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BQP and the Polynomial Hierarchy∗ BQP and the Polynomial Hierarchy∗ Scott Aaronson† MIT ABSTRACT 1. INTRODUCTION The relationship between BQP and PH has been an open A central task of quantum computing theory is to un- problem since the earliest days of quantum computing. We derstand how BQP—Bounded-Error Quantum Polynomial- present evidence that quantum computers can solve prob- Time, the class of all problems feasible for a quantum computer— lems outside the entire polynomial hierarchy, by relating this fits in with classical complexity classes. In their original question to topics in circuit complexity, pseudorandomness, 1993 paper defining BQP, Bernstein and Vazirani [9] showed #P 1 and Fourier analysis. that BPP BQP P . Informally, this says that quan- ⊆ ⊆ First, we show that there exists an oracle relation problem tum computers are at least as fast as classical probabilistic (i.e., a problem with many valid outputs) that is solvable in computers and no more than exponentially faster (indeed, BQP, but not in PH. This also yields a non-oracle relation they can be simulated using an oracle for counting). Bern- problem that is solvable in quantum logarithmic time, but stein and Vazirani also gave evidence that BPP = BQP, 6 not in AC0. by exhibiting an oracle problem called Recursive Fourier Ω(log n) Second, we show that an oracle decision problem separat- Sampling that requires n queries on a classical com- ing BQP from PH would follow from the Generalized Linial- puter but only n queries on a quantum computer. The Nisan Conjecture, which we formulate here and which is evidence for the power of quantum computers became dra- likely of independent interest. The original Linial-Nisan matically stronger a year later, when Shor [23] (building on Conjecture (about pseudorandomness against constant-depth work of Simon [24]) showed that Factoring and Discrete circuits) was recently proved by Braverman, after being open Logarithm are in BQP. On the other hand, Bennett et al. for twenty years. [8] gave oracle evidence that NP BQP, and while no one regards such evidence as decisive,6⊂ today it seems extremely Categories and Subject Descriptors unlikely that quantum computers can solve NP-complete problems in polynomial time. A vast body of research, con- F.1.3 [Theory of Computation]: Computation by Ab- tinuing to the present, has sought to map out the detailed stract Devices—Complexity Measures and Classes boundary between those NP problems that are feasible for quantum computers and those that are not. General Terms However, there is a complementary question that—despite being universally recognized as one of the “grand challenges” Theory of the field—has had essentially zero progress over the last sixteen years: Keywords BQP NP BQP constant-depth circuits, Fourier analysis, Linial-Nisan Con- Is in ? More generally, is con- tained anywhere in the polynomial hierarchy PH = jecture, quantum complexity classes NP NP NPNP NPNP ? ∗We regret that, because of space limitations, we are un- ∪ ∪ ∪··· able to prove anything in this extended abstract. Readers The “default” conjecture is presumably BQP PH, since interested in the proofs should stop reading now and go to no one knows what a simulation of BQP in PH6⊂would look www.scottaaronson.com/papers/bqpph.pdf. like. Before this work, however, there was no formal evi- †Based upon work supported by the National Science Foun- dation under Grant No. 0844626. Also supported by a dence for or against that conjecture. Almost all the prob- DARPA YFA grant and the Keck Foundation. lems for which we have quantum algorithms—including Fac- toring and Discrete Logarithm—are easily seen to be in NP coNP.2 One notable exception is Recursive Fourier Sampling∩ , the problem that Bernstein and Vazirani [9] orig- inally used to construct an oracle A relative to which BPPA = 6 Permission to make digital or hard copies of all or part of this work for 1 BQP PP personal or classroom use is granted without fee provided that copies are The upper bound was later improved to by Adleman, DeMarrais, and Huang [2]. ⊆ not made or distributed for profit or commercial advantage and that copies 2 bear this notice and the full citation on the first page. To copy otherwise, to Here we exclude BQP-complete problems such as approxi- republish, to post on servers or to redistribute to lists, requires prior specific mating the Jones polynomial [3], which, by the very fact of permission and/or a fee. being BQP-complete, seem hard to interpret as “evidence” for BQP PH. Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$5.00. 