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Leibniz international proceedings in informatics

Quantum Closest-Pair Search for Biological Sequences via k-Mer Distribution Statistics

Zhezheng Xander Song, Carl Kingsford

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Finding highly similar pairs of biological sequences is a fundamental task in bioinformatics. For alignment-free k-mer distributional similarities, the induced feature space is high-dimensional and lacks the low-dimensional geometric structure used by classical exact closest-pair algorithms. Thus, for a collection of N sequences in the pairwise-score setting, exhaustive evaluation over the binom(N,2) candidate pairs is the natural classical baseline. We present the first quantum framework targeting alignment-free closest-pair search in biological sequence collections using distributional k-mer statistics. The central technical contribution is the construction of a coherent pairwise-score estimation circuit for this similarity measure. It encodes empirical k-mer distributions as square-root amplitude states and provides a sparse prefix-tree construction for preparing these states, under which the state overlap is exactly the Bhattacharyya coefficient. Standard SWAP-test and quantum-amplitude-estimation subroutines provide a coherent bounded-precision estimator for the squared Bhattacharyya overlap. We analyze maximum finding under an explicit assumption that a fixed ε-resolved total order over all legal pairs admits an efficient clean coherent implementation. Under this assumption, the procedure returns, with probability at least 2/3, a pair whose squared Bhattacharyya score is within ε of the optimal score, using O(N) expected comparison-oracle calls. If the optimal score is separated from every strictly suboptimal score by more than ε, the returned pair is exactly optimal. Combining this comparison-order assumption with an idealized qRAM-style data-access model gives the conditional sequential gate complexity Õ(NL/ε), whereas explicit multiplexed indexed loading gives Õ(N²L/ε). We also provide a proof-of-concept Q#implementation that integrates coherent indexed loading, SWAP-test-based score estimation, finite-precision marking, and Grover-style search, providing circuit-level validation of the main computational components.

