Quantum computation represents a computational paradigm whose distinctive attributes confer the ability to devise algorithms with asymptotic performance levels significantly superior to those achievable via classical computation. Recent strides have been taken to apply this computational framework in tackling and resolving various issues related to text processing. The resultant solutions demonstrate marked advantages over their classical counterparts. This study employs quantum computation to efficaciously surmount text processing challenges, particularly those involving string comparison. The focus is on the alignment of fixed-length substrings within two input strings. Specifically, given two input strings, $x$ and $y$, both of length $n$, and a value $d \leq n$, we want to verify the following conditions: the existence of a common prefix of length $d$, the presence of a common substring of length $d$ beginning at position $j$ (with $0 \leq j < n$) and, the presence of any common substring of length $d$ beginning in both strings at the same position. Such problems find applications as sub-procedures in a variety of problems concerning text processing and sequence analysis. Notably, our approach furnishes polylogarithmic solutions, a stark contrast to the linear complexity inherent in the best classical alternatives.
Quantum computation represents a computational paradigm whose distinctive attributes confer the ability to devise algorithms with asymptotic performance levels significantly superior to those achievable via classical computation. Recent strides have been taken to apply this computational framework in tackling and resolving various issues related to text processing. The resultant solutions demonstrate marked advantages over their classical counterparts. This study employs quantum computation to efficaciously surmount text processing challenges, particularly those involving string comparison. The focus is on the alignment of fixed-length substrings within two input strings. Specifically, given two input strings, x and y, both of length n, and a value d≤n, we want to verify the following conditions: the existence of a common prefix of length d, the presence of a common substring of length d beginning at position j (with 0≤j<n) and, the presence of any common substring of length d beginning in both strings at the same position. Such problems find applications as sub-procedures in a variety of problems concerning text processing and sequence analysis. Notably, our approach furnishes polylogarithmic solutions, a stark contrast to the linear complexity inherent in the best classical alternatives.
String matching stands as a cornerstone problem in computer science, finding extensive application in many fields, such as data mining, text retrieval, and bioinformatics, just to cite a few. Its essence lies in identifying all occurrences of a specified pattern, typically denoted by length m, within a text of length n. Despite its seemingly straightforward formulation, the problem has sparked a plethora of solutions, stemming from diverse computational paradigms and approaches. Classical solutions exhibit a worst-case time complexity of Θ(n), contrasting with the capability of quantum computation to achieve a [EQUATION] complexity using Grover search, thereby realizing a quadratic speed-up compared to classical methods. This article presents a first practical implementation of a quantum circuit tailored to address string matching, particularly focusing on binary strings. The exposition delves into the technical intricacies of the implementation, which leverages the Qiskit open-source toolkit developed by IBM. By elucidating various algorithmic nuances overlooked in prior theoretical formulations, our solution serves as a conduit between the realms of text processing and quantum computing, fostering cross-disciplinary dialogue and innovation. While current real-world hardware lacks the capability to execute the proposed circuit, particularly when dealing with lengthy texts, this study presents experimental results from a simulation of the circuit on classical hardware, serving to validate the efficacy of the proposed implementation.
The Longest Common Substring (LCS) and Longest Palindromic Substring (LPS) problems are fundamental challenges in string processing, traditionally solved in linear time using classical computation through suffix trees. Recent breakthroughs by Le Gall and Seddighin [1] introduced sublinear quantum query algorithms, while Akmal and Jin [2] further improved the LCS complexity to O ˜ ( n 2 / 3 ) . While these results are remarkable in the quantum query model , their practical implementation on real quantum hardware remains elusive. In this paper, we bridge this gap by presenting the first O ˜ ( n ) quantum algorithms for both LCS and LPS in the circuit model of computation . Our circuits are explicitly constructed and analyzed in terms of size and depth, achieving polylogarithmic overheads while preserving the O ˜ ( n ) depth bound. This provides, for the first time, concrete circuit-level blueprints and resource estimates for quantum solutions to LCS and LPS.
Text searching is a fundamental problem in computer science, and it finds applications in several scientific fields. The string matching problem is the common framework for the class of text searching problems where the objective is to find all, possibly approximate, occurrences of a given pattern of length m within a larger text of length n . This paper introduces the quantum path parallelism approach, that is, a general strategy based on quantum computation, which is easily adapted to a variety of nonstandard text-searching problems. Our method translates a text-searching problem into an automata-based string-recognition problem, associating each possible approximation of the pattern with a different path accepted by the automaton. Under favorable conditions, our algorithm solves an approximate search problem in O ( n m log ( n ) ) -time, yielding a quadratic speed-up over classical solutions. In other cases, such a speed-up is achieved only for short patterns, i.e. whenever m = O ( log ( n ) ) . We show the flexibility of our method, giving explicit adaptations to some specific approximate string matching problems.
Longest Common Substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decisional version of this problem, LCS with threshold d, asks whether two length-n input strings have a common substring of length d. The two extreme cases, d=1 and d=n, correspond respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case 1≪d≪n was not fully understood. We show that the complexity of LCS with threshold d smoothly interpolates between the two extreme cases up to no(1) factors: LCS with threshold d has a quantum algorithm in n2/3+o(1)/d1/6 query complexity and time complexity, and requires at least Ω(n2/3/d1/6) quantum query complexity. Our result improves upon previous upper bounds O~(min{n/d1/2,n2/3}) (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples n/τ1−o(1) synchronizing positions in the string depending on their length-Θ(τ) contexts, and each synchronizing position can be reported by a quantum algorithm in O~(τ1/2+o(1)) time. As another application of our quantum string synchronizing set, we study the k-mismatch Matching problem, which asks if the pattern has an occurrence in the text with at most k Hamming mismatches. Using a structural result of Charalampopoulos, Kociumaka, and Wellnitz (FOCS 2020), we obtain a quantum algorithm for k-mismatch matching with k3/4n1/2+o(1) query complexity and O~(kn1/2) time complexity.
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.
Longest common substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decision version of this problem, LCS with threshold d , asks whether two length- n input strings have a common substring of length d . The two extreme cases, d=1 and d=n , correspond, respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case 1≪d≪n was not fully understood. We show that the complexity of LCS with threshold d smoothly interpolates between the two extreme cases up to no(1) factors: — LCS with threshold d has a quantum algorithm in n2/3+o(1)/d1/6 query complexity and time complexity, and requires at least Ω(n2/3/d1/6) quantum query complexity. Our result improves upon previous upper bounds O(min{n/d1/2,n2/3}) (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples n/τ1−o(1) synchronizing positions in the string depending on their length- Θ(τ) contexts, and each synchronizing position can be reported by a quantum algorithm in O(τ1/2+o(1)) time. Our quantum string synchronizing set also yields a near-optimal LCE data structure in the quantum setting. As another application of our quantum string synchronizing set, we study the k -mismatch Matching problem, which asks if the pattern has an occurrence in the text with at most k Hamming mismatches. Using a structural result of Charalampopoulos et al. (FOCS 2020), we obtain: — k -mismatch matching has a quantum algorithm with k3/4n1/2+o(1) query complexity and O(kn1/2) time complexity. We also observe a non-matching quantum query lower bound of Ω(kn) .