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arXiv:2407.17810 · Physical Review Research

Scalable circuit depth reduction in feedback-based quantum optimization with a quadratic approximation

D. Arai, Ken N. Okada, Yuichiro Nakano, Kosuke Mitarai, Keisuke Fujii

Published Jan 9, 2025• 7 citations
arXivHTMLPDF

Abstract

Combinatorial optimization problems are one of the areas where near-term noisy quantum computers may have practical advantage against classical computers. Recently, a feedback-based quantum optimization algorithm has been proposed by Magann , . The method explicitly determines quantum circuit parameters by feeding back measurement results thus avoids classical parameter optimization that is known to cause significant trouble in quantum approximate optimization algorithm, the well-studied near-term algorithm. Meanwhile, a significant drawback of the feedback-based quantum optimization is that it requires deep circuits, rendering the method unsuitable to noisy quantum devices. In this study we propose a feedback law for parameter determination by introducing the second-order approximation with respect to time interval, a hyperparameter in the feedback-based quantum optimization. This allows one to take larger time interval, leading to acceleration of convergence to solutions. In numerical simulations on the maximum cut problem we demonstrate that our proposal significantly reduces circuit depth, with its linear scaling with the problem size smaller by more than an order of magnitude. We expect that the feedback law proposed in this work may pave the way for feedback-based quantum optimization with near-term noisy quantum computers.

DOI
10.1103/physrevresearch.7.013035

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Combinatorial optimization problems are one of the areas where near-term noisy quantum computers may have practical advantage against classical computers. Recently, a feedback-based quantum optimization algorithm has been proposed by Magann , . The method explicitly determines quantum circuit parameters by feeding back measurement results thus avoids classical parameter optimization that is known to cause significant trouble in quantum approximate optimization algorithm, the well-studied near-term algorithm. Meanwhile, a significant drawback of the feedback-based quantum optimization is that it requires deep circuits, rendering the method unsuitable to noisy quantum devices. In this study we propose a feedback law for parameter determination by introducing the second-order approximation with respect to time interval, a hyperparameter in the feedback-based quantum optimization. This allows one to take larger time interval, leading to acceleration of convergence to solutions. In numerical simulations on the maximum cut problem we demonstrate that our proposal significantly reduces circuit depth, with its linear scaling with the problem size smaller by more than an order of magnitude. We expect that the feedback law proposed in this work may pave the way for feedback-based quantum optimization with near-term noisy quantum computers.
The prospect of using quantum computers to solve combinatorial optimization problems via the quantum approximate optimization algorithm (QAOA) has attracted considerable interest in recent years. However, a key limitation associated with QAOA is the need to classically optimize over a set of quantum circuit parameters. This classical optimization can have significant associated costs and challenges. Here we provide an expanded description of Lyapunov control-inspired strategies for quantum optimization, as presented in [Magann et al., Phys. Rev. Lett. 129, 250502 (2022)], that do not require any classical optimization effort. Instead, these strategies utilize feedback from qubit measurements to assign values to the quantum circuit parameters in a deterministic manner, such that the combinatorial optimization problem solution improves monotonically with the quantum circuit depth. Numerical analyses are presented that investigate the utility of these strategies towards MaxCut on weighted and unweighted 3-regular graphs, both in ideal implementations and in the presence of measurement noise. We also discuss how how these strategies compare with QAOA and how they may be used to seed QAOA optimizations in order to improve performance for near-term applications, and we explore connections to quantum annealing.
Feedback-based ALgorithm for Quantum OptimizatioN (FALQON) is a hybrid quantum-classical algorithm for solving combinatorial optimization problems, which circumvents classical parameter optimization but requires a deep quantum circuit. To reduce circuit depth, Arai et al. proposed second-order FALQON (SO-FALQON), achieving the best depth reduction among existing approaches. However, SO-FALQON brings a 2.3 times per-step measurement overhead, as it needs to calculate an additional second-order control coefficient. In this paper, inspired by the Backtracking Line Search (BLS) theory, we propose another method called BLS-FALQON, which not only reduces circuit depth to a comparable extent, but also achieves fewer measurements than SO-FALQON. Numerical simulations on max-cut problem with 8 to 20 vertices demonstrate that BLS-FALQON reduces the total measurement count by 37.7% compared to SO-FALQON, while maintaining a comparable circuit depth. Furthermore, we conduct real quantum hardware experiments on the Tianyan-176 quantum computer, which uses the zuchongzhi2 superconducting quantum processor, confirming that BLS-FALQON remains effective under real quantum hardware conditions.
The Feedback-based Algorithm for Quantum Optimization (FALQON) offers a deterministic alternative to variational quantum algorithms by bypassing classical optimization loops. However, maintaining convergence on large problem instances often requires restricting the time step, necessitating quantum circuit depths that exceed Noisy Intermediate-Scale Quantum (NISQ) hardware capabilities. This paper investigates the parameter transferability of second-order FALQON applied to the Max-Cut problem on 3-regular graphs. Through numerical experiments evaluating quantum circuits up to 16 layers on graphs up to 24 nodes, we demonstrate a highly advantageous scaling behavior: transferring feedback parameters optimized on small instances to larger target graphs yields significantly higher approximation ratios than natively optimizing the parameters directly on the larger graphs. This performance advantage arises because parameters trained on smaller instances can safely adopt aggressively larger time steps. By offloading the expensive parameter discovery phase to small-scale instances, this transfer strategy simultaneously reduces computational overhead and enhances the approximation ratio, thereby bringing FALQON closer to practical viability on near-term quantum architectures.
It is hoped that quantum computers will offer advantages over classical computers for combinatorial optimization. Here, we introduce a feedback-based strategy for quantum optimization, where the results of qubit measurements are used to constructively assign values to quantum circuit parameters. We show that this procedure results in an estimate of the combinatorial optimization problem solution that improves monotonically with the depth of the quantum circuit. Importantly, the measurement-based feedback enables approximate solutions to the combinatorial optimization problem without the need for any classical optimization effort, as would be required for the quantum approximate optimization algorithm. We demonstrate this feedback-based protocol on a superconducting quantum processor for the graph-partitioning problem MaxCut, and present a series of numerical analyses that further investigate the protocol's performance.
Feedback-based methods have gained significant attention as an alternative training paradigm for the Quantum Approximate Optimization Algorithm (QAOA) in solving combinatorial optimization problems such as MAX-CUT. In particular, Quantum Lyapunov Control (QLC) employs feedback-driven control laws that guarantee monotonic non-decreasing objective values, can substantially reduce the training overhead of QAOA, and mitigate barren plateaus. However, these methods might require long control sequences, leading to sub-optimal convergence rates. In this work, we propose a hybrid method that incorporates per-layer gradient estimation to accelerate the convergence of QLC while preserving its low training overhead and stability guarantees. By leveraging layer-wise gradient information, the proposed approach selects near-optimal control parameters, resulting in significantly faster convergence and improved robustness. We validate the effectiveness of the method through extensive numerical experiments across a range of problem instances and optimization settings.
Feedback-based quantum algorithms recently gained attention by demonstrating how quantum computers can be used to solve optimization problems without resorting to hybrid quantum-classical architectures. Recent progress made with Feedback-based Algorithm for Quantum Optimization (FALQON) variants significantly reduced the depth of the required circuits. In this work, we merge the time-rescaled and second-order FALQON variants into a unified framework. The results show an important improvement over past modifications, reducing circuit depth and allowing a more flexible choice of time-steps, while maintaining stable solutions. This FALQON variant makes it suitable for the NISQ era.