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arXiv:2007.03402 · Leibniz international proceedings in informatics

Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language

Andris Ambainis, Kaspars Balodis, Jānis Iraids, Kamil Ravilevich Khadiev, Vladislavs Kļevickis, Krišjānis Prūsis, Yixin Shen, Juris Smotrovs, Jevgēnijs Vihrovs

Published Jan 1, 2020• 20 citations
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Abstract

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 $k$. We call this the $Dyck_{k,n}$ problem. We prove a lower bound of $Ω(c^k \sqrt{n})$, showing that the complexity of this problem increases exponentially in $k$. Here $n$ is the length of the word. When $k$ is a constant, this is interesting as a representative example of star-free languages for which a surprising $\tilde{O}(\sqrt{n})$ query quantum algorithm was recently constructed by Aaronson et al. Their proof does not give rise to a general algorithm. When $k$ is not a constant, $Dyck_{k,n}$ is not context-free. We give an algorithm with $O\left(\sqrt{n}(\log{n})^{0.5k}\right)$ quantum queries for $Dyck_{k,n}$ for all $k$. This is better than the trival upper bound $n$ for $k=o\left(\frac{\log(n)}{\log\log n}\right)$. 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 $Ω(n^{1.5-ε})$ for the directed 2D grid and $Ω(n^{2-ε})$ 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.

DOI
10.4230/lipics.mfcs.2020.8

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Quantum Lower Bounds for 2D-Grid and Dyck Language

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We show quantum lower bounds for 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 $k$. It has been known that, for any $k$, $\tilde{O}(\sqrt{n})$ queries suffice, with a $\tilde{O}$ term depending on $k$. We prove a lower bound of $Ω(c^k \sqrt{n})$, showing that the complexity of this problem increases exponentially in $k$. This is interesting as a representative example of star-free languages for which a surprising $\tilde{O}(\sqrt{n})$ query quantum algorithm was recently constructed by Aaronson et al. Second, we consider connectivity problems on directed/undirected grid 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 $Ω(n^{1.5-ε})$ for the directed 2D grid and $Ω(n^{2-ε})$ 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.

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We consider the problem of determining if a sequence of parentheses is well parenthesized, with a depth of at most h. We denote this language as $Dyck_h$. We study the quantum query complexity of this problem for different h as function of the length n of the word. It has been known from a recent paper by Aaronson et al. that, for any constant h, since $Dyck_h$ is star-free, it has quantum query complexity $\tildeΘ(\sqrt{n})$, where the hidden logarithm factors in $\tildeΘ$ depend on h. Their proof does not give rise to an algorithm. When h is not a constant, $Dyck_h$ is not even context-free. We give an algorithm with $O\left(\sqrt{n}\log(n)^{0.5h}\right)$ quantum queries for $Dyck_h$ for all h. This is better than the trival upper bound $n$ when $h=o(\frac{\log(n)}{\log\log n})$. We also obtain lower bounds: we show that for every $00$ such that $Q(\text{Dyck}_{c\log(n)}(n))=Ω(n^{1-ε})$. When $h=ω(\log(n))$, the quantum query complexity is close to $n$, i.e. $Q(\text{Dyck}_h(n))=ω(n^{1-ε})$ for all $ε>0$. Furthermore when $h=Ω(n^ε)$ for some $ε>0$, $Q(\text{Dyck}_{h}(n))=Θ(n)$.

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A Quantum Query Complexity Trichotomy for Regular Languages

