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arXiv:1605.00619

A Note on Oracle Separations for BQP

Lijie Chen

Published May 2, 2016• 8 citations
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Abstract

In 2009, using the $\textsf{Fourier Checking}$ problem, Aaronson claimed to construct the relativized worlds such that $\textsf{BQP} \not\subset \mathsf{BPP_{path}}$ and $\textsf{BQP} \not\subset \textsf{SZK}$. However, there are subtle errors in the original proof. In this paper, we point out the issues, and rescue these two separations by using more sophisticated constructions. Meanwhile, we take the opportunity to study the complexity classes $\mathsf{BPP_{path}}$ and $\textsf{SZK}$. We give general ways to construct functions which are hard for $\textsf{SZK}$ and $\mathsf{BPP_{path}}$ (in the query complexity sense). Using these techniques, we give alternative construction for the oracle separation $\textsf{BQP} \not\subset \textsf{SZK}$, using only Simon's problem. We also give new oracle separations for $\textsf{P}^{\textsf{SZK}}$ from $\mathsf{BPP_{path}}$ and $\textsf{P}^{\textsf{SZK}}$ from $\textsf{QSK}$. The latter result suggests that $\textsf{P}^{\textsf{SZK}}$ might be strictly larger than $\textsf{SZK}$.

DOI
10.48550/arxiv.1605.00619

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In 2009, using the Fourier Checking\textsf{Fourier Checking}Fourier Checking problem, Aaronson claimed to construct the relativized worlds such that BQP⊄BPPpath\textsf{BQP} \not\subset \mathsf{BPP_{path}}BQP⊂BPPpath​ and BQP⊄SZK\textsf{BQP} \not\subset \textsf{SZK}BQP⊂SZK. However, there are subtle errors in the original proof. In this paper, we point out the issues, and rescue these two separations by using more sophisticated constructions. Meanwhile, we take the opportunity to study the complexity classes BPPpath\mathsf{BPP_{path}}BPPpath​ and SZK\textsf{SZK}SZK. We give general ways to construct functions which are hard for SZK\textsf{SZK}SZK and BPPpath\mathsf{BPP_{path}}BPPpath​ (in the query complexity sense). Using these techniques, we give alternative construction for the oracle separation BQP⊄SZK\textsf{BQP} \not\subset \textsf{SZK}BQP⊂SZK, using only Simon's problem. We also give new oracle separations for PSZK\textsf{P}^{\textsf{SZK}}PSZK from BPPpath\mathsf{BPP_{path}}BPPpath​ and PSZK\textsf{P}^{\textsf{SZK}}PSZK from QSK\textsf{QSK}QSK. The latter result suggests that PSZK\textsf{P}^{\textsf{SZK}}PSZK might be strictly larger than SZK\textsf{SZK}SZK.
BQP⊆IP\mathsf{BQP} \subseteq \mathsf{IP}BQP⊆IP Does Not Relativize
We construct an oracle relative to which BQP⊈IP\mathsf{BQP} \not\subseteq \mathsf{IP}BQP⊆IP, resolving a long-standing open question in quantum complexity theory. Together with recent work due to Aaronson et al., our work also gives the first oracle separation between IP\mathsf{IP}IP and MIP\mathsf{MIP}MIP, answering a question dating back to Fortnow's thesis. Our separation is based on the Forrelation problem, where given Boolean functions fff and ggg, the goal is to determine if fff is correlated with the Fourier spectrum of ggg. While this task is solvable by a query-efficient quantum algorithm, we show that it admits no classical interactive protocol with polynomial communication and a polynomial-query verifier. Our proof is based on (i) a new structural result showing how to approximate Avg-Max circuits (which are well-known to capture the power of interactive proofs in the oracular setting) by convex functions with small first and second derivatives and (ii) a novel analysis establishing that the Forrelation distribution suggested by Aaronson and Ambainis fools such functions. Our results imply that any prover-efficient classical interactive protocol for BQP\mathsf{BQP}BQP must rely on non-relativizing techniques. This might serve as a partial explanation for the lack of progress towards doubly-efficient, unconditionally sound classical verification of quantum computation.
The Fourier hierarchy FH0⊆FH1⊆FH2⊆⋯\mathrm{FH}_0\subseteq\mathrm{FH}_1\subseteq\mathrm{FH}_2\subseteq\cdotsFH0​⊆FH1​⊆FH2​⊆⋯, introduced by Shi (TCS 2005), measures a quantum computation by the number of Hadamard layers it uses. Between two layers the circuit may permute basis states and attach phases, but it may not create superposition; the layers are its only source of interference. The first level is exactly BPP\mathrm{BPP}BPP, while the second already solves Simon's problem and, through phase estimation, factors integers. Shi conjectured that every additional layer strictly increases computational power, and asked, as a first step, for oracle separations between consecutive levels. To our knowledge, the question was open at every level k≥2k\ge2k≥2. We prove that for every constant k≥2k\ge2k≥2 there is an oracle relative to which FHk⊊FHk+1\mathrm{FH}_k\subsetneq\mathrm{FH}_{k+1}FHk​⊊FHk+1​. The separating problem is built from Forrelation (Aaronson and Ambainis, STOC 