has much more consciousness than I do, then I say, well, what do you even mean anymore by the word consciousness? I don't think you're talking about the same thing that I was trying to talk about. Right, right. Yeah. Okay. And then the pretty hard problem consciousness is. Yeah, that's just the problem of, okay, and then the pretty hard problem with consciousness is... explain how subjective experience can arise out of mere matter or mere physical systems obeying the laws of physics. And in some sense, this is a problem that goes all the way back to Democritus in like 400 BC. And you have a book on this. I do. I do. Yeah. I mean, Democritus already stated this problem in a remarkably modern way, right? And certainly Descartes asked this question centuries ago. So this is one of the great problems of

philosophy or the great problems of all of human existence, right? And it's not even clear what an answer would look like or what kind of thing could be an answer, you know, let alone how to find that answer. Okay, but so then, I said, you know, whenever you encounter something that is just way too hard, whether it's the hard problem of consciousness, or whether it's P versus NP, right, then, you know, it's natural to want to scale back and say, well, you know, is there an easier related problem that we can make progress on, right? And so, what I called the pretty hard problem is just the question, which physical systems are associated with consciousness and which are not, right?

And so, you know, you could imagine answering that without having to answer, like, you know, among the ones that are conscious, like, you know, why are they conscious or how, right? You could imagine just giving a criterion that, you know, agrees with intuition in all the cases, you know, that we can think of and that, you know, and that actually calculates, you know, some measure of consciousness, you know, from some simple starting point. You know, that is precisely what IIT was trying to do. And I respect that it was trying to do that. I just don't think that it succeeds. Well, Professor, thank you for spending so long with me. Sure. It was so much fun. Yeah. All right. I hope to speak with you again. Yeah. All right. Take care. Talk to you later. Bye bye. If you enjoyed this podcast, then one that I recommend is the one with Tim Palmer and again. Yeah, all right. Take care. Talk to you later. Bye-bye. then please do so at patreon.com slash kurtjaimungal, or the PayPal link is in the description, as well as a crypto link.

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