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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.
We show quantum lower bounds for 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. It has been known that, for any kkk, O~(n)\tilde{O}(\sqrt{n})O~(n​) queries suffice, with a O~\tilde{O}O~ term depending on kkk. 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. 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. Second, we consider connectivity problems on directed/undirected grid 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.
We consider the problem of determining if a sequence of parentheses is well parenthesized, with a depth of at most h. We denote this language as DyckhDyck_hDyckh​. We study the quantum query complexity of this problem for different h as function of the length n of the word. It has been known from a recent paper by Aaronson et al. that, for any constant h, since DyckhDyck_hDyckh​ is star-free, it has quantum query complexity Θ~(n)\tildeΘ(\sqrt{n})Θ~(n​), where the hidden logarithm factors in Θ~\tildeΘΘ~ depend on h. Their proof does not give rise to an algorithm. When h is not a constant, DyckhDyck_hDyckh​ is not even context-free. We give an algorithm with O(nlog⁡(n)0.5h)O\left(\sqrt{n}\log(n)^{0.5h}\right)O(n​log(n)0.5h) quantum queries for DyckhDyck_hDyckh​ for all h. This is better than the trival upper bound nnn when h=o(log⁡(n)log⁡log⁡n)h=o(\frac{\log(n)}{\log\log n})h=o(loglognlog(n)​). We also obtain lower bounds: we show that for every 000000 such that Q(Dyckclog⁡(n)(n))=Ω(n1−ε)Q(\text{Dyck}_{c\log(n)}(n))=Ω(n^{1-ε})Q(Dyckclog(n)​(n))=Ω(n1−ε). When h=ω(log⁡(n))h=ω(\log(n))h=ω(log(n)), the quantum query complexity is close to nnn, i.e. Q(Dyckh(n))=ω(n1−ε)Q(\text{Dyck}_h(n))=ω(n^{1-ε})Q(Dyckh​(n))=ω(n1−ε) for all ε>0ε>0ε>0. Furthermore when h=Ω(nε)h=Ω(n^ε)h=Ω(nε) for some ε>0ε>0ε>0, Q(Dyckh(n))=Θ(n)Q(\text{Dyck}_{h}(n))=Θ(n)Q(Dyckh​(n))=Θ(n).
The Tarski fixed-point problem asks for a fixed point of a monotone map on a kkk-dimensional grid lattice of width nnn and has applications in game theory, economics, and verification. Substantial gaps remain between the known upper and lower bounds on its quantum query complexity. We prove two hardness bootstrapping theorems for quantum query lower bounds. Starting from a hard instance family whose instances have pair-wise incomparable unique fixed points, and a lower bound established by the nonnegative adversary method, we bootstrap this lower bound by (i) a factor of Ω(t)Ω(t)Ω(t) when the dimension is multiplied by ttt and the width remains unchanged, and (ii) a factor of Ω(log⁡m)Ω(\log m)Ω(logm) when the width is multiplied by mmm and the dimension is doubled. Applying these theorems, we obtain three consequences: (i) an Ω(klog⁡2n)Ω(k\log^2 n)Ω(klog2n) lower bound for k≥4k\geq 4k≥4, improving previously best-known classical and quantum lower bounds; (ii) for every fixed constant α>0α>0α>0, there exists a constant δ(α)>0δ(α)>0δ(α)>0 such that Qε(Tarski(n,k))=Ω(knδ(α))Q_ε(\text{Tarski}(n,k))=Ω(kn^{δ(α)})Qε​(Tarski(n,k))=Ω(knδ(α)) whenever k>nαk>n^αk>nα, and in particular, an Ω(kn0.0072)Ω(kn^{0.0072})Ω(kn0.0072) lower bound for k>n2/203k>n^{2/203}k>n2/203, the first lower bound with simultaneous linear dependence on the dimension and polynomial dependence on the width; and (iii) a superpolynomial lower bound of (k/log⁡k)Ω(log⁡k)(k/\log k)^{Ω(\log k)}(k/logk)Ω(logk) for exponentially wide grids, where n=2Θ(k)n=2^{Θ(k)}n=2Θ(k). The last result rules out a route to polynomial-time algorithms for simple stochastic games and ARRIVAL through generic polynomial-query black-box Tarski solvers.
We study the quantum query complexity of several basic graph problems. 1) We prove a tight Θ(n3/2)Θ(n^{3/2})Θ(n3/2) bound for finding a maximum-out-degree vertex in an nnn-vertex tournament, resolving a conjecture of Mande, Paraashar, and Saurabh [FSTTCS'23]. 2) Using the same proof idea, we give a direct proof that finding a maximum-degree vertex in an nnn-vertex undirected graph requires Θ(n3/2)Θ(n^{3/2})Θ(n3/2) quantum queries, a bound implicit in the work of Apers and Lee [CCC'21]. Both lower bounds above hold even for computing only the maximum degree or maximum out-degree. 3) We also study Maximum (k,n−k)(k,n-k)(k,n−k)-Cut, which asks for the largest cut with one side of size kkk. For every constant k≥1k\geq 1k≥1, we give an O(n2−1/(k+1))O(n^{2-1/(k+1)})O(n2−1/(k+1))-query quantum algorithm. 4) We establish a reduction to Maximum (k,n−k)(k,n-k)(k,n−k)-Cut involving kkk-Graph Collision, which asks whether the marked vertices of a known graph contain a kkk-clique. For k≥3k\geq 3k≥3, this yields unconditional lower bounds with exponents strictly larger than 3/23/23/2, showing that this problem is strictly harder than computing maximum degree. Additionally, under a conjecture we propose for kkk-Graph Collision, the reduction yields a lower bound matching our upper bound above.
We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity Θ(1), Θ̃(√ n), or Θ(n). The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we show can have query complexity Θ(nc) for all computable c [1/2,1]. Our result implies an equivalent trichotomy for the approximate degree of regular languages, and a dichotomy-either Θ(1) or Θ(n)-for sensitivity, block sensitivity, certificate complexity, deterministic query complexity, and randomized query complexity. The heart of the classification theorem is an explicit quantum algorithm which decides membership in any star-free language in Õ(√n) time. This well-studied family of the regular languages admits many interesting characterizations, for instance, as those languages expressible as sentences in first-order logic over the natural numbers with the less-than relation. Therefore, not only do the star-free languages capture functions such as OR, they can also express functions such as "there exist a pair of 2's such that everything between them is a 0." Thus, we view the algorithm for star-free languages as a nontrivial generalization of Grover's algorithm which extends the quantum quadratic speedup to a much wider range of string-processing algorithms than was previously known. We show a variety of applications-new quantum algorithms for dynamic constant-depth Boolean formulas, balanced parentheses nested constantly many levels deep, binary addition, a restricted word break problem, and path-discovery in narrow grids-all obtained as immediate consequences of our classification theorem.
We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity Theta(1), ~Theta(sqrt n), or Theta(n). The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we show can have query complexity Theta(n^c) for all computable c in [1/2,1]. Our result implies an equivalent trichotomy for the approximate degree of regular languages, and a dichotomy---either Theta(1) or Theta(n)---for sensitivity, block sensitivity, certificate complexity, deterministic query complexity, and randomized query complexity. The heart of the classification theorem is an explicit quantum algorithm which decides membership in any star-free language in ~O(sqrt n) time. This well-studied family of the regular languages admits many interesting characterizations, for instance, as those languages expressible as sentences in first-order logic over the natural numbers with the less-than relation. Therefore, not only do the star-free languages capture functions such as OR, they can also express functions such as "there exist a pair of 2's such that everything between them is a 0." Thus, we view the algorithm for star-free languages as a nontrivial generalization of Grover's algorithm which extends the quantum quadratic speedup to a much wider range of string-processing algorithms than was previously known. We show a variety of applications---new quantum algorithms for dynamic constant-depth Boolean formulas, balanced parentheses nested constantly many levels deep, binary addition, a restricted word break problem, and path-discovery in narrow grids---all obtained as immediate consequences of our classification theorem.