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arXiv:2407.06872

Quantum Query-Space Lower Bounds Using Branching Programs

Debajyoti Bera, Tharrmashastha Sapv

Published Jul 9, 2024• 0 citations
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Abstract

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 OR$_n$ decision problem, given a promise that at most one position of an $n$-sized Boolean array is a 1, satisfies the bound $Q^2 s = Ω(n^2)$, where $Q$ denotes the number of queries and $s$ 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 $Ω(\sqrt{n})$-lower bound on the query complexity of any non-constant symmetric Boolean function.

DOI
10.48550/arxiv.2407.06872

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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.
The Gap-Hamming distance problem is the promise problem of deciding if the Hamming distance hhh between two strings of length nnn is greater than aaa or less than bbb, where the gap g=∣a−b∣≥1g=|a-b|\geq 1g=∣a−b∣≥1 and aaa and bbb could depend on nnn. In this short note, we give a lower bound of Ω(n/g)Ω( \sqrt{n/g})Ω(n/g​) on the quantum query complexity of computing the Gap-Hamming distance between two given strings of lenght nnn. The proof is a combinatorial argument based on block sensitivity and a reduction from a threshold function.
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.
The weight decision problem, which requires to determine the Hamming weight of a given binary string, is a natural and important problem, with applications in cryptanalysis, coding theory, fault-tolerant circuit design and so on. In particular, both Deutsch-Jozsa problem and Grover search problem can be interpreted as special cases of weight decision problems. In this work, we investigate the exact quantum query complexity of weight decision problems, where the quantum algorithm must always output the correct answer. More specifically we consider a partial Boolean function which distinguishes whether the Hamming weight of the length-nnn input is kkk or it is lll. Our contribution includes both upper bounds and lower bounds for the precise number of queries. Furthermore, for most choices of (kn,ln)(\frac{k}{n},\frac{l}{n})(nk​,nl​) and sufficiently large nnn, the gap between our upper and lower bounds is no more than one. To get the results, we first build the connection between Chebyshev polynomials and our problem, then determine all the boundary cases of (kn,ln)(\frac{k}{n},\frac{l}{n})(nk​,nl​) with matching upper and lower bounds, and finally we generalize to other cases via a new \emph{quantum padding} technique. This quantum padding technique can be of independent interest in designing other quantum algorithms.
One of the main reasons for query model's prominence in quantum complexity is the presence of concrete lower bounding techniques: polynomial method and adversary method. There have been considerable efforts to not just give lower bounds using these methods but even to compare and relate them. We explore the value of these bounds on quantum query complexity for the class of symmetric functions, arguably one of the most natural and basic set of Boolean functions. We show (using the recently introduced spectral sensitivity) that both these bounds (positive adversary and approximate degree) give the same value for every total symmetric Boolean function. We also look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. In addition, we study how large certificate complexity and block sensitivity can be as compared to sensitivity (even up to constant factors) for symmetric functions. We show tight separations, i.e., give upper bound on possible separations and construct functions achieving the same.
We prove the first non-trivial upper bound for the quantum query complexity of Boolean Matrix Product Verification (BMPV\mathsf{BMPV}BMPV), answering a longstanding open question in quantum query complexity. For n×nn\times nn×n matrices, our upper bound is O~(n17/12)\widetilde O(n^{17/12})O(n17/12), improving on the standard O(n3/2)O(n^{3/2})O(n3/2) bound obtained using Grover search by Buhrman and Špalek [SODA 2006]. We complement this result by showing an Ω(n5/4)Ω(n^{5/4})Ω(n5/4) lower bound, which improves over the previous best known lower bound of Ω~(n19/18)\widetildeΩ(n^{19/18})Ω(n19/18) by Childs, Kimmel, and Kothari [ESA 2012]. Our approach centers on a connection with Orthogonal Vectors (OV\mathsf{OV}OV), which asks whether an indexed list of nnn Boolean vectors of dimension nnn contains two vectors with disjoint supports. In particular, we prove equivalences between OV\mathsf{OV}OV and BMPV\mathsf{BMPV}BMPV and establish the above bounds for OV\mathsf{OV}OV. We also prove a tight Θ~(n3/2)\widetilde Θ(n^{3/2})Θ(n3/2) bound for a variant of BMPV\mathsf{BMPV}BMPV that asks whether the product contains a given row vector. Together, these results imply a polynomial separation between the quantum query complexities of deciding whether a graph has radius at most two and whether it has diameter at most two.
We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most kkk. We call this the Dyckk,nDyck_{k,n}Dyckk,n​ problem. We prove a lower bound of Ω(ckn)Ω(c^k \sqrt{n})Ω(ckn​), showing that the complexity of this problem increases exponentially in kkk. Here nnn is the length of the word. When kkk is a constant, this is interesting as a representative example of star-free languages for which a surprising O~(n)\tilde{O}(\sqrt{n})O~(n​) query quantum algorithm was recently constructed by Aaronson et al. Their proof does not give rise to a general algorithm. When kkk is not a constant, Dyckk,nDyck_{k,n}Dyckk,n​ is not context-free. We give an algorithm with O(n(log⁡n)0.5k)O\left(\sqrt{n}(\log{n})^{0.5k}\right)O(n​(logn)0.5k) quantum queries for Dyckk,nDyck_{k,n}Dyckk,n​ for all kkk. This is better than the trival upper bound nnn for k=o(log⁡(n)log⁡log⁡n)k=o\left(\frac{\log(n)}{\log\log n}\right)k=o(loglognlog(n)​). Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of Ω(n1.5−ε)Ω(n^{1.5-ε})Ω(n1.5−ε) for the directed 2D grid and Ω(n2−ε)Ω(n^{2-ε})Ω(n2−ε) for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.