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arXiv:2506.09214 · Physical review. E

Classical algorithm inspired by the feedback-based algorithm for quantum optimization and local counterdiabatic driving

Takuya Hatomura

Published Oct 8, 2025• 0 citations
arXivHTMLPDF

Abstract

We propose a quantum-inspired classical algorithm for combinatorial optimization problems, named the counterdiabaticity-assisted classical algorithm for optimization (CACAO). In this algorithm, a solution of a given combinatorial optimization problem is heuristically searched with classical spin dynamics based on quantum Lyapunov control of local counterdiabatic driving. We compare the performance of CACAO with that of quantum time-evolution algorithms, i.e., quantum annealing, the feedback-based algorithm for quantum optimization (known as FALQON), and the counterdiabatic feedback-based quantum algorithm (known as CD-FQA). We also study the performance of CACAO applied to large systems up to 10000 spins.

DOI
10.1103/v7t1-42gh

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We propose a quantum-inspired classical algorithm for combinatorial optimization problems, named the counterdiabaticity-assisted classical algorithm for optimization (CACAO). In this algorithm, a solution of a given combinatorial optimization problem is heuristically searched with classical spin dynamics based on quantum Lyapunov control of local counterdiabatic driving. We compare the performance of CACAO with that of quantum time-evolution algorithms, i.e., quantum annealing, the feedback-based algorithm for quantum optimization (known as FALQON), and the counterdiabatic feedback-based quantum algorithm (known as CD-FQA). We also study the performance of CACAO applied to large systems up to 10000 spins.
In recent quantum algorithmic developments, a feedback-based approach has shown promise for preparing quantum many-body system ground states and solving combinatorial optimization problems. This method utilizes quantum Lyapunov control to iteratively construct quantum circuits. Here, we propose a substantial enhancement by implementing a protocol that uses ideas from quantum Lyapunov control and the counterdiabatic driving protocol, a key concept from quantum adiabaticity. Our approach introduces an additional control field inspired by counterdiabatic driving. We apply our algorithm to prepare ground states in one-dimensional quantum Ising spin chains. Comprehensive simulations demonstrate a remarkable acceleration in population transfer to low-energy states within a significantly reduced time frame compared to conventional feedback-based quantum algorithms. This acceleration translates to a reduced quantum circuit depth, a critical metric for potential quantum computer implementation. We validate our algorithm on the IBM cloud computer, highlighting its efficacy in expediting quantum computations for many-body systems and combinatorial optimization problems. Published by the American Physical Society 2024
Feedback-based quantum optimization is a quantum approach to combinatorial optimization. In this paper, we introduce the classical counterpart of feedback-based quantum optimization by using the quantum-classical correspondence of spin systems to discuss the possibility of quantum advantage. It also enables us to develop higher-order theory of a previously proposed classical approach to feedback-based quantum optimization. First, we compare the feedback-based algorithm for quantum optimization (FALQON) and its variant with their classical counterparts. Then, we perform benchmark tests of various quantum and classical algorithms with small-scale instances, and of classical algorithms with large-scale instances. Main findings are that (i) quantum algorithms can be advantageous to classical algorithms in terms of the quality of solutions, while classical algorithms tend to show faster convergence than quantum ones, and (ii) one of the classical algorithms discussed in this paper shows significant scalability for higher-order unconstrained binary optimization problems. These findings highlight the importance of quantumness and the usefulness of classical approaches.
Abstract We introduce a quantum algorithm that integrates counterdiabatic (CD) protocols with quantum Lyapunov control (QLC) to address combinatorial optimization problems. This approach offers versatility, allowing implementation as either a digital-analog or purely digital algorithm based on selected control strategies. By examining spin-glass Hamiltonians, we illustrate how the algorithm can explore alternative paths to enhance solution outcomes compared to conventional CD techniques. This method reduces dependence on extensive higher-order CD terms and on classical optimization techniques, making it more suitable for existing quantum computing platforms. The combination of digital compression via CD protocols and the adaptable nature of QLC methods positions this approach as a promising candidate for near-term quantum devices.
The quantum approximate optimization algorithm (QAOA) has proved to be an effective classical-quantum algorithm serving multiple purposes, from solving combinatorial optimization problems to finding the ground state of many-body quantum systems. Since the QAOA is an Ansatz-dependent algorithm, there is always a need to design Ans\"atze for better optimization. To this end, we propose a digitized version of the QAOA enhanced via the use of shortcuts to adiabaticity. Specifically, we use a counterdiabatic (CD) driving term to design a better Ansatz, along with the Hamiltonian and mixing terms, enhancing the global performance. We apply our digitized-CD QAOA to Ising models, classical optimization problems, and the PPP-spin model, demonstrating that it outperforms the standard QAOA in all cases we study.
We introduce a new hybrid quantum-classical algorithm inspired by an advanced control strategy known as model predictive control (MPC). This algorithm unifies the optimization-based design of variational quantum algorithms (VQAs) with the feedback-based design of feedback-based quantum algorithms (FQAs). Variational circuit parameters are optimized using a layer-wise receding horizon strategy, where observable measurements after every layer initialize a classically simulated dynamic model used to predict quantum state evolution and optimize over future parameterized gates. This hybrid algorithm can be used for applications such as ground state preparation and approximate combinatorial optimization, and presents an ideal use case for the integration of quantum computers with high-performance computing, where the latter resource can be used to increase the scale and efficiency of the predictions critical to MPC. We show through mathematical proof and numerical evidence that the MPC-based algorithm can be guaranteed to at least match the performance of FQAs. Through simulations on Max-Cut problems and a two-dimensional transverse-field Ising model, we demonstrate that relaxed implementations of the MPC-based algorithm can also provide improved performance in practice compared to an FQA.
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.