15 Skepticism of quantum computing

Transcription

1 15 Skepticism of quantum computing Last chapter, we talked about whether quantum states should be thought of as exponentially long vectors, and I brought up class BQP/qpoly and concepts like quantum advice. Actually, I d say that the main reason why I care is something I didn t mention last time, which is that it relates to whether we should expect quantum computing to be fundamentally possible or not. There are people, like Leonid Levin and Oded Goldreich, who just take it as obvious that quantum computing must be impossible. 1 Part of their argument is that it s extravagant to imagine a world where describing the state of 200 particles takes more bits then there are particles in the universe. To them, this is a clear indication something is going to break down. So part of the reason that I like to study the power of quantum proofs and quantum advice is that it helps us answer the question of whether we really should think of a quantum state as encoding an exponential amount of information. So, on to the Eleven Objections. 1. Works on paper, not in practice. 2. Violates Extended Church Turing Thesis. 3. Not enough real physics. 4. Small amplitudes are unphysical. 5. Exponentially large states are unphysical. 6. Quantum computers are just souped-up analog computers. 7. Quantum computers aren t like anything we ve ever seen before. 8. Quantum mechanics is just an approximation to some deeper theory. 1 See, for example, and weizmann.ac.il/ oded/on-qc.html

2 218 quantum computing since democritus 9. Decoherence will always be worse than the fault-tolerance threshold. 10. We don t need fault-tolerance for classical computers. 11. Errors aren t independent. What I did is to write out every skeptical argument against the possibility of quantum computing that I could think of. We ll just go through them, and make commentary along the way. Let me just start by saying that my point of view has always been rather simple: it s entirely conceivable that quantum computing is impossible for some fundamental reason. If so, then that s by far the most exciting thing that could happen for us. That would be much more interesting than if quantum computing were possible, because it changes our understanding of physics. To have a quantum computer capable of factoring digit integers is the relatively boring outcome the outcome that we d expect based on the theories we already have. I like to engage skeptics for several reasons. First of all, because I like arguing. Second, often I find that the best way to come up with new results is to find someone who s saying something that seems clearly, manifestly wrong to me, and then try to think of counterarguments. Wrong claims are a fertile source of research ideas. So what are some of the skeptical arguments that I ve heard? The one I hear more than any other argument is well, it works formally, on paper, but it s not gonna work in the real world. People actually say this, and they actually treat it like it was an argument. For me, the fallacy here is not that people can have ideas that don t work in the real world, but rather that if they don t work in the real world, they can still work on paper. Of course, there could be assumptions such that an idea only works if the assumptions are satisfied. Thus, the question becomes if the assumptions are stated clearly or not. I was happy to find out that I wasn t the first person to point out this particular fallacy. Immanuel Kant wrote an entire treatise

3 skepticism of quantum computing 219 demolishing it: On the Common Saying: That may be right in theory but does not work in practice. The second argument is that quantum computing must be impossible because it violates the Extended Church Turing Thesis: Anything that is efficiently computable in the physical world is computable in polynomial time on a standard Turing machine. That is, we know that quantum computing can t be possible (assuming BPP BQP), because we know that BPP defines the limit of the efficiently computable. So, we have this thesis, and quantum computing violates the thesis, so (if you have faith in the thesis) it must be impossible On the other hand, if you replaced factoring with NP-complete problems, then this argument would actually become more plausible to me, because I would think that any world in which we could solve NP-complete problems efficiently would not look much like our world. For NP-intermediate problems like factoring and Graph Isomorphism, I m not willing to take some sort of a-priori theological position. But the diagram below shows how I think things most likely stand. NP-complete 3SAT Graph Isomorphism Factoring P

