We're going to start by beating a retreat from QuantumLand, back onto the safe territory of computational complexity. In particular, we're going to see how, in the 1980s and 1990s, computational complexity theory reinvented the millennia-old concept of mathematical proof – making it probabilistic, interactive, and cryptographic. But then, having fashioned our new pruning-hooks (proving-hooks?), we're going to return to QuantumLand and reap the harvest. In particular, I’ll show you why, if you could see the entire trajectory of a hidden variable, then you could efficiently solve any problem that admits a “statistical zero-knowledge proof protocol,” including problems like Graph Isomorphism for which no efficient quantum algorithm is yet known.

What is a proof?

Historically, mathematicians have had two very different notions of “proof.”

The first is that a proof is something that induces in the audience (or at least the prover!) an intuitive sense of certainty that the result is correct. In this view, a proof is an inner transformative experience – a way for your soul to make contact with the eternal verities of Platonic heaven.

So, in this chapter, we're going to ask – and hopefully answer – this question of whether there's free will or not. If you want to know where I stand, I’ll tell you: I believe in free will. Why? Well, the neurons in my brain just fire in such a way that my mouth opens and I say I have free will. What choice do I have?

Before we start, there are two common misconceptions that we have to get out of the way. The first one is committed by the free will camp, and the second by the anti-free-will camp.

The misconception committed by the free will camp is the one I alluded to before: if there’s no free will, then none of us are responsible for our actions, and hence (for example) the legal system would collapse. Well, I know of only one trial where the determinism of the laws of physics was actually invoked as a legal defense. It’s the Leopold and Loeb trial in 1926. Have you heard of this? It was one of the most famous trials in American history, next to the OJ trial. So, Leopold and Loeb were these brilliant students at the University of Chicago (one of them had just finished his undergrad at 18), and they wanted to prove that they were Nietzschean supermen who were so smart that they could commit the perfect murder and get away with it. So they kidnapped this 14-year-old boy and bludgeoned him to death. And they got caught – Leopold dropped his glasses at the crime scene.

Answers to puzzles from Chapter 7

Puzzle 1. We are given a biased coin that comes up heads with probability p. Using this coin, construct an unbiased coin.

Solution. The solution is the “von Neumann trick”: flip the biased coin twice, interpreting HT as heads and TH as tails. If the flips come up HH or TT, then try again. Under this scheme, “heads” and “tails” are equiprobable, each occurring with probability p(1 - p) in any given trial. Conditioned on either HT or TH occurring, it follows that the simulated coin is unbiased.

Puzzle 2. n people sit in a circle. Each person wears either a red hat or a blue hat, chosen independently and uniformly at random. Each person can see the hats of all the other people, but not his/her own hat. Based only upon what they see, each person votes on whether or not the total number of red hats is odd. Is there a scheme by which the outcome of the vote is correct with probability greater than ½?

Solution. Each person decides his/her vote as follows: if the number of visible blue hats is larger than the number of visible red hats, then vote according to the parity of the number of visible red hats. Otherwise, vote the opposite of the parity of the number of visible red hats. If the number of red hats differs from the number of blue hats by at least 2, then this scheme succeeds with certainty. Otherwise, the scheme might fail. However, the probability that the number of red hats differs from the number of blue hats by less than 2 is small – O(1/√N).

In the last chapter, we talked about the rules for first-order logic. There's an amazing result called Gödel's Completeness Theorem that says that these rules are all you ever need. In other words: if, starting from some set of axioms, you can't derive a contradiction using these rules, then the axioms must have a model (i.e., they must be consistent). Conversely, if the axioms are inconsistent, then the inconsistency can be proved using these rules alone.

Think about what that means. It means that Fermat's Last Theorem, the Poincaré Conjecture, or any other mathematical achievement you care to name can be proved by starting from the axioms for set theory, and then applying these piddling little rules over and over again. Probably 300 million times, but still...

