Despite the premise of quantum computing for Combinatorial Optimization (CO), near-term quantum computers still lack the physical resources, especially the number of qubits, to address problem instances of the same scale as their classical counterparts. Various methods to scale the problem instances addressable with few qubits have been popular and promising. In this paper, we introduce a method for CO utilizing few qubits. The method consists of applications of Quantum Random Access Code (QRAC) for encoding variables into qubits, Classical Shadow (CS) for decoding the variables, and Coordinate Descent (CD) for optimizing quantum circuits that generate desirable quantum states for the CO instances. We show that QRAC is useful to reveal the caveats of encoding more variables into few qubits and how to alleviate them. We also show that CS is effective to reduce quantum resources for decoding variables from the qubits. We then argue that CD methods are useful not only for optimizing parameters of quantum circuits encoding the solution of the CO instance, but also for explaining why utilizing activation function like tanh in the postprocessing of observing quantum states and in the objective function help achieve higher solution quality. We demonstrate the effectiveness of our proposed method by experiments on parameterized quantum circuits for solving instances of NP-hard graph partitioning task called minimum bisection.
Despite the premise of quantum computing for Combinatorial Optimization (CO), near-term quantum computers still lack the physical resources, especially the number of qubits, to address problem instances of the same scale as their classical counterparts. Various methods to scale the problem instances addressable with few qubits have been popular and promising. In this paper, we introduce a method for CO utilizing few qubits. The method consists of applications of Quantum Random Access Code (QRAC) for encoding variables into qubits, Classical Shadow (CS) for decoding the variables, and Coordinate Descent (CD) for optimizing quantum circuits that generate desirable quantum states for the CO instances. We show that QRAC is useful to reveal the caveats of encoding more variables into few qubits and how to alleviate them. We also show that CS is effective to reduce quantum resources for decoding variables from the qubits. We then argue that CD methods are useful not only for optimizing parameters of quantum circuits encoding the solution of the CO instance, but also for explaining why utilizing activation function like tanh in the postprocessing of observing quantum states and in the objective function help achieve higher solution quality. We demonstrate the effectiveness of our proposed method by experiments on parameterized quantum circuits for solving instances of NP-hard graph partitioning task called minimum bisection.
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) 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.
Combinatorial optimization problems play a vital role in computer science and engineering, with applications in logistics, network design, finance, and resource allocation. However, many of these problems are NP-hard, making them computationally expensive to solve using classical optimization techniques, especially for large-scale instances. In recent years, quantum computing has emerged as a promising paradigm capable of addressing these limitations by leveraging quantum mechanical principles such as superposition and entanglement. This study investigates the application of quantum computing for solving combinatorial optimization problems, with a focus on algorithm design, complexity analysis, and real-world applicability. The research primarily employs the Quantum Approximate Optimization Algorithm (QAOA) and Grover's search algorithm to solve representative problems such as Max-Cut and the Knapsack problem. The performance of these algorithms is evaluated using key metrics including accuracy, execution time, and approximation ratio, and is compared with classical optimization techniques. The results demonstrate that QAOA is capable of generating near-optimal solutions, achieving accuracy levels of up to 93% for small problem instances. However, the study also highlights the challenges associated with current quantum systems, including noise, limited qubit availability, and increased execution time. Comparative analysis reveals that while classical methods remain more efficient for small-scale problems, quantum algorithms show significant potential for scalability and handling complex optimization tasks. The findings suggest that hybrid quantum-classical approaches offer a practical solution in the current Noisy Intermediate-Scale Quantum (NISQ) era. The study concludes that although quantum computing is still evolving, it holds strong potential for transforming combinatorial optimization in the future.
Quantum computing offers an alternative paradigm for addressing combinatorial optimization problems compared to classical computing. Despite recent hardware improvements, the execution of empirical quantum optimization experiments at scales known to be hard for state-of-the-art classical solvers is not yet in reach. In this work, we offer a different way to approach combinatorial optimization with near-term quantum computing. Motivated by the promising results observed in using quantum-enhanced Markov chain Monte Carlo (QeMCMC) for approximating complicated probability distributions, we combine ideas of sampling from the device with QeMCMC together with warm-starting and parallel tempering, in the context of combinatorial optimization. We demonstrate empirically that our algorithm recovers the global optima for instances of the Maximum Independent Set problem (MIS) up to 117 decision variables using 117 qubits on IBM quantum hardware. We show early evidence of a scaling advantage of our algorithm compared to similar classical methods for the chosen instances of MIS. MIS is practically relevant across domains like financial services and molecular biology, and, in some cases, already difficult to solve to optimality classically with only a few hundred decision variables.
Relaxation is a common way for dealing with combinatorial optimization problems. QRAO (Quantum Random Access Optimization) is a quantum-relaxation based optimizer that uses fewer qubits than the number of bits in the original problem, by encoding multiple variables per qubit using QRAC (Quantum Random Access Code). Reducing the number of qubits will alleviate physical noise (typically, decoherence) and, as a result, the quality of the binary solution of QRAO may be robust against noise, which is however unknown. In this paper, we numerically demonstrate that the mean approximation ratio of the (3, 1)-QRAC Hamiltonian, i.e., the Hamiltonian utilizing the encoding of 3 bits into 1 qubit by QRAC, is less affected by noise compared to the conventional Ising Hamiltonian used in quantum annealer and QAOA (Quantum Approximate Optimization Algorithm). Based on this observation, we discuss a plausible mechanism behind the robustness of QRAO under depolarizing noise. Finally, we assess the number of shots required to estimate the values of binary variables correctly under depolarizing noise and show that the (3, 1)-QRAC Hamiltonian requires less shots to achieve the same accuracy compared to the Ising Hamiltonian.
The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.
Variational quantum algorithms, and in particular variants of the variational quantum eigensolver, have been proposed as approaches to combinatorial optimization (CO) problems. With only shallow ansatz circuits, these methods are considered suitable for current noisy intermediate-scale quantum hardware. Yet, the resources required to train variational circuits often scale superpolynomially with problem size. In this study, we numerically investigate the practical implications of this scaling for CO problems, using Max-Cut and random QUBO instances as benchmarks. For a fixed computational budget, we compare the average performance of training shallow variational circuits, sampling with replacement, and greedy local search. We identify the minimum problem size at which the quantum algorithms consistently outperform random sampling, and for each size, we characterize their separation from greedy local search. Beyond average-case performance, we analyze correlations between algorithms across individual problem instances. These results strengthen the case for intuitive yet meaningful benchmarks of variational quantum algorithms for CO problems under realistic resource constraints.