4 220 quantum computing since democritus So that was the second argument. On to the third: I m suspicious of all these quantum computing papers because there isn t enough of the real physics that I learned in school. There s too many unitaries and not enough Hamiltonians. There s all this entanglement, but my professor told me not to even think about entanglement, because it s all just kind of weird and philosophical, and has nothing to do with the structure of the helium atom. What can one say to this? Certainly, this argument succeeds in establishing that we have a different way of talking about quantum mechanics now, in addition to the ways people have had for many years. Those making this argument are advancing an additional claim, though, which is that the new way of talking about quantum mechanics is wrong.and that claim, of course, requires a separate argument. I don t know if any further response is needed. The fourth argument is that these exponentially small amplitudes are clearly unphysical. This is another argument that Leonid Levin has made. Consider some state of 1000 qubits, such that each component has an amplitude of We don t know of any physical law that holds to more than about a dozen decimal places, and you re asking for accuracy to hundreds of decimal points. Why should someone even imagine that makes any sense whatsoever? The obvious repudiation of argument 4, then, is that I can take a classical coin and flip it a thousand times. Then, the probability of any particular sequence is , which is far smaller than any constant we could ever measure in nature. Does this mean that probability theory is some mere approximation of a deeper theory, or that it s going to break down if I start flipping the coin too many times? For me the key point is that amplitudes evolve linearly, and in that respect are similar to probabilities. We ve got minus signs, and so we ve got interference, but maybe if we really thought about why probabilities are okay, we could argue that it s not just that we re always in a deterministic state and just don t know what it is, but that this property of linearity is something more general. Linearity is the thing that prevents small errors from creeping up on us. If we

5 skepticism of quantum computing 221 have a bunch of small errors, the errors add rather than multiplying. That s linearity. Argument 5 gets back to what we were talking about in the previous chapter: it s obvious that quantum states are these extravagant objects; you can t just take 2 n bits and pack them into n qubits. Actually, I was arguing with Paul Davies, and he was making this argument, appealing to the holographic principle and saying that we have a finite upper bound on the number of bits that can be stored in a finite region of spacetime. If you have some 1000-qubit quantum state, it requires bits, and according to Davies, we ve just violated the holographic bound. 2 So how should one respond to that? First of all, this information, whether or not we think it s there, can t generally be read out. This is the content of results like Holevo s theorem. In some sense, you might be able to pack 2 n bits into a state, but the number of bits that you can reliably get out is only n. The holographic bound says, informally, that the maximum number of bits that can be stored in any finite region is proportional to the region s surface area, at roughly the rate of one bit per Planck area, or bits per meter squared. Why should the maximum number of bits grow like the surface area, rather than the volume? That s a very profound question that people like Ed Witten and Juan Maldacena probably stay up at night worrying about. The doofus answer is that if you try to take lots and lots of bits and pack them into some volume (such as a cubical hard disk), then at some point, your cubical hard disk will collapse and form a black hole. A flat drive will also collapse, but a one-dimensional drive won t collapse. Here s the thing: there seem to be all these bits near the event horizon of the black hole. Why the event horizon? Because if you re standing outside a black hole, then you never actually see anything fall through the event horizon. Instead, because of time dilation, all 2 Davies subsequently published this argument; see

6 222 quantum computing since democritus the infalling objects will seem to get eerily frozen just outside the event horizon approaching it, Zeno-like, but never reaching it. Then, if you want to preserve unitarity, and not have pure states evolve into mixed states when something gets dropped into a black hole, you say that when the black hole evaporates via Hawking radiation, then the bits get peeled off like scales, and go flying out into space. Again, this is not something that people really understand. People treat the holographic bound (rightfully) as the one of the few clues we have for a quantum theory of gravity, but they don t yet have the detailed theory that implements the bound, except for some special model systems. The other funny thing about this is that, in classical general relativity, the event horizon doesn t play a particularly special role. You could pass through it and you wouldn t even notice. Eventually, you ll know you passed through it, because you ll be sucked into the singularity, but while you re passing through it, it doesn t feel special. On the other hand, this information point of view says that as you pass through, you ll pass a lot of bits near the event horizon. What is it that singles out the event horizon as being special in terms of information storage? It s very strange, and I wish I understood it (see Chapter 22 for further discussion). There actually is an interesting question here. The holographic principle says that you can store only so much information within a region of space, but what does it mean to have stored that information? Do you have to have random access to the information? Do you have to be able to access whatever bit you want and get the answer in a reasonable amount of time? In the case that these bits are stored in a black hole, apparently if there are n bits on the surface, then it takes on the order of n 3/2 time for the bits to evaporate via Hawking radiation. So, the time-order of retrieval is polynomial in the number of bits, but it still isn t particularly efficient. A black hole should not be one s first choice for a hard disk. Argument 6: a quantum computer would merely be a souped-up analog computer. This I ve heard again and again, from people like