2015): the level above solves it with a constant number of queries, whereas at level kkk it stays hard even for circuits making exponentially many queries. This holds for both of the usual ways of giving a circuit access to an oracle, the phase oracle and the standard oracle, which writes its answer into a register. The two are not interchangeable: relative to an oracle, the standard oracle is strictly more powerful at the same number of layers. We also separate the union of all the levels from BQP\mathrm{BQP}BQP relative to an oracle. The lower bounds rest on a structural property of the hierarchy: the number of Hadamard layers limits how adaptively a circuit can query its oracle. With a phase oracle, a circuit with kkk layers is reproduced exactly by an algorithm making only k−1k-1k−1 rounds of parallel queries, which brings known lower bounds for such algorithms to bear. The standard oracle lets a circuit branch on earlier answers, and that case needs a separate argument.
Complexity class containments involving interactive proof classes are famously nonrelativizing: although IP=PSPACE\mathsf{IP} = \mathsf{PSPACE}IP=PSPACE, Fortnow and Sipser showed that that there exists an oracle relative to which coNP⊈IP\mathsf{coNP} \not\subseteq \mathsf{IP}coNP⊆IP. In contrast, the question of whether the containment BQP⊆IP\mathsf{BQP} \subseteq \mathsf{IP}BQP⊆IP is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment BQP⊆MIP\mathsf{BQP} \subseteq \mathsf{MIP}BQP⊆MIP holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle OOO, a PCP\mathsf{PCP}PCP proof system for BQPO\mathsf{BQP}^{O}BQPO where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle OOO. Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an IP\mathsf{IP}IP for BQP\mathsf{BQP}BQP in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.
In recent years, the quantum oracle model introduced by Aaronson and Kuperberg (2007) has found a lot of use in showing oracle separations between complexity classes and cryptographic primitives. It is generally assumed that proof techniques that do not relativize with respect to quantum oracles will also not relativize with respect to classical oracles. Aaronson (2009) showed that this is not the case by showing a complexity class containment that relativizes with respect to classical oracles but not with quantum oracles. However, their result only works for zero-error quantum complexity classes and they leave open the problem for bounded-error complexity classes. We show that there is a quantum oracle problem that is contained in the class QMA, but not in a class we call polyQCPH. However, with respect to classical oracles, QMA is contained in polyQCPH, because polyQCPH is equal to PSPACE with respect to classical oracles. Our result works for polyQCPH, which is a bounded-error complexity class, thus it resolves the open problem from Aaronson (2009). We also show that the same separation holds relative to a distributional oracle, which is a model introduced by Natarajan and Nirkhe (2024). We believe our findings show the need for some caution when using these non-standard oracle models, particularly when showing separations between quantum and classical resources.
Recent works have demonstrated that quantum oracles have subtle behavior, as access to inverse, conjugate or controlled queries can exponentially change the query complexity of certain tasks. Inspired by these works, we introduce the notion of meta-complexity of quantum relativization. We ask: for any two quantum complexity classes, under which "types" of quantum oracles are they equal or separated? Different pantheons of oracles (or quantum oracle types, e.g. unitary vs state, poly- vs superpoly-dimensional, closed under inverse or not) form a partially ordered set based on their power in separating complexity classes. Moreover, two oracle pantheons A and B are separated if there exists a pair of complexity classes that are separated under an oracle from pantheon A but yet the complexity classes are equivalent under all oracles from pantheon B. We show that this meta-complexity can be nontrivial by giving a complete classification, within the family of oracle pantheons defined in this paper, of which models can separate the complexity classes PostBQP\mathsf{PostBQP}PostBQP and PreciseBQP\mathsf{PreciseBQP}PreciseBQP, the exponentially precise variant of BQP\mathsf{BQP}BQP. Both classes equal PP\mathsf{PP}PP in the unrelativized setting. Within our taxonomy, they remain equal relative to real or polynomial-dimensional unitary oracles and whenever inverse or conjugate access is supplied. In contrast, we give a separation relative to forward-only complex diagonal unitaries of superpolynomial dimension, as well as a separation relative to single-qubit state-preparation oracles. We view this as a test case for the meta-complexity of oracles which underscores the subtlety inherent to the relativization of quantum complexity classes.