How does Gödel prove the Completeness Theorem? The proof has been described as “extracting semantics from syntax.” We simply cook up objects to order as the axioms request them! And if we ever run into an inconsistency, that can only be because there was an inconsistency in the original axioms.

One immediate consequence of the Completeness Theorem is the Löwenheim–Skolem Theorem: every consistent set of axioms has a model of at most countable cardinality. (Note: One of the best predictors of success in mathematical logic is having an umlaut in your name.) Why? Because the process of cooking up objects to order as the axioms request them can only go on for a countably infinite number of steps!

In the last two chapters, we talked about computational complexity up till the early 1970s. Here, we'll add a new ingredient to our already simmering stew – something that was thrown in around the mid-1970s, and that now pervades complexity to such an extent that it's hard to imagine doing anything without it. This new ingredient is randomness.

Certainly, if you want to study quantum computing, then you first have to understand randomized computing. I mean, quantum amplitudes only become interesting when they exhibit some behavior that classical probabilities don't: contextuality, interference, entanglement (as opposed to correlation), etc. So we can't even begin to discuss quantum mechanics without first knowing what it is that we're comparing against.

Alright, so what is randomness? Well, that’s a profound philosophical question, but I’m a simpleminded person. So, you’ve got some probability p, which is a real number in the unit interval [0, 1]. That’s randomness.

But wasn’t it a big achievement when Kolmogorov put probability on an axiomatic basis in the 1930s? Yes, it was! But in this chapter, we’ll only care about probability distributions over finitely many events, so all the subtle questions of integrability, measurability, and so on won’t arise. In my view, probability theory is yet another example where mathematicians immediately go to infinite-dimensional spaces, in order to solve the problem of having a nontrivial problem to solve! And that’s fine – whatever floats your boat. I’m not criticizing that. But in theoretical computer science, we’ve already got our hands full with 2n choices. We need 2ℵ0 choices like we need a hole in the head.

So why Democritus? First of all, who was Democritus? He was this Ancient Greek dude. He was born around 450 BC in this podunk Greek town called Abdera, where people from Athens said that even the air causes stupidity. He was a disciple of Leucippus, according to my source, which is Wikipedia. He's called a “pre-Socratic,” even though actually he was a contemporary of Socrates. That gives you a sense of how important he's considered: “Yeah, the pre-Socratics – maybe stick ’em in somewhere in the first week of class.” Incidentally, there's a story that Democritus journeyed to Athens to meet Socrates, but then was too shy to introduce himself.

Almost none of Democritus’s writings survive. Some survived into the Middle Ages, but they’re lost now. What we know about him is mostly due to other philosophers, like Aristotle, bringing him upin order to criticize him.

So, what did they criticize? Democritus thought the whole universe is composed of atoms in a void, constantly moving around according to determinate, understandable laws. These atoms can hit each other and bounce off, or they can stick together to make bigger things. They can have different sizes, weights, and shapes – maybe some are spheres, some are cylinders, whatever. On the other hand, Democritus says that properties like color and taste are not intrinsic to atoms, but instead emerge out of the interactions of many atoms. For if the atoms that made up the ocean were “intrinsically blue,” then how could they form the white froth on waves?

Here, we're gonna talk about sets. What will these sets contain? Other sets! Like a bunch of cardboard boxes that you open only to find more cardboard boxes, and so on all the way down.

You might ask “how is this relevant to a book on quantum computing?”

Well, hopefully we’ll see a few answers later. For now, suffice it to say that math is the foundation of all human thought, and set theory – countable, uncountable, etc. – that’s the foundation of math. So regardless of what a book is about, it seems like a fine place to start.

I probably should tell you explicitly that I’m compressing a whole math course into this chapter. On the one hand, that means I don’t really expect you to understand everything. On the other hand, to the extent you do understand – hey! You got a whole math course in one chapter! You’re welcome.

