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arXiv:2502.02245 · Advanced Quantum Technologies

Qubit‐Efficient Quantum Local Search for Combinatorial Optimization

Mikhail Podobrii, Viacheslav Kuzmin, Vladimir Voloshinov, Margarita Veshchezerova, Michael Romanovich Perelshtein

Published Mar 1, 2026• 1 citation
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Abstract

ABSTRACT An essential component of many sophisticated metaheuristics for solving combinatorial optimization problems is some variation of a local search routine that iteratively searches for a better solution within a chosen set of immediate neighbors. The size of this set is limited due to the computational costs required to run the method on classical processing units. We present a qubit‐efficient variational quantum algorithm that implements a quantum version of local search with only qubits and, therefore, can potentially work with classically intractable neighborhood sizes. Increasing the amount of quantum resources employed in the algorithm allows for a larger neighborhood size, improving the quality of obtained solutions. This trade‐off is crucial for present and near‐term quantum devices characterized by a limited number of logical qubits. Numerically simulating our algorithm, we successfully solved the largest graph coloring instance that was tackled by a quantum method. This achievement highlights the algorithm's potential for solving large‐scale combinatorial optimization problems on near‐term quantum devices.

DOI
10.1002/qute.202500438

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ABSTRACT An essential component of many sophisticated metaheuristics for solving combinatorial optimization problems is some variation of a local search routine that iteratively searches for a better solution within a chosen set of immediate neighbors. The size of this set is limited due to the computational costs required to run the method on classical processing units. We present a qubit‐efficient variational quantum algorithm that implements a quantum version of local search with only qubits and, therefore, can potentially work with classically intractable neighborhood sizes. Increasing the amount of quantum resources employed in the algorithm allows for a larger neighborhood size, improving the quality of obtained solutions. This trade‐off is crucial for present and near‐term quantum devices characterized by a limited number of logical qubits. Numerically simulating our algorithm, we successfully solved the largest graph coloring instance that was tackled by a quantum method. This achievement highlights the algorithm's potential for solving large‐scale combinatorial optimization problems on near‐term quantum devices.
Quantum computers hold the promise of more efficient combinatorial optimization solvers, which could be game-changing for a broad range of applications. However, a bottleneck for materializing such advantages is that, in order to challenge classical algorithms in practice, mainstream approaches require a number of qubits prohibitively large for near-term hardware. Here we introduce a variational solver for MaxCut problems over m=O(nk)m={{\mathcal{O}}}({n}^{k})m=O(nk) binary variables using only n qubits, with tunable k > 1. The number of parameters and circuit depth display mild linear and sublinear scalings in m, respectively. Moreover, we analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature. Altogether, this leads to high quantum-solver performances. For instance, for m = 7000, numerical simulations produce solutions competitive in quality with state-of-the-art classical solvers. In turn, for m = 2000, experiments with n = 17 trapped-ion qubits feature MaxCut approximation ratios estimated to be beyond the hardness threshold 0.941. Our findings offer an interesting heuristics for quantum-inspired solvers as well as a promising route towards solving commercially-relevant problems on near-term quantum devices. Combinatorial optimization problems might be solvable more efficiently with quantum computing, but near-term quantum optimization solvers are limited in qubit size. Here, the authors propose and demonstrate a variational quantum algorithm that polynomially reduces the qubit overhead for applications like MaxCut.
We present a new hybrid, local search algorithm for quantum approximate optimization of constrained combinatorial optimization problems. We focus on the Maximum Independent Set problem and demonstrate the ability of quantum local search to solve large problem instances on quantum devices with few qubits. This hybrid algorithm iteratively finds independent sets over carefully constructed neighborhoods and combines these solutions to obtain a global solution. We study the performance of this algorithm on 3-regular, Community, and Erdős-Rényi graphs with up to 100 nodes.
