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arXiv:2609.29497

Quantum Query Advantage Requires Space

Minbo Gao, Zhengfeng Ji, Ziyi Xie

quant-phcs.CCSubmitted Sep 4, 2026• Announced Sep 7, 2026• 0 citations• Revised Oct 2, 2026 (v2)
v1 · Sep 4, 2026v2 · Oct 2, 2026
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Abstract

Hao, Huang, and Liu (STOC'26) recently showed that optimal quantum query complexity may require large workspace even for short-output problems, and asked whether a quantum query advantage over classical computation can itself require space. We resolve this question by exhibiting an explicit total Boolean function with quantum query complexity $Q=\widetildeΘ(M)$, and randomized query complexity $R=Θ(M^{21/20})$, whereas, for every fixed $0<η<1/100$ and $S\le O(M^{1/100-η})$, its $S$-space quantum query complexity satisfies $Q_S=ω(M^{21/20})$. Consequently, $$ Q<R<Q_S, $$ so the unrestricted quantum query advantage disappears under sufficiently small workspace. The separation is obtained through a one-bit filtered-parity construction and a space-sensitive quantum lower bound based on compressed-oracle capacity and a new parity-to-capacity inequality, which may be of independent interest.

Comments
21 pages, update introduction and fix grammar mistakes

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Hao, Huang, and Liu (STOC'26) recently showed that optimal quantum query complexity may require large workspace even for short-output problems, and asked whether a quantum query advantage over classical computation can itself require space. We resolve this question by exhibiting an explicit total Boolean function with quantum query complexity Q=Θ~(M)Q=\widetildeΘ(M)Q=Θ(M), and randomized query complexity R=Θ(M21/20)R=Θ(M^{21/20})R=Θ(M21/20), whereas, for every fixed 0<η<1/1000<η<1/1000<η<1/100 and S≤O(M1/100−η)S\le O(M^{1/100-η})S≤O(M1/100−η), its SSS-space quantum query complexity satisfies QS=ω(M21/20)Q_S=ω(M^{21/20})QS​=ω(M21/20). Consequently, Q<R<QS,Q<R<Q_S,Q<R<QS​, so the unrestricted quantum query advantage disappears under sufficiently small workspace. The separation is obtained through a one-bit filtered-parity construction and a space-sensitive quantum lower bound based on compressed-oracle capacity and a new parity-to-capacity inequality, which may be of independent interest.
It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these measures are possible. In particular, 1) We employ the cheatsheet framework to obtain an unbounded parallel quantum query advantage over its randomized analogue for a total function, falsifying a conjecture of [https://arxiv.org/abs/1309.6116]. 2) We strengthen 1 by constructing a total function which exhibits an unbounded parallel quantum query advantage despite having no sequential advantage, suggesting that genuine quantum advantage could occur entirely due to parallelism. 3) We construct a total function that exhibits a polynomial separation between 2-round quantum and randomized query complexities, contrasting a result of [https://arxiv.org/abs/1001.0018] that there is at most a constant separation for 1-round (nonadaptive) algorithms. 4) We develop a new technique for deriving parallel quantum lower bounds from sequential upper bounds. We employ this technique to give lower bounds for Boolean symmetric functions and read-once formulas, ruling out large parallel query advantages for them. We also provide separations between randomized and deterministic parallel query complexities analogous to items 1-3.
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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.
Branching programs are quite popular for studying time-space lower bounds. Bera et al. recently introduced the model of generalized quantum branching program aka. GQBP that generalized two earlier models of quantum branching programs. In this work we study a restricted version of GQBP with the motivation of proving bounds on the query-space requirement of quantum-query circuits. We show the first explicit query-space lower bound for our restricted version. We prove that the well-studied ORn_nn​ decision problem, given a promise that at most one position of an nnn-sized Boolean array is a 1, satisfies the bound Q2s=Ω(n2)Q^2 s = Ω(n^2)Q2s=Ω(n2), where QQQ denotes the number of queries and sss denotes the width of the GQBP. We then generalize the problem to show that the same bound holds for deciding between two strings with a constant Hamming distance; this gives us query-space lower bounds on problems such as Parity and Majority. Our results produce an alternative proof of the Ω(n)Ω(\sqrt{n})Ω(n​)-lower bound on the query complexity of any non-constant symmetric Boolean function.
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting Q(f∘g)=Θ(Q(f)Q(g))Q(f\circ g)=Θ(Q(f)Q(g))Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions fff and ggg; a direct sum theorem asserting that computing kkk copies of a function (or search problem) costs Ω(k)Ω(k)Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires Ω(k)Ω(k)Ω(k) times the cost of computing one copy to bounded error. However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution μμμ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing kkk copies of fff when the input comes from the product distribution μkμ^kμk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions. In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the γ2γ_2γ2​ norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.