The most trivial theorem I’ve ever written up

Theorem: Suppose NP-complete problems are efficiently solvable by quantum computers. Then either the polynomial hierarchy collapses, or else BQP ⊄ AM (that is, quantum computations can’t be efficiently simulated by Arthur-Merlin protocols).

Proof: Suppose NP ⊆ BQP and BQP ⊆ AM. Then coNP ⊆ BQP ⊆ AM, and hence the polynomial hierarchy collapses to the second level by a result of Boppana, Håstad, and Zachos.

Note: If only we could delete the weasel phrase “or else BQP ⊄ AM” from my Most Trivial Theorem, we would’ve achieved a long-sought breakthrough in quantum computing theory. In particular, we would’ve shown that any fast quantum algorithm to solve NP-complete problems would imply an unlikely collapse of classical complexity classes. But while the weasel phrase is weaselly enough to make the Most Trivial Theorem a triviality, I don’t think it’s infinitely weaselly. The reason is my growing suspicion that BQP ⊆ AM in the unrelativized world.

Second Note: When I call this my “Most Trivial Theorem,” obviously I’m excluding homework exercises.

21 Responses to “The most trivial theorem I’ve ever written up”

  1. Peter Shor Says:

    I have to bite. Why do you think BQP is inside AM in the unrelativized world?

  2. Scott Says:

    Why do you think BQP is inside AM in the unrelativized world?

    Because every attempt to construct a candidate problem in BQP\AM has — unexpectedly to me! — been such a total failure. Whenever we find an efficient quantum algorithm — for example, for HSP, for walking on two conjoined binary trees, for the problems recently considered by Harrow & Hallgren, for various combinations of the last three — we also seem to find global structure that can be detected using approximate counting. Why does the one property always seem to go with the other?

    To be clear, I do think we’re likely to succeed eventually in showing BQP⊄AM relative to an oracle. Probably Recursive Fourier Sampling already gives a suitable oracle, though it could only lead to an nlog n versus n separation, not an exponential one. If not, Greg Kuperberg and I have an alternative proposal based on “pseudorandom unitaries” (see here for more), which could lead to an exponential separation. However, both types of oracle are so contrived — in contrast (for example) to the oracles relative to which P≠BQP and NP⊄BQP — that I don’t know if anything can be learned from them about the unrelativized case.

    Alright, let me be more conservative: I see strong reasons to believe P≠NP, P≠BQP, and NP⊄BQP. I expected to find similarly strong reasons to believe BQP⊄AM, and haven’t found any yet.

  3. Greg Kuperberg Says:

    I disagree with Scott. It is possible that BQP is in AM, unrelativized. But of course, everyone believes that NP = AM, unrelativized, so that would imply that BQP is in NP. I see no strong evidence that it has to be so. I have no strong evidence that it is false, either, but to me it more indicates our primitive understanding of BQP. The apparent size of a complexity class typically changes as you understand it better, and I still think that BQP will look larger in the future.

    Of course some oracle separations should still be taken seriously as evidence for the unrelativized world. The question is whether the oracle can reasonably be replaced by an instance.

  4. mick Says:

    Scott, just for your info, the “subset” character isn’t appearing properly in IE.

    Oh, and nice theorem etc.

  5. Mustela nivalis Says:

    This is actually just the first of the “Wary Weasel” theorems. As I recall, the second is something like this:

    Wary Weasel theorem 2. There is an infinite series of propositions of the following form -

    Proposition N: Suppose NP-complete theorems are efficiently solvable by quantum computers, and that (Escape Clause N-1) is not true; then the polynomial hierarchy collapses, or else (Escape Clause N).

    - such that:

    (i) for N>1, the Escape Clause refers to properties of the series of propositions;
    (ii) for N=1,2, proposition N implies the truth of Wary Weasel theorem N;
    (iii) for N>=3, the shortest possible statement of the Escape Clause grows as fast as a Nabutovsky-Weinberger number.

  6. Mustela nivalis Says:

    That should be NP-complete problems, of course.

  7. Peter Shor Says:

    Greg: While I believe that P = BPP, I’m much more wary about AM = NP. So your statement that everybody believes it has an exception.

    Scot: How about a weaker conjecture: BQP is in IP(log n). That is, interactive proofs with log n rounds of communication. Does this avoid even the potential recursive Fourier sampling counterexample?


  8. Scott Says:

    Peter: Yes, BQP in IP[log] avoids even the RFS counterexample, and I correspondingly attach a greater probability to its truth.

    Regarding AM=NP, I wouldn’t have believed it either before the Klivans-van Melkebeek paper. But I certainly believe there are languages in NE that require exponential-size nondeterministic circuits.

  9. Greg Kuperberg Says:

    So your statement that everybody believes it has an exception.

    Well, I am not sure that it was a particularly informed statement that NP = AM is so likely. But I can point to what Scott said.

    I have thought some about trying to prove that AM contains BQP. Given the examples such as IP = PSPACE, even a non-relativizing inclusion may not be as hard as P vs NP. But it seemed hopeless, although I concede that it deserves another try. In fact, the mechanics of showing that AM contains BQP was one of the motivations of the oracle that Scott and I propose to place BQP outside of AM (and do other things). Namely, as Scott said, we think that you can make a “limited randomness” quantum oracle from a random classical oracle, post-conditioned on the promise that it gives bounded-error answers. I think that such an oracle would be incomprehensible to an AM computer.

    But on the other hand, it is possible that such an oracle would be unrealistic for the same reason that the random oracle is unrealistic for the question of IP vs PSPACE.

  10. Peter Shor Says:

    I also tried to show that BQP was in IP(log n), but there didn’t seem to be any promising avenues of attack.

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    BQP related: you can play integer factorization online:

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