Designing Quantum Solutions to the Closest String Problem with the Non-Variational QWOA

Abstract

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 $\mathcal{O}(nm\log k)$ and a circuit depth of $\mathcal{O}(pm\log^2 n + pn\log n)$, 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, $\gamma$, that depends on problem size but requires no instance-specific optimization. This prescription mitigates excessive phase wrapping modulo $2\pi$, 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.