We present a full-scale implementation and experimental evaluation of a quantum algorithm for the Longest Common Substring (LCS) problem in the circuit model, bridging the gap between recent theoretical advances and practical realization. Building upon a previously proposed \(\tilde{O}(\sqrt {n})\)-depth quantum circuit, we develop a modular implementation in Qiskit that supports non-binary alphabets and incorporates several key enhancements, including a deterministic BBHT-inspired Grover search, domain expansion via ancillary qubits to stabilize amplitude amplification, and circuit-level optimizations that reduce overhead. Our approach is validated through an extensive experimental campaign over a binary alphabet augmented with two termination symbols and length 16 demonstrating an overall accuracy of 98.4%. The results show that errors are both rare and small, with a consistent conservative bias toward underestimation, and that the algorithm maintains high performance across a wide range of input configurations. We further analyze the behavior of the algorithm under realistic noise models, showing a progressive degradation of accuracy and identifying a structural asymmetry in the error patterns induced by the oracle. These findings provide concrete evidence that circuit-based quantum algorithms for string processing can achieve reliable behavior in ideal settings, while highlighting key challenges for their deployment on noisy quantum devices.

We investigate and compare the performance of two quantum optimization approaches, the Quantum Approximate Optimization Algorithm (QAOA) and quantum annealing, applied to the Minimum Vertex Cover (MVC) problem. The problem is encoded as an and Ising model, and experiments are conducted on IBM’s GenericBackendV2 noisy superconducting qubit simulator and the D-Wave Advantage2 quantum annealer. Performance is evaluated in terms of solution quality, measurement probability, and proportion of valid solutions. The results we obtained show that, within our experimental setting, quantum annealing consistently outperforms its classical counterpart on small instances, while QAOA, though currently limited by simulation constraints, shows promising behavior that improves with increasing circuit depth. As problem size grows, both approaches exhibit sensitivity to parameter choices such as the penalty term and graph density, underscoring the need for careful tuning. These findings suggest that while both paradigms hold potential for combinatorial optimization, further advances in hardware capabilities and parameter calibration will be necessary to achieve reliable performance on larger instances.

Pattern matching with swaps is a well-known non-standard variant of the string matching problem in which adjacent characters of the pattern may be swapped, under the constraint that each character participates in at most one swap. This problem has received considerable attention in the stringology literature due to its applications in areas such as computational biology, error-tolerant text processing, and pattern discovery. In this paper we investigate the problem in the quantum computing model. Building upon the recently introduced framework of quantum bit-parallelism, which provides a systematic way to translate classical bit-parallel string matching algorithms into quantum circuits, we develop the first quantum algorithm for swap matching based on this paradigm. Our approach adapts the classical cross-sampling technique for swap matching into a reversible quantum procedure that simulates the evolution of two interacting automata configurations using quantum registers and bitwise quantum operations. While the resulting quantum circuit preserves the near-linear behavior of its classical counterpart in terms of circuit depth, its integration with Grover’s search enables a quadratic speedup. By embedding the verification procedure within a quantum search framework and carefully controlling the size of the search space, we obtain a quantum algorithm that solves the swap matching problem in \(\tilde{O}(\sqrt {n})\) time. This result extends the applicability of quantum bit-parallelism to non-standard string matching problems and provides further evidence that a broad class of classical text searching techniques can be systematically lifted to the quantum setting while preserving their structural efficiency.

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.

Following the success of the so-called algebraic approach to the study of decision constraint satisfaction problems (CSPs), exact optimization of valued CSPs, and most recently promise CSPs, we propose an algebraic framework for valued promise CSPs.

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions.

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami (SODA’20) for promise (non-valued) CSPs (on finite domains).

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. It is desirable to classify the computational complexity of VCSPs depending on a fixed set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified in this sense. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear homogeneous cost functions. We show that such VCSPs can be solved in polynomial time when the cost functions are additionally submodular, and that this is indeed a maximally tractable class: adding any cost function that is not submodular leads to an NP-hard VCSP.