Parameterized complexity enables the practical solution of generally intractable NP-hard problems when certain parameters are small, making it particularly useful in real-world applications. The study of string problems in this framework has been particularly fruitful, yielding many state-of-the-art classical algorithms that run efficiently in certain parameter regimes contrary to their worst- or average-case performance. Motivated by the dramatic increase in genomic data and its growing computational demands, we initiate the study of the quantum parameterized complexity of the Closest String Problem (CSP) and the related Closest Substring Problem (CSSP). We present three quantum algorithms for the CSP and one for the CSSP. Each algorithm demonstrates improved performance over classical counterparts in specific parameter regimes, highlighting the promise of quantum approaches in structured combinatorial settings. We also derive a conditional lower bound for the CSP with binary alphabets, showing that our first algorithm is tight in its dominant scaling factor.
Parameterized complexity enables the practical solution of generally intractable NP-hard problems when certain parameters are small, making it particularly useful in real-world applications. The study of string problems in this framework has been particularly fruitful, yielding many state-of-the-art classical algorithms that run efficiently in certain parameter regimes contrary to their worst- or average-case performance. Motivated by the dramatic increase in genomic data and its growing computational demands, we initiate the study of the quantum parameterized complexity of the Closest String Problem (CSP) and the related Closest Substring Problem (CSSP). We present three quantum algorithms for the CSP and one for the CSSP. Each algorithm demonstrates improved performance over classical counterparts in specific parameter regimes, highlighting the promise of quantum approaches in structured combinatorial settings. We also derive a conditional lower bound for the CSP with binary alphabets, showing that our first algorithm is tight in its dominant scaling factor.
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.
Quantum computing offers promising approaches to combinatorial optimization problems whose search spaces grow exponentially with problem size. The Closest String problem belongs to this category, which seeks a string that minimizes the maximum Hamming distance to any string in a given collection of reference strings. In this work, we present an end-to-end quantum implementation of the Closest String problem using the Non-Variational Quantum Walk Optimization Algorithm (NV-QWOA), evaluating its resource requirements and solution amplification capabilities. Our automated workflow converts arbitrary problem instances into quantum circuits expressed in one- or two-qubit gates that encode candidate solutions, evaluate the objective function, and apply NV-QWOA optimisation to increase the measurement probabilities of high-quality solutions. Analytical resource estimates yield a qubit complexity of O(nmlogk) and a circuit depth of O(pmlog2n+pnlogn), where n is the string length, m is the number of reference strings, k is the alphabet size, and p is the number of NV-QWOA iterations. We introduce a revised prescription for the phase-separation parameter, γ, that depends on problem size but requires no instance-specific optimization. This prescription mitigates excessive phase wrapping modulo 2π, which can weaken the relationship between objective values and the phases guiding the optimization dynamics. Across all simulated problem sizes, the algorithm consistently amplifies optimal and near-optimal candidates. This work provides an explicit gate-level implementation, an improved phase-scaling method, and a foundation for hardware experiments and benchmarking against classical algorithms.
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.
Smoothed analysis is a central framework in classical algorithms for explaining the performance of algorithms beyond the worst case, often explaining why algorithms perform well in practice. We initiate a systematic study of its quantum counterpart and show the following results. (1) We show that there is a total function whose smoothed quantum query complexity is exponentially smaller than its classical query complexity. (2) We give near-tight characterizations of smoothed randomized and quantum query complexities for symmetric Boolean functions, unifying the worst-case complexity results of [Beals et al, FOCS'98] and average-case complexity results of [Ambainis and de Wolf, STACS'00]. (3) We study string problems such as pattern matching and edit distance and, in various regimes, give polynomial to superpolynomial quantum speedups. Our main technical ingredients include a near-tight quantum algorithm for ε-approximating the number of collisions between two non-repetitive strings, improving the result of Le Gall and Ng [QIC'22]. Together, our results show that smoothing can reveal larger quantum speedups than worst-case analysis suggests, opening a path towards quantum advantage on more realistic inputs.
Graph coloring is a well-known combinatorial optimization problem with broad applications in network security, scheduling, and register allocation. Being an NP-hard problem, exact solutions for large instances are computationally infeasible, calling for other solution strategies. In this study, parameterized and approximation methods to tackle the graph coloring problem are investigated. While approximation algorithms yield near-optimal solutions with theoretical guarantees of performance that are well-suited for large-scale applications, parameterized algorithms utilize graph-specific structural properties to manage constrained instances efficiently. We examine the trade-offs in these approaches, highlighting their strengths and weaknesses. In addition, we introduce a hybrid approach that combines both parameterized and approximation methods to best optimize computational resources and balance accuracy versus feasibility. Furthermore, we discuss the newly realized potential of quantum computing—specifically quantum annealing and the Quantum Approximate Optimization Algorithm (QAOA)—to speed up graph coloring by leveraging quantum parallelism and state superposition. Such quantum-based techniques represent a promising line of attack for breaking up large and hard instances that overwhelm classical approaches. Future research will aim to tighten approximation guarantees, create quantum-augmented hybrid models, and test theoretical results against real-world datasets.