Emerging quantum processors provide an opportunity to explore new approaches for solving traditional problems in the post Moore’s law supercomputing era. However, the limited number of qubits makes it infeasible to tackle massive real-world datasets directly in the near future, leading to new challenges in utilizing these quantum processors for practical purposes. Hybrid quantum-classical algorithms that leverage both quantum and classical types of devices are considered as one of the main strategies to apply quantum computing to large-scale problems. In this article, we advocate the use of multilevel frameworks for combinatorial optimization as a promising general paradigm for designing hybrid quantum-classical algorithms. To demonstrate this approach, we apply this method to two well-known combinatorial optimization problems, namely, the Graph Partitioning Problem, and the Community Detection Problem. We develop hybrid multilevel solvers with quantum local search on D-Wave’s quantum annealer and IBM’s gate-model based quantum processor. We carry out experiments on graphs that are orders of magnitude larger than the current quantum hardware size, and we observe results comparable to state-of-the-art solvers in terms of quality of the solution. Reproducibility : Our code and data are available at Reference [1].
We present Quantum Local Search (QLS) approach and demonstrate its efficacy by applying it to the problem of community detection in real-world networks. QLS is a hybrid algorithm that combines a classical machine with a small quantum device. QLS starts with an initial solution and searches its neighborhood, iteratively trying to find a better candidate solution. One of the main challenges of the quantum computing in NISQ era is the small number of available qubits. QLS addresses this challenge by using the quantum device only for the neighborhood search, which can be restricted to be small enough to fit on near-term quantum device. We implement QLS for modularity maximization graph clustering using QAOA on IBM Q Experience as a quantum local solver. We demonstrate the potential for quantum acceleration by showing that existing state-of-the-art optimization solvers cannot find a good solution to the local problems quickly and provide an estimate of how larger quantum devices can improve the performance of QLS.
Combinatorial optimization is a promising application for near-term quantum computers, however, identifying performant algorithms suited to noisy quantum hardware remains as an important goal to potentially realizing quantum computational advantages. To address this we present a variational quantum algorithm for solving combinatorial optimization problems with linear-depth circuits. Our algorithm uses an ansatz composed of Hamiltonian generators designed to control each term in the target combinatorial function, along with parameter updates following a modified version of quantum imaginary time evolution. We evaluate this ansatz in numerical simulations that target solutions to the MAXCUT problem. The state evolution is shown to closely mimic imaginary time evolution, and its optimal-solution convergence is further improved using adaptive transformations of the classical Hamiltonian spectrum. With these innovations, the algorithm consistently converges to optimal solutions, with interesting highly-entangled dynamics along the way. We further demonstrate the success of this approach by performing optimization of a truncated version of our ansatz with up to 32 qubits in a trapped-ion quantum computer, using measurements from the quantum computer to train the ansatz and prepare optimal solutions with high fidelity in the majority of cases we consider. The success of these large-scale quantum circuit optimizations, in the presence of realistic hardware constraints and without error mitigation, mark significant progress on the path towards solving combinatorial problems using quantum computers. We conclude our performant and resource-minimal approach is a promising candidate for potential quantum computational advantages.
Abstract MaxCut is a key NP-hard combinatorial optimization problem. Quantum computing offers methods to solve such problems potentially better than classical counterparts, with the Quantum Approximate Optimization Algorithm (QAOA) being a state-of-the-art example. However, the performance of quantum methods is currently hindered by hardware noise and limited qubit volumes. We present a variational Qubit-Efficient MaxCut (QEMC) algorithm that requires only O(log⁡N)O(\log N)O(logN) O ( log N ) qubits to tackle graphs of size N , an exponential reduction compared to QAOA. We demonstrate cutting-edge performance for 32-node graph instances (5 qubits) on real superconducting hardware, and for graphs with up to 2048 nodes (11 qubits) via classical simulations. The QEMC algorithm is based on an innovative encoding scheme, with potentially broad applicability, that empowers it with strong noise resilience, but also enables its efficient classical simulation. As such, the QEMC algorithm provides a challenging benchmark for QAOA on noisy devices and offers a novel quantum-inspired approach.