So let’s start with the empty set and see how far we get.

THE EMPTY SET.

Any questions so far?

Actually, before we talk about sets, we need a language for talking about sets. The language that Frege, Russell, and others developed is called first-order logic. It includes Boolean connectives (and, or, not), the equals sign, parentheses, variables, predicates, quantifiers (“there exists” and “for all”) – and that’s about it. I’m told that the physicists have trouble with these. Hey, I’m just ribbin’ ya. If you haven’t seen this way of thinking before, then you haven’t seen it. But maybe, for the benefit of the physicists, let’s go over the basic rules of logic.

In the last chapter, we talked about free will, superintelligent predictors, and Dr. Evil planning to destroy the Earth from his moon base. Now I’d like to talk about a more down-to-earth topic: time travel. The first point I have to make is one that Carl Sagan made: we're all time travelers – at the rate of one second per second! Har har! Moving on, we have to distinguish between time travel into the distant future and into the past. Those are very different.

Travel into the distant future is by far the easier of the two. There are several ways to do it.

• Cryogenically freeze yourself and thaw yourself out later.

• Travel at relativistic speed.

• Go close to a black hole horizon.

This suggests one of my favorite proposals for how to solve NP-complete problems in polynomial time: why not just start your computer working on an NP-complete problem, then board a spaceship traveling at close to the speed of light and return to Earth to pick up the solution? If this idea worked, it would let us solve much more than just NP. It would also let us solve PSPACE-complete and EXP-complete problems – maybe even all computable problems, depending on how much speedup you want to assume is possible. So what are the problems with this approach?

By any objective standard, the theory of computational complexity ranks as one of the greatest intellectual achievements of humankind – along with fire, the wheel, and computability theory. That it isn't taught in high schools is really just an accident of history. In any case, we'll certainly need complexity theory for everything else we're going to do in this book, which is why the next five or six chapters will be devoted to it. So before we dive in, let's step back and pontificate about where we're going.

What I’ve been trying to do is show you the conceptual underpinnings of the universe, before quantum mechanics comes on the scene. The amazing thing about quantum mechanics is that, despite being a grubby empirical discovery, it changes some of the underpinnings! Others it doesn't change, and others it's not so clear whether it changes them or not. But if we want to debate how things are changed by quantum mechanics, then we'd better understand what they looked like before quantum mechanics.

• 1950s: Late Turingzoic

• 1960s: Dawn of the Asymptotic Age

• 1971: The Cook–Levin Asteroid; extinction of the Diagonalosaurs

• Early 1970s: The Karpian Explosion

• 1978: Early Cryptozoic

• 1980s: Randomaceous Era

• 1993: Eruption of Mt Razborudich; extinction of the Combinataurs

• 1994: Invasion of the Quantodactyls

• Mid-1990s to present: Derandomaceous Era

To remind you, this book is based on a course I taught in 2006. On the last day of class, I followed the great tradition pioneered by Richard Feynman, in which the last class should be one where you can ask the teacher anything. Feynman's rule was that you could ask about anything except politics, religion, or the final exam. In my case, there was no final exam, and I didn't even make politics or religion off-limits. This chapter collects some of the questions people asked me, together with my responses.

1. Student: Do you often think about using computer science to limit or give us a hint about physical theories? Do you think that we'll be able to discover physical theories which give more powerful models than quantum computation?

2. Scott: Is BQP the end of the road, or is there more to be found? That's a fantastic question, and I wish more people would think about it. I’m being a bit of a politician here and not answering directly, because obviously the answer is “I don't know.” I guess the whole idea with science is that if we don't know the answer, we don't try to sprout one out of our butt or something. We try to base our answers on something. So, everything we know is consistent with the idea that quantum computing is the end of the road. Greg Kuperberg had an analogy I really liked. He said that there are people who keep saying that we've gone from classical to quantum mechanics so what other surprises are in store? But maybe that's like first assuming the Earth is flat, and then on discovering that it's round, saying who knows, maybe it has the topology of a Klein bottle. There's a surprise in a given direction, but once you've assimilated it, there may not be any further surprise in that same direction.