DOI
10.4230/lipics.wabi.2026.9

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Finding highly similar pairs of biological sequences is a fundamental task in bioinformatics. For alignment-free k-mer distributional similarities, the induced feature space is high-dimensional and lacks the low-dimensional geometric structure used by classical exact closest-pair algorithms. Thus, for a collection of N sequences in the pairwise-score setting, exhaustive evaluation over the binom(N,2) candidate pairs is the natural classical baseline. We present the first quantum framework targeting alignment-free closest-pair search in biological sequence collections using distributional k-mer statistics. The central technical contribution is the construction of a coherent pairwise-score estimation circuit for this similarity measure. It encodes empirical k-mer distributions as square-root amplitude states and provides a sparse prefix-tree construction for preparing these states, under which the state overlap is exactly the Bhattacharyya coefficient. Standard SWAP-test and quantum-amplitude-estimation subroutines provide a coherent bounded-precision estimator for the squared Bhattacharyya overlap. We analyze maximum finding under an explicit assumption that a fixed ε-resolved total order over all legal pairs admits an efficient clean coherent implementation. Under this assumption, the procedure returns, with probability at least 2/3, a pair whose squared Bhattacharyya score is within ε of the optimal score, using O(N) expected comparison-oracle calls. If the optimal score is separated from every strictly suboptimal score by more than ε, the returned pair is exactly optimal. Combining this comparison-order assumption with an idealized qRAM-style data-access model gives the conditional sequential gate complexity Õ(NL/ε), whereas explicit multiplexed indexed loading gives Õ(N²L/ε). We also provide a proof-of-concept Q#implementation that integrates coherent indexed loading, SWAP-test-based score estimation, finite-precision marking, and Grover-style search, providing circuit-level validation of the main computational components.
Motivation: The exponential growth of publicly available genomic data has created unprecedented opportunities for sequence-based discovery. Locating specific k-mers is fundamental to diverse applications, including metagenomic classification, pathogen and cancer detection, and variant calling yet efficient identification of multiple k-mer patterns across large sequencing data and massive databases remains a significant computational challenge. Method: We implement two quantum algorithms for DNA multi-pattern string matching for k-mer detection, leveraging Grover's amplitude amplification under the idealized quantum random access memory (QRAM) framework. The first algorithm uses an enumerate-m oracle that sequentially checks a loaded text substring against all m patterns achieving O (√S) query complexity for S text positions but requiring O (m · L) work per oracle call. The second algorithm employs nested Grover search with an outer loop over text positions and an inner loop over pattern space, reducing oracle complexity to O(L) while performing O (√S · √m) in total. These asymptotic gains highlight the potential advantages that could be unlocked by future large-scale, low-noise QRAM architectures, positioning our results as a promising proof-of-concept foundation. Results: This work introduces two quantum implementations of multi-pattern string matching tailored for k-mer detection. Leveraging quantum parallelism and Grover-inspired search primitives, our methods accelerate dictionary-based pattern matching, particularly in contexts involving large sequences, such as genomic data, and extensive pattern sets. Conclusion: While implementation challenges such as QRAM overhead remain, this study demonstrates both the promise and current limitations of quantum-enhanced string matching, establishing a foundational step toward quantum readiness in bioinformatics. Availability and Implementation: To maximize accessibility and practical use, we provide our methodology at: https://github.com/Georgakopoulos-Soares-lab/quantum-multi-motif-finder.
Motivation: The exponential growth of publicly available genomic data has created unprecedented opportunities for sequence-based discovery. Locating specific k-mers is fundamental to diverse applications, including metagenomic classification, pathogen and cancer detection, and variant calling yet efficient identification of multiple k-mer patterns across large sequencing data and massive databases remains a significant computational challenge. Method: We implement two quantum algorithms for DNA multi pattern string matching for k-mer detection based on Grover's amplitude amplification with quantum random access memory (QRAM). The first algorithm uses an enumerate-m oracle that sequentially checks a loaded text substring against all m patterns achieving O(√S) query complexity for S text positions but requiring O(m·L) work per oracle call. The second algorithm employs nested Grover search with an outer loop over text positions and an inner loop over pattern space, reducing oracle complexity to O(L) while performing O(√S · √m) in total. Results: We present two quantum implementations of multi-pattern string matching tailored for k-mer detection. Leveraging quantum parallelism and Grover-inspired search primitives, our methods accelerate dictionary-based pattern matching, particularly in contexts involving large sequences, such as genomic data, and extensive pattern sets. Conclusions: While implementation challenges such as QRAM overhead remain, this study demonstrates both the promise and current limitations of quantum-enhanced string matching, establishing a foundational step toward quantum readiness in bioinformatics.
Sequence Alignment is the process of aligning biological sequences in order to identify similarities between multiple sequences. In this paper, a Quantum Algorithm for finding the optimal alignment between DNA sequences has been demonstrated which works by mapping the sequence alignment problem into a path-searching problem through a 2D graph. The transition, which converges to a fixed path on the graph, is based on a proposed oracle for profit calculation. By implementing Grover's search algorithm, our proposed approach is able to align a pair of sequences and figure out the optimal alignment within linear time, which hasn't been attained by any classical deterministic algorithm. In addition to that, the proposed algorithm is capable of quadratic speeding up to any unstructured search problem by finding out the optimal paths accurately in a deterministic manner, in contrast to existing randomized algorithms that frequently sort out the sub-optimal alignments, therefore, don't always guarantee of finding out the optimal solutions.
Multiple sequence alignment (MSA) is a core operation for comparing genome sequences and is widely used in bio-informatics. MAFFT, a practical MSA tool, repeatedly shifts a pair of sequences and computes a distance. Because the number of sequence pairs grows quadratically with the number of sequences, this procedure can become a bottleneck. We propose Quantum Shift-based Sequence Alignment (QShift-SA), which implements this ``shift-wise score computation'' as a gate-based quantum circuit and searches over shift amounts and sequence pairs using Grover algorithm. QShift-SA constructs an oracle circuit that compute the Hamming distance (the number of mismatches) between two sequences with data encoding, controlled shift, comparison, and addition. This oracle can search for candidates with small distances. QShift-SA does not aim to replace the full MSA workflow; instead, it targets the screening steps that often dominate the runtime in classical MAFFT as stated above. We evaluate circuit resources (number of qubits, gate count, and depth) and benchmark simulation time across multiple quantum circuit simulators. We find that a decision diagram (DD)-based quantum circuit simulator runs more than 1,000×\times× faster than state-vector and MPS simulators and can handle larger circuits.
Quantum search algorithms are considered in the context of protein sequence comparison in bioinformatics. Given a sample protein sequence of length m (i.e., m residues), the problem considered is to find an optimal match in a large database containing N residues. Initially, Grover's quantum search algorithm is applied to a simple illustrative case-namely, where the database forms a complete set of states over the 2(m) basis states of a m qubit register, and thus is known to contain the exact sequence of interest. This example demonstrates explicitly the typical O(square root of [N]) speedup on the classical O(N) requirements. An algorithm is then presented for the (more realistic) case where the database may contain repeat sequences, and may not necessarily contain an exact match to the sample sequence. In terms of minimizing the Hamming distance between the sample sequence and the database subsequences the algorithm finds an optimal alignment, in O(square root of [N]) steps, by employing an extension of Grover's algorithm, due to Boyer et al. for the case when the number of matches is not a priori known.
We present a quantum algorithm for estimating the amplitude content of user-specified sequency bands in quantum-encoded signals. The method employs a sequency-ordered Quantum Walsh-Hadamard Transform (QWHT), a comparator-based oracle that coherently marks basis states within an arbitrary sequency range, and Quantum Amplitude Estimation (QAE) to estimate the total probability mass in the selected band. This enables the detection of structured signal components, including both high- and low-sequency features, as well as the identification of rapid sign-change behavior associated with noise or anomalies. The proposed method can be embedded as a module within a larger quantum algorithm; in this setting, both the input and output remain fully quantum, enabling seamless integration with upstream and downstream quantum operations. We show that the sequency-ordered QWHT can be implemented with circuit depth O(log⁡2N)O(\log_2 N)O(log2​N) (equivalently O(n)O(n)O(n) for N=2nN=2^nN=2n) when acting on an amplitude-encoded quantum state, whereas computing the full Walsh-Hadamard spectrum of an explicit length-NNN classical signal requires O(Nlog⁡2N)O(N\log_2 N)O(Nlog2​N) operations via the fast Walsh-Hadamard transform. This results in an exponential quantum advantage when the QWHT is used as a modular block within a larger quantum algorithm, relative to classical fast Walsh-Hadamard transform-based approaches operating on explicit data. From an application perspective, the proposed sequency band-energy estimation may be interpreted as a structure-based anomaly indicator, enabling the detection of unexpected high-sequency components relative to a nominal low-sequency signal class. The algorithm is applicable to quantum-enhanced signal processing tasks such as zero-crossing analysis, band-limited noise estimation, and feature extraction in the Walsh basis.