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arXiv:2603.14744

Towards Exponential Quantum Improvements in Solving Cardinality-Constrained Binary Optimization

Haomu Yuan, Hanqing Wu, Kuan-Cheng Chen, Bin Cheng, Crispin H. W. Barnes

quant-phcs.CCmath.OCmath.STSubmitted Mar 16, 2026• Announced Mar 17, 2026• 0 citations
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Abstract

Cardinality-constrained binary optimization is a fundamental computational primitive with broad applications in machine learning, finance, and scientific computing. In this work, we introduce a Grover-based quantum algorithm that exploits the structure of the fixed-cardinality feasible subspace under a natural promise on solution existence. For quadratic objectives, our approach achieves ${O}\left(\sqrt{\frac{\binom{n}{k}}{M}}\right)$ Grover rotations for any fixed cardinality $k$ and degeneracy of the optima $M$, yielding an exponential reduction in the number of Grover iterations compared with unstructured search over $\{0,1\}^n$. Building on this result, we develop a hybrid classical--quantum framework based on the alternating direction method of multipliers (ADMM) algorithm. The proposed framework is guaranteed to output an $ε$-approximate solution with a consistency tolerance $ε+ δ$ using at most $ {O}\left(\sqrt{\binom{n}{k}}\frac{n^{6}k^{3/2} }{ \sqrt{M}ε^2 δ}\right)$ queries to a quadratic oracle, together with ${O}\left(\frac{n^{6}k^{3/2}}{ε^2δ}\right)$ classical overhead. Overall, our method suggests a practical use of quantum resources and demonstrates an exponential improvements over existing Grover-based approaches in certain parameter regimes, thereby paving the way toward quantum advantage in constrained binary optimization.

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Cardinality-constrained binary optimization is a fundamental computational primitive with broad applications in machine learning, finance, and scientific computing. In this work, we introduce a Grover-based quantum algorithm that exploits the structure of the fixed-cardinality feasible subspace under a natural promise on solution existence. For quadratic objectives, our approach achieves O((nk)M){O}\left(\sqrt{\frac{\binom{n}{k}}{M}}\right)O(M(kn​)​​) Grover rotations for any fixed cardinality kkk and degeneracy of the optima MMM, yielding an exponential reduction in the number of Grover iterations compared with unstructured search over {0,1}n\{0,1\}^n{0,1}n. Building on this result, we develop a hybrid classical--quantum framework based on the alternating direction method of multipliers (ADMM) algorithm. The proposed framework is guaranteed to output an εεε-approximate solution with a consistency tolerance ε+δε+ δε+δ using at most O((nk)n6k3/2Mε2δ) {O}\left(\sqrt{\binom{n}{k}}\frac{n^{6}k^{3/2} }{ \sqrt{M}ε^2 δ}\right)O((kn​)​M​ε2δn6k3/2​) queries to a quadratic oracle, together with O(n6k3/2ε2δ){O}\left(\frac{n^{6}k^{3/2}}{ε^2δ}\right)O(ε2δn6k3/2​) classical overhead. Overall, our method suggests a practical use of quantum resources and demonstrates an exponential improvements over existing Grover-based approaches in certain parameter regimes, thereby paving the way toward quantum advantage in constrained binary optimization.
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Optimization problems in machine-learning (ML) applications can be computationally challenging for classical methods, particularly when they involve large, unstructured search spaces. Quantum computing offers a promising approach to combinatorial optimization by utilizing quantum superposition and amplitude amplification. This paper presents a new methodology using logic-based quantum machine learning (QML) as a complete framework for employing a multi-solution ternary Grover’s algorithm to minimize incomplete binary functions represented by Pseudo-Kronecker Reed–Muller (PKRO) expansions, consistent with Occam’s razor principle. Unlike previous quantum approaches based on Kronecker Reed–Muller (KRO) or fixed-polarity Reed–Muller (FPRM) representations, our work introduces the first Grover-based optimization framework for PKRO forms. This framework formulates logic minimization as a quantum search problem to identify minimum-cost AND-XOR representations with the fewest nonzero coefficients. A hybrid binary–ternary quantum architecture is developed to explore the enlarged PKRO search space, enabling optimization of both completely and incompletely specified binary functions. Owing to the greater flexibility of PKRO expansions, our framework produces more compact AND–XOR representations than KRO- and FPRM-based approaches. Across all 256 3-variable binary functions, PKRO provides less nonzero coefficients for 4.688% of functions compared with KRO and for 37.5% compared with FPRM, with average reductions of 0.047 and 0.484 nonzero coefficients, respectively. Our multi-solution ternary Grover search reduces the number of Grover iterations by more than 91% compared with the single-solution approach.
Grover's algorithm is a fundamental quantum algorithm that achieves a quadratic speedup for unstructured search problems of size NNN. Recent studies have reformulated this task as a maximization problem on the unitary manifold and solved it via linearly convergent Riemannian gradient ascent (RGA) methods, resulting in a complexity of O(N/Mlog⁡(1/ε))O(\sqrt{N/M}\log (1/\varepsilon))O(N/M​log(1/ε)), where MMM denotes the number of target items and ε\varepsilonε denotes the success probability error. In this work, we adopt the Riemannian modified Newton (RMN) method to solve the quantum search problem, under the assumption that the ratio $ M/N$ is known. We show that, in this setting, the Riemannian Newton direction is collinear with the Riemannian gradient in the sense that the Riemannian gradient is always an eigenvector of the corresponding Riemannian Hessian. This structure removes the overhead of Hessian inversion and allows the proposed RMN method to retain the local quadratic convergence in terms of the error ε\varepsilonε. More precisely, we rigorously prove an overall complexity of O(N/M+log⁡log⁡(1/ε))O(\sqrt{N/M}+\log\log(1/\varepsilon))O(N/M​+loglog(1/ε)). Furthermore, our approach remains Grover-compatible, namely, it relies exclusively on the standard Grover diffusion and oracle operators to ensure algorithmic implementability, and its parameter update process can be efficiently precomputed on classical computers.
Quantum search is one of the most important algorithmic developments in quantum computing because it demonstrates how quantum principles can improve the efficiency of solving unstructured search problems. Grover’s algorithm provides a quadratic speed-up over classical linear search by reducing query complexity from O(N) to O(√N), where N represents the size of the search space. This review examines the principles, performance, and applications of Grover’s algorithm, with emphasis on its theoretical foundation and practical relevance. The discussion covers the roles of superposition, quantum interference, oracle operations, diffusion operators, and amplitude amplification in increasing the probability of measuring a desired solution. The manuscript also reviews the computational performance of quantum search in comparison with classical search methods and considers its relevance to database searching, combinatorial optimisation, cryptography, machine learning, bioinformatics, quantum chemistry, and materials science. The review highlights that Grover’s algorithm is not limited to database retrieval, as many computationally intensive problems can be reformulated as search or optimisation tasks. However, practical implementation remains constrained by current quantum hardware limitations, including decoherence, quantum noise, limited qubit availability, circuit depth, and the difficulty of designing efficient oracles. Despite these challenges, continuing progress in quantum hardware, error correction, cloud-based quantum platforms, and hybrid quantum-classical architectures suggests that quantum search will remain an important area of research. Overall, Grover’s algorithm represents a foundational quantum algorithm with significant theoretical value and potential long-term application in computational domains involving large and unstructured search spaces.