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arXiv:2405.10451 · Physical Review A

Simulation of a feedback-based algorithm for quantum optimization for a realistic neutral-atom system with an optimized small-angle controlled-phase gate

S. X. Li, W. L. Mu, Jia-Bin You, X. Q. Shao

Published Jun 10, 2024• 7 citations
arXivHTMLPDF

Abstract

In contrast to the classical optimization process required by the quantum approximate optimization algorithm, FALQON, a feedback-based algorithm for quantum optimization [A. B. Magann et al., Phys. Rev. Lett. 129, 250502 (2022)], enables one to obtain approximate solutions to combinatorial optimization problems without any classical optimization effort. In this study, we leverage the specifications of a recent experimental platform for the neutral-atom system [Z. Fu et al., Phys. Rev. A 105, 042430 (2022)] and present a scheme to implement an optimally tuned small-angle controlled-phase gate. By examining the two- to four-qubit FALQON algorithms in the MaxCut problem and considering the spontaneous emission of the neutral atomic system, we observe that the performance of FALQON implemented with small-angle controlled-phase gates exceeds that of FALQON utilizing controlled-$Z$ gates. This approach has the potential to significantly simplify the logic circuit required to simulate FALQON and effectively address the MaxCut problem, which may pave the way for the experimental implementation of near-term noisy intermediate-scale quantum algorithms with neutral-atom systems.

DOI
10.1103/physreva.109.062603

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In contrast to the classical optimization process required by the quantum approximate optimization algorithm, FALQON, a feedback-based algorithm for quantum optimization [A. B. Magann et al., Phys. Rev. Lett. 129, 250502 (2022)], enables one to obtain approximate solutions to combinatorial optimization problems without any classical optimization effort. In this study, we leverage the specifications of a recent experimental platform for the neutral-atom system [Z. Fu et al., Phys. Rev. A 105, 042430 (2022)] and present a scheme to implement an optimally tuned small-angle controlled-phase gate. By examining the two- to four-qubit FALQON algorithms in the MaxCut problem and considering the spontaneous emission of the neutral atomic system, we observe that the performance of FALQON implemented with small-angle controlled-phase gates exceeds that of FALQON utilizing controlled-ZZZ gates. This approach has the potential to significantly simplify the logic circuit required to simulate FALQON and effectively address the MaxCut problem, which may pave the way for the experimental implementation of near-term noisy intermediate-scale quantum algorithms with neutral-atom systems.
Feedback-based quantum algorithms have recently emerged as potential methods for approximating the ground states of Hamiltonians . One such algorithm, the feedback-based algorithm for quantum optimization (FALQON), is specifically designed to solve quadratic unconstrained binary optimization problems. Its extension, the feedback-based algorithm for quantum optimization with constraints (FALQON-C), was introduced to handle constrained optimization problems with equality and inequality constraints . In this work, we extend the feedback-based quantum algorithms framework to address a broader class of constraints known as invalid configuration (IC) constraints, which explicitly prohibit specific configurations of decision variables. We first present a transformation technique that converts the constrained optimization problem with invalid configuration constraints into an equivalent unconstrained problem by incorporating a penalizing term into the cost function. Then, leaning upon control theory, we propose an alternative method tailored for feedback-based quantum algorithms that directly tackles IC constraints without requiring slack variables. Our approach introduces a new operator that encodes the optimal feasible solution of the constrained optimization problem as its ground state. Then, a controlled quantum system based on the Lyapunov control technique is designed to ensure convergence to the ground state of this operator. Two approaches are introduced in the design of this operator to address IC constraints: the folded spectrum approach and the deflation approach. These methods eliminate the need for slack variables, significantly reducing the quantum circuit depth and the number of qubits required. We show the effectiveness of our proposed algorithms through numerical simulations.
Feedback-based adaptive quantum optimization (FALQON) is a promising approach for solving combinatorial problems on noisy intermediate-scale quantum (NISQ) devices, requiring only single circuit evaluations per layer. However, standard FALQON relies on fixed hyperparameters that severely limit convergence speed, requiring hundreds to thousands of layers for acceptable solutions. This paper proposes Optimal FALQON, an optimization-based formulation that treats the per-layer time step (δkδ_kδk​) and scaling factor (MkM_kMk​) as decision variables optimized via classical methods. We present a comprehensive empirical study on all 94 non-isomorphic 3-regular graphs with 12 vertices, comparing Optimal FALQON with standard FALQON and multiple QAOA variants. Results demonstrate statistically significant improvements in success probability, evaluation efficiency, and depth-normalized cost across the evaluated benchmarks. Furthermore, initializing QAOA with parameters from Optimal FALQON yields superior warm-start performance compared to fixed initialization.
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.
The feedback-based algorithm for quantum optimization (FALQON), which is an extension of the quantum approximate optimization algorithm (QAOA), is a promising approach for finding approximate solutions to combinatorial optimization problems. The FALQON avoids the optimization of variational parameters by using information derived from iterative measurements. To solve a constrained combinatorial optimization problem, it is sometimes more efficient to use an XY mixer rather than the conventional X mixer. We applied the FALQON with the XY mixer to graph coloring and investigated the expectation value of the problem Hamiltonian, success probability of graph coloring, and fair sampling for several small-scale problems. The FALQON with the XY mixer exhibited better performance than that with the X mixer.
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 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.