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arXiv:2205.04633 · Leibniz international proceedings in informatics

An Optimal Oracle Separation of Classical and Quantum Hybrid Schemes

Atsuya Hasegawa, François Le Gall

Published Jan 1, 2022• 4 citations
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Abstract

Recently, Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) have shown that there exists an oracle $\mathcal{O}$ such that $\mathsf{BQP}^\mathcal{O} \neq (\mathsf{BPP^{BQNC}})^\mathcal{O} \cup (\mathsf{BQNC^{BPP}})^\mathcal{O}$. In fact, Chia et al. proved a stronger statement: for any depth parameter $d$, there exists an oracle that separates quantum depth $d$ and $2d+1$, when polynomial-time classical computation is allowed. This implies that relative to an oracle, doubling quantum depth gives classical and quantum hybrid schemes more computational power. In this paper, we show that for any depth parameter $d$, there exists an oracle that separates quantum depth $d$ and $d+1$, when polynomial-time classical computation is allowed. This gives an optimal oracle separation of classical and quantum hybrid schemes. To prove our result, we consider $d$-Bijective Shuffling Simon's Problem (which is a variant of $d$-Shuffling Simon's Problem considered by Chia et al.) and an oracle inspired by an "in-place" permutation oracle.

DOI
10.4230/lipics.isaac.2022.6

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Recently, Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) have shown that there exists an oracle O\mathcal{O}O such that BQPO≠(BPPBQNC)O∪(BQNCBPP)O\mathsf{BQP}^\mathcal{O} \neq (\mathsf{BPP^{BQNC}})^\mathcal{O} \cup (\mathsf{BQNC^{BPP}})^\mathcal{O}BQPO=(BPPBQNC)O∪(BQNCBPP)O. In fact, Chia et al. proved a stronger statement: for any depth parameter ddd, there exists an oracle that separates quantum depth ddd and 2d+12d+12d+1, when polynomial-time classical computation is allowed. This implies that relative to an oracle, doubling quantum depth gives classical and quantum hybrid schemes more computational power. In this paper, we show that for any depth parameter ddd, there exists an oracle that separates quantum depth ddd and d+1d+1d+1, when polynomial-time classical computation is allowed. This gives an optimal oracle separation of classical and quantum hybrid schemes. To prove our result, we consider ddd-Bijective Shuffling Simon's Problem (which is a variant of ddd-Shuffling Simon's Problem considered by Chia et al.) and an oracle inspired by an "in-place" permutation oracle.
An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models, that combine poly-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a d-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a d-depth quantum computer with the ability to change the sequence of gates being applied depending on measurement outcomes processed by a classical computation. Chia, Chung & Lai (STOC 2020) and Coudron & Menda (STOC 2020) showed that these models (with d=log^O(1) (n)) are strictly weaker than BQP (the class of problems solvable by poly-time quantum computation), relative to an oracle, disproving a conjecture of Jozsa in the relativised world. We show that, despite the similarities between CQ_d and QC_d, the two models are incomparable, i.e. CQ_d ⊈\nsubseteq⊈ QC_d and QC_d ⊈\nsubseteq⊈ CQ_d relative to an oracle. In other words, there exist problems that one model can solve but not the other and vice versa. We do this by considering new oracle problems that capture the distinctions between the two models and by introducing the notion of an intrinsically stochastic oracle, an oracle whose responses are inherently randomised, which is used for our second result. While we leave showing the second separation relative to a standard oracle as an open problem, we believe the notion of stochastic oracles could be of independent interest for studying complexity classes which have resisted separation in the standard oracle model. Our constructions also yield simpler oracle separations between the hybrid models and BQP, compared to earlier works.
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.
Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “ Any quantum polynomial-time algorithm can be implemented with only O (log n ) quantum depth interspersed with polynomial-time classical computations. ” This can be formalized as asserting the equivalence of BQP and “ BQNC BPP .” However, Aaronson conjectured that “ there exists an oracle separation between BQP and BPP BQNC . ” BQNC BPP and BPP BQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d , there exists an oracle that separates quantum depth d and 2 d +1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.