This chapter is about Roger Penrose's arguments against the possibility of artificial intelligence, as famously set out in his books The Emperor's New Mind and Shadows of the Mind. It would be strange for a book like this one not to discuss these arguments, since, agree with them or not, they're some of the most prominent landmarks at the intersection of math, CS, physics, and philosophy. The reason we're discussing them now is that we finally have all the prerequisites (computability, complexity, quantum mechanics, and quantum computing).

Penrose's views are complicated: they involve speculations about an “objective collapse” of quantum states, which would arise from an as-yet-undiscovered quantum theory of gravity. Even more controversially, this hypothesized objective collapse would play a role in human intelligence, through its influence on cellular structures called microtubules in the brain.

But what is it that leads Penrose to make these exotic speculations in the first place? The core of Penrose’s thesis is a certain argument purporting to show that human intelligence can’t be algorithmic, for reasons related to Gödel’s Incompleteness Theorem. And therefore, some nonalgorithmic element must be sought in human brain function, and the only plausible source of such an element is new physics (coming, for example, from quantum gravity). The “Gödel argument” itself didn’t originate with Penrose: Gödel himself apparently believed some version of it (though he never published his views), and even in 1950 it was well enough known for Alan Turing to rebut it in his famous paper “Computing machinery and intelligence.” Probably the first detailed presentation of the Gödel argument in print came in 1961, from the philosopher John Lucas. Penrose’s main innovation is that he takes the argument seriously enough to explore, at length, what the universe and our brains would actually need to be like – or better, what they could possibly be like – if the argument were valid. Hence, all the stuff about quantum gravity and microtubules.

Puzzle from last chapter: What can you compute with “narrow” CTCs that only send one bit back in time?

Solution: let x be a chronology-respecting bit, and let y be a CTC bit. Then, set x := x ⊕ y and y := x. Suppose that Pr[x = 1] = p and Pr[y = 1] = q. Then, causal consistency implies p = q. Hence, Pr[x ⊕ y = 1] = p(1 - q) + q(1 - p) = 2p(1 - p).

So we can start with p exponentially small, and then repeatedly amplify it. We can thereby solve NP-complete problems in polynomial time (and indeed PP ones also, provided we have a quantum computer).

I’ll start with the “New York Times model” of cosmology – that is, the thing that you read about in popular articles until fairly recently – which says that everything depends on the density of matter in the universe. There’s this parameter Ω which represents the mass density of the universe, and if it’s greater than unity, the universe is closed. That is, the matter density of the universe is high enough that, after the Big Bang, there has to be a Big Crunch. Furthermore, if Ω > 1, spacetime has a spherical geometry (positive curvature). If Ω = 1, the geometry of spacetime is flat and there’s no Big Crunch. If Ω < 1, then the universe is open, and has a hyperbolic geometry. The view was that these are the three cases.

The puzzle from last chapter is known as Hume's Problem of Induction.

Puzzle: If you observe 500 black ravens, what basis do you have for supposing that the next one you observe will also be black?

Many people’s answer would be to apply Bayes’s Theorem. For this to work, though, we need to make some assumption such as that all the ravens are drawn from the same distribution. If we don’t assume that the future resembles the past at all, then it’s very difficult to get anything done. This kind of problem has led to lots of philosophical arguments like the following.

Suppose you see a bunch of emeralds, all of which are green. This would seem to lend support to the hypothesis that all emeralds are green. But then, define the word grue to mean “green before 2050 and blue afterwards.” Then, the evidence equally well supports the hypothesis that all emeralds are grue, not green. This is known as the grue paradox.