6⊂ BQPA. One can show, without too much difficulty, that polynomial hierarchy collapses? At first glance, this seems Recursive Fourier Sampling yields oracles A relative to like a wild hope; certainly we have no idea at present how which BQPA NPA and indeed BQPA MAA. How- to prove anything of the kind. However, notice that if ever, while it6⊂ is reasonable to conjecture6⊂ that Recursive BQP AM, then the desired implication would follow imme- Fourier Sampling (as an oracle problem) is not in PH, it diately!⊆ For in that case, coNP BQP, hence coNP AM, P ⊆ ⊆ is open even to show that this problem (or any other BQP hence PH = Σ2 , where the last implication was shown by oracle problem) is not in AM! Recall that AM = NP un- Boppana, H˚astad, and Zachos [10]. der plausible derandomization assumptions. Thus, until we solve the problem of constructing an oracle A such that 1.2 Our Results BQPA AMA, we cannot even claim to have oracle evi- This paper presents the first formal evidence for the possi- 6⊂ dence (which is itself, of course, a weak form of evidence) bility that BQP PH. Perhaps more importantly, it places that BQP NP. the relativized BQP6⊂ versus PH question at the frontier of 6⊂ Before going further, we should clarify that there are two (classical) circuit lower bounds. The heart of the problem, questions here: whether BQP PH and whether PromiseBQP we will find, is to extend Braverman’s spectacular recent ⊆ ⊆ PromisePH. In the unrelativized world, it is entirely pos- proof [11] of the Linial-Nisan Conjecture, in ways that would sible that quantum computers can solve promise problems reveal a great deal of information about small-depth circuits outside the polynomial hierarchy, but that all languages in independent of the implications for quantum computing. BQP are nevertheless in PH. However, for the specific pur- We have two main contributions. First, we achieve an pose of constructing an oracle A such that BQPA PHA, oracle separation between BQP and PH for the case of re- 6⊂ the two questions are equivalent, basically because one can lation problems. A relation problem is simply a problem “offload” a promise into the construction of the oracle A. where the desired output is an n-bit string (rather than a single bit), and any string from some nonempty set S is 1.1 Motivation acceptable. Relation problems arise often in theoretical There are at least four reasons why the BQP versus PH computer science; one well-known example is finding a Nash question is so interesting. At a basic level, it is both theoret- equilibrium (shown to be PPAD-complete by Daskalakis et ically and practically important to understand what classical al. [13]). Within quantum computing, there is consider- resources are needed to simulate quantum physics. For ex- able precedent for studying relation problems as a warmup ample, when a quantum system evolves to a given state, is to the harder case of decision problems. For example, in there always a short classical proof that it does so? Can 2004 Bar-Yossef, Jayram, and Kerenidis [4] gave a relation one estimate quantum amplitudes using approximate count- problem with quantum one-way communication complexity ing (which would imply BQP BPPNP)? If something like O (log n) and randomized one-way communication complex- this were true, then while the⊆ exponential speedup of Shor’s ity Ω (√n). It took several more years for Gavinsky et al. factoring algorithm might stand, quantum computing would [15] to achieve the same separation for decision problems, nevertheless seem much less different from classical comput- and the proof was much more complicated.3 ing than previously thought. Formally, our result is as follows: Second, if BQP PH, then many possibilities for new quantum algorithms6⊂ might open up to us. One often hears Theorem 1. There exists an oracle A relative to which A PHA the complaint that there are too few quantum algorithms, FBQP FBPP , where FBQP and FBPP are the rela- 6⊂ 4 or that progress on quantum algorithms has slowed since the tion versions of BQP and BPP respectively. mid-1990s. In our opinion, the real issue here has nothing to do with quantum computing, and is simply that there are Interestingly, the oracle A in Theorem 1 can be taken to too few natural NP-intermediate problems for which there be a random oracle. By contrast, Aaronson and Ambainis plausibly could be quantum algorithms! In other words, in- [1] have shown that, under a plausible conjecture about in- stead of focussing on Graph Isomorphism and a small num- fluences in low-degree polynomials, one cannot separate the NP decision classes BQP and BPP by a random oracle without ber of other -intermediate problems, it might be fruitful #P to look for quantum algorithms solving completely different also separating P from P in the unrelativized world. In types of problems—problems that are not necessarily even other words, the use of relation problems (rather than deci- in PH. In this paper, we will see a new example of such a sion problems) seems essential if one wants a random oracle separation.
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