The 222-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating BQP\mathsf{BQP}BQP and PH\mathsf{PH}PH relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that 222-Forrelation can be solved using Instantaneous Quantum Polynomial-time (IQP\mathsf{IQP}IQP) circuits, a restricted model of quantum computation in which all gates commute. Concretely, signed 222-Forrelation can be solved by a classical random choice between two one-query IQP\mathsf{IQP}IQP circuits, while the absolute-value variant uses two independent executions of this randomized procedure. This answers a recent open question of Girish (STOC 2026) on the power of commuting quantum computations. For the Raz-Tal distribution, this randomization is unnecessary. We use this to show that there is an oracle OOO such that IQPO⊈PHO\mathsf{IQP}^O \not\subseteq \mathsf{PH}^OIQPO⊆PHO, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between IQP\mathsf{IQP}IQP and DQC1\mathsf{DQC}_1DQC1​. We prove Fourier growth bounds for multi-query IQP\mathsf{IQP}IQP circuits, including bounds in terms of the size of their accepting set. Our results suggest a possible route toward decision-based quantum advantage within the restricted IQP\mathsf{IQP}IQP model. The key ingredient is an algebraic identity of the quadratic function Q(x)=∑i<jxixjQ(x) = \sum_{i < j} x_ix_jQ(x)=∑i<j​xi​xj​ that allows extracting inner-product phases within an IQP\mathsf{IQP}IQP circuit.
In this paper, we introduce the hybrid query complexity, denoted as Q(f;q)\mathrm{Q}(f;q)Q(f;q), which is the minimal query number needed to compute fff, when a classical decision tree is allowed to call q′q'q′-query quantum subroutines for any q′≤qq'\leq qq′≤q. We present the following results: ∙\bullet∙ There exists a total Boolean function fff such that Q(f;1)=O~(R(f)4/5)\mathrm{Q}(f;1) = \widetilde{\mathcal{O}}(\mathrm{R}(f)^{4/5})Q(f;1)=O(R(f)4/5). ∙\bullet∙ Q(f;q)=Ω(bs(f)/q+bs(f))\mathrm{Q}(f;q) = Ω(\mathrm{bs}(f)/q + \sqrt{\mathrm{bs}(f)})Q(f;q)=Ω(bs(f)/q+bs(f)​) for any Boolean function fff; the lower bound is tight when fff is the OR{\rm O{\small R}}OR function. ∙\bullet∙ Q(g∘XORClog⁡n;1)=Ω~(n)\mathrm{Q}(g \circ {\rm X{\small OR}}_{C \log n};1) = \widetildeΩ(\sqrt{n})Q(g∘XORClogn​;1)=Ω(n​) for some sufficiently large constant CCC, where g:=BOOLSIMONng := {\rm B{\small OOL}S{\small IMON}}_ng:=BOOLSIMONn​ is a variant of Simon's problem. Note that Q(g∘XORClog⁡n)=O(polylog  n)\mathrm{Q}(g\circ {\rm X{\small OR}}_{C \log n}) = \mathcal{O}(\mathrm{polylog}\; n)Q(g∘XORClogn​)=O(polylogn). Therefore an exponential separation is established. Furthermore, this open the road to prove the conjecture ∀k, Q(g∘XORClog⁡k+1n;log⁡kn)=Ω~(n)\forall k,\,\mathrm{Q}(g \circ {\rm X{\small OR}}_{C \log^{k+1} n};\log^{k} n) = \widetildeΩ(\sqrt{n})∀k,Q(g∘XORClogk+1n​;logkn)=Ω(n​), which would imply the oracle separation HP(QSIZE(nα))O⊊BQPO\mathsf{HP}(\mathsf{QSIZE}(n^α))^\mathfrak{O} \subsetneq \mathsf{BQP}^\mathfrak{O}HP(QSIZE(nα))O⊊BQPO for any ααα, where HP(QSIZE(nα))\mathsf{HP}(\mathsf{QSIZE}(n^α))HP(QSIZE(nα)) is a complexity class that contains BQTIME(nα)BPP\mathsf{BQTIME}(n^α)^{\mathsf{BPP}}BQTIME(nα)BPP and BPPBQTIME(nα)\mathsf{BPP}^{\mathsf{BQTIME}(n^α)}BPPBQTIME(nα) in any relativized world.
We study the quantum-classical polynomial hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers followed by a quantum verifier. Our main result is that QCPH is infinite relative to a random oracle (previously, this was not even known relative to any oracle). We further prove that higher levels of PH are not contained in lower levels of QCPH relative to a random oracle; this is a strengthening of the somewhat recent result that PH is infinite relative to a random oracle (Rossman, Servedio, and Tan 2016). The oracle separation requires lower bounding a certain type of low-depth alternating circuit with some quantum gates. To establish this, we give a new switching lemma for quantum algorithms which may be of independent interest. Our lemma says that for any ddd, if we apply a random restriction to a function fff with quantum query complexity Q(f)≤n1/3\mathrm{Q}(f)\le n^{1/3}Q(f)≤n1/3, the restricted function becomes exponentially close (in terms of ddd) to a depth-ddd decision tree. Our switching lemma works even in a "worst-case" sense, in that only the indices to be restricted are random; the values they are restricted to are chosen adversarially. Moreover, the switching lemma also works for polynomial degree in place of quantum query complexity.