If you want to delve even “deeper,” then consider the “gavagai” paradox. Suppose that you’re trying to learn a language, and you’re an anthropologist visiting an Amazon tribe speaking the language. (Alternatively, maybe you’re a baby in the tribe. Either way, suppose you’re trying to learn the language from the tribe.) Then, suppose that some antelope runs by and some tribesman points to it and shouts “gavagai!” It seems reasonable to conclude from this that the word “gavagai” means “antelope” in their language, but how do you know that it doesn’t refer to just the antelope’s horn? Or it could be the name of the specific antelope that ran by. Worse still, it could mean that a specific antelope ran by on some given day of the week! There’s any number of situations that the tribesman could be using the word to refer to, and so we conclude that there is no way to learn the language, even if we spend an infinite amount of time with the tribe.

A Critical Review of Scott Aaronson's Quantum Computing since Democritus by Scott Aaronson

Quantum Computing since Democritus is a candidate for the weirdest book ever to be published by Cambridge University Press. The strangeness starts with the title, which conspicuously fails to explain what this book is about. Is this another textbook on quantum computing – the fashionable field at the intersection of physics, math, and computer science that's been promising the world a new kind of computer for two decades, but has yet to build an actual device that can do anything more impressive than factor 21 into 3 × 7 (with high probability)? If so, then what does this book add to the dozens of others that have already mapped out the fundamentals of quantum computing theory? Is the book, instead, a quixotic attempt to connect quantum computing to ancient history? But what does Democritus, the Greek atomist philosopher, really have to do with the book's content, at least half of which would have been new to scientists of the 1970s, let alone of 300 BC?

Having now read the book, I confess that I’ve had my mind blown, my worldview reshaped, by the author's truly brilliant, original perspectives on everything from quantum computing (as promised in the title) to Gödel's and Turing's theorems to the P versus NP question to the interpretation of quantum mechanics to artificial intelligence to Newcomb's Paradox to the black-hole information loss problem. So, if anyone were perusing this book at a bookstore, or with Amazon's “Look Inside” feature, I would certainly tell that person to buy a copy immediately. I’d also add that the author is extremely handsome.

There are two ways to teach quantum mechanics. The first way – which for most physicists today is still the only way – follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then, you learn about the “blackbody paradox” and various strange experimental results, and the great crisis these things posed for physics. Next, you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you're lucky, after years of study, you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.

Look, obviously the physicists had their reasons for teaching quantum mechanics that way, and it works great for a certain kind of student. But the “historical” approach also has disadvantages, which in the quantum information age are becoming increasingly apparent. For example, I’ve had experts in quantum field theory – people who've spent years calculating path integrals of mind-boggling complexity – ask me to explain the Bell inequality to them, or other simple conceptual things like Grover's algorithm. I felt as if Andrew Wiles had asked me to explain the Pythagorean Theorem.

Why have so many great thinkers found quantum mechanics so hard to swallow? To hear some people tell it, the whole source of the trouble is that “God plays dice with the universe” – that, whereas classical mechanics could in principle predict the fall of every sparrow, quantum mechanics gives you only statistical predictions.

Well, you know what? Whoop-de-doo! If indeterminism were the only mystery about quantum mechanics, quantum mechanics wouldn't be mysterious at all. We could imagine, if we liked, that the universe did have a definite state at any time, but that some fundamental principle (besides the obvious practical difficulties) kept us from knowing the whole state. This wouldn't require any serious revision of our worldview. Sure, “God would be throwing dice,” but in such a benign way that not even Einstein could have any real beef with it.

The real trouble in quantum mechanics is not that the future trajectory of a particle is indeterministic – it’s that the past trajectory is also indeterministic! Or more accurately, the very notion of a “trajectory” is undefined, since until you measure, there’s just an evolving wavefunction. And crucially, because of the defining feature of quantum mechanics – interference between positive and negative amplitudes – this wavefunction can’t be seen as merely a product of our ignorance, in the same way that a probability distribution can.