Abstract Lexicographically minimal string rotation (LMSR) is a problem to find the minimal one among all rotations of a string in the lexicographical order, which is widely used in equality checking of graphs, polygons, automata and chemical structures. In this paper, we propose an $$O(n^{3/4})$$ O(n3/4) quantum query algorithm for LMSR. In particular, the algorithm has average-case query complexity $$O(\sqrt{n} \log n)$$ O(nlogn) , which is shown to be asymptotically optimal up to a polylogarithmic factor, compared to its $$\Omega \left( \sqrt{n/\log n}\right) $$ Ωn/logn lower bound. Furthermore, we show that our quantum algorithm outperforms any (classical) randomized algorithms in both worst and average cases. As an application, it is used in benzenoid identification and disjoint-cycle automata minimization.
Abstract Lexicographically minimal string rotation (LMSR) is a problem to find the minimal one among all rotations of a string in the lexicographical order, which is widely used in equality checking of graphs, polygons, automata and chemical structures. In this paper, we propose an O(n3/4) O(n3/4) quantum query algorithm for LMSR. In particular, the algorithm has average-case query complexity O(nlogn) O(nlogn) , which is shown to be asymptotically optimal up to a polylogarithmic factor, compared to its Ω(n/logn) Ωn/logn lower bound. Furthermore, we show that our quantum algorithm outperforms any (classical) randomized algorithms in both worst and average cases. As an application, it is used in benzenoid identification and disjoint-cycle automata minimization.
Lexicographically minimal string rotation is a fundamental problem in string processing that has recently garnered significant attention in quantum computing. Near-optimal quantum algorithms have been proposed for solving this problem, utilizing a divide-and-conquer structure. In this note, we show that its quantum query complexity is n ⋅ 2 O ( log n ) , improving the prior result of n ⋅ 2 ( log n ) 1 / 2 + ε by Akmal and Jin (2022). Notably, this improvement is quasi-polylogarithmic, which is achieved by only logarithmic level-wise optimization using fault-tolerant quantum minimum finding.
We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: Longest Common Substring can be solved by a quantum algorithm in Õ(n2/3) time, improving upon the recent Õ(n5/6)-time algorithm by Le Gall and Seddighin (2020). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, 2019) and generalized difference covers. Lexicographically Minimal String Rotation can be solved by a quantum algorithm in n1/2 + o(1) time, improving upon the recent Õ(n3/4)-time algorithm by Wang and Ying (2020). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding. Longest Square Substring can be solved by a quantum algorithm in time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2020) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies. Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix.
We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: - Longest Common Substring can be solved by a quantum algorithm in O~(n2/3) time, improving upon the recent O~(n5/6)-time algorithm by Le Gall and Seddighin (2020). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, 2019) and generalized difference covers. - Lexicographically Minimal String Rotation can be solved by a quantum algorithm in n1/2+o(1) time, improving upon the recent O~(n3/4)-time algorithm by Wang and Ying (2020). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding. - Longest Square Substring can be solved by a quantum algorithm in O~(n) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2020) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies. Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix.
Abstract We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: Longest Common Substring can be solved by a quantum algorithm in O~(n2/3) O ~ ( n 2 / 3 ) time, improving upon the recent O~(n5/6) O ~ ( n 5 / 6 ) -time algorithm by Le Gall and Seddighin (in: Proceedings of the 13th innovations in theoretical computer science conference (ITCS 2022), pp 97:1–97:23, 2022. https://doi.org/10.4230/LIPIcs.ITCS.2022.97 ). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, in: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing (STOC 2019), ACM, pp 756–767, 2019. https://doi.org/10.1145/3313276.3316368 ) and generalized difference covers. Lexicographically Minimal String Rotation can be solved by a quantum algorithm in n1/2+o(1) n 1 / 2 + o ( 1 ) time, improving upon the recent O~(n3/4) O ~ ( n 3 / 4 ) -time algorithm by Wang and Ying (in: Quantum algorithm for lexicographically minimal string rotation. CoRR, 2020. arXiv:2012.09376 ). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding. Longest Square Substring can be solved by a quantum algorithm in O~(n) O ~ ( n ) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2022) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies. Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix.
In this thesis we provide new upper and lower bounds on the quantum query complexity of a diverse set of problems. Specifically, we study quantum algorithms for Hamiltonian simulation, matrix multiplication, oracle identification, and graph-property recognition. \n \nFor the Hamiltonian simulation problem, we provide a quantum algorithm with query complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Our algorithm is based on a new quantum algorithm for implementing unitary matrices that can be written as linear combinations of efficiently implementable unitary gates. This algorithm uses a new form of ``oblivious amplitude amplification'' that can be applied even though the reflection about the input state is unavailable. \n \nIn the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set of size M, and our task is to identify x. We present the first quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered gamma_2-norm semidefinite program. \n \nWe then study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular. Our main result is an output-sensitive algorithm for Boolean matrix multiplication that multiplies two n x n Boolean matrices with query complexity O(n sqrt{l}), where l is the sparsity of the output matrix. The algorithm is based on a reduction to the graph collision problem and a new algorithm for graph collision. \n \nFinally, we study the quantum query complexity of minor-closed graph properties and show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Theta(n^{3/2}) and those that do have such a characterization can be solved strictly faster, with o(n^{3/2}) queries. Our lower bound is based on a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.
In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string x∈{−1,1}n. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(nlogn) and a near-matching lower bound of Ω(n). Belovs [Comput.~Comp.'15] subsequently showed a tight O(n) upper bound.
We consider a natural generalization of this problem, parametrized by a subset Q⊆2[n], where an algorithm may test whether xS=b for an arbitrary S∈Q and b∈{−1,1}S of its choice, at unit cost. We show near-tight bounds when Q is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set.
All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning x is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions f:{−1,1}n→R the ratio of the maximum value of f to the maximum (over T∈Q) standard deviation of f on a subcube whose free variables are exactly T.'
To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.