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arXiv:2408.00557 · Physical Review Research

End-to-end protocol for high-quality quantum approximate optimization algorithm parameters with few shots

Tianyi Hao, Zichang He, Ruslan Shaydulin, Jeffrey M. Larson, Marco Pistoia

Published Aug 1, 2025• 3 citations
arXivHTMLPDF

Abstract

The quantum approximate optimization algorithm (QAOA) is a quantum heuristic for combinatorial optimization that has been demonstrated to scale better than state-of-the-art classical solvers for some problems. For a given problem instance, QAOA performance depends crucially on the choice of the parameters. While average-case optimal parameters are available in many cases, meaningful performance gains can be obtained by fine-tuning these parameters for a given instance. This task is especially challenging, however, when the number of circuit executions (shots) is limited. In this work, we develop an end-to-end protocol that combines multiple parameter settings and fine-tuning techniques. We use large-scale numerical experiments to optimize the protocol for the shot-limited setting and observe that optimizers with the simplest internal model (linear) perform best. We implement the optimized pipeline on a trapped-ion processor using up to 32 qubits and 5 QAOA layers, and we demonstrate that the pipeline is robust to small amounts of hardware noise. To the best of our knowledge, these are the largest demonstrations of QAOA parameter fine-tuning on a trapped-ion processor in terms of two-qubit gate count.

DOI
10.1103/24gg-7p8z

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The quantum approximate optimization algorithm (QAOA) is a quantum heuristic for combinatorial optimization that has been demonstrated to scale better than state-of-the-art classical solvers for some problems. For a given problem instance, QAOA performance depends crucially on the choice of the parameters. While average-case optimal parameters are available in many cases, meaningful performance gains can be obtained by fine-tuning these parameters for a given instance. This task is especially challenging, however, when the number of circuit executions (shots) is limited. In this work, we develop an end-to-end protocol that combines multiple parameter settings and fine-tuning techniques. We use large-scale numerical experiments to optimize the protocol for the shot-limited setting and observe that optimizers with the simplest internal model (linear) perform best. We implement the optimized pipeline on a trapped-ion processor using up to 32 qubits and 5 QAOA layers, and we demonstrate that the pipeline is robust to small amounts of hardware noise. To the best of our knowledge, these are the largest demonstrations of QAOA parameter fine-tuning on a trapped-ion processor in terms of two-qubit gate count.
Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.
The quantum approximate optimization algorithm (QAOA) is a powerful heuristic that seeks to solve combinatorial optimization problems using quantum hardware and classical optimization in tandem. Various methods exist to train the parameterized quantum circuits that serve as an ansatz in QAOA. However, which method works best to identify optimal angles for a given problem instance remains poorly understood, especially at utility-scale, i.e., 100100100 qubits or more. In this work, we address this challenge through utility-scale benchmarks from which we distill operational guidance for QAOA practitioners. First, we investigate approximation techniques, such as matrix product states and Pauli propagation, to find optimal angles. Second, we train QAOA on small-scale representative problems and transfer the angles to larger ones. We then validate the results on quantum hardware for utility-scale problem instances that can be meaningfully executed. In this way, we identify insights for QAOA angle setting strategies that work best for problems at the utility scale, including as a function of resource cost for the search. Crucially, the operational implications we draw from our benchmarks will help quantum optimization practitioners execute QAOA end-to-end pipelines efficiently on current and future hardware.
Quantum Approximate Optimization Algorithm (QAOA) is one of the most promising quantum heuristics for combinatorial optimization. While QAOA has been shown to perform well on small-scale instances and to provide an asymptotic speedup over state-of-the-art classical algorithms for some problems, fault-tolerance is understood to be required to realize this speedup in practice. The low resource requirements of QAOA make it particularly suitable to benchmark on early fault-tolerant quantum computing (EFTQC) hardware. However, the performance of QAOA depends crucially on the choice of the free parameters in the circuit. The task of setting these parameters is complicated in the EFTQC era by the large overheads, which preclude extensive classical optimization. In this paper, we summarize recent advances in parameter setting in QAOA and show that these advancements make EFTQC experiments with QAOA practically viable.
Quantum approximate optimization algorithm (QAOA) is a promising quantum-classical hybrid technique to solve NP-hard problems in near-term gate-based noisy quantum devices. In QAOA, the gate parameters of a parameterized quantum circuit (PQC) are varied by a classical optimizer to generate a quantum state with a significant support to the optimal solution. The existing analysis fails to consider nonidealities in the qubit quality i.e., short lifetime and imperfect gate operations in a realistic quantum hardware. In this article, we study the impact of various noise sources on the performance of QAOA both in simulation and on a real quantum computer from IBM. Our analysis indicates that QAOA performance is noise-sensitive (especially higher-depth QAOA instances). Therefore, the optimal number of stages (p-value) for any QAOA instance is limited by the noise in the target hardware as opposed to the current perception that QAOA will provide monotonically better performance with higher-depth. We show that the two-qubit gate error has to be decreased by more than 75% of the current state-of-the-art levels to attain a performance within 10% of the maximum value for the lowest-depth QAOA.
We introduce a novel quantum optimization paradigm: the Fixed-Parameter-Count Quantum Approximate Optimization Algorithm (FPC-QAOA). It is a scalable variational framework that maintains a constant number of trainable parameters regardless of the number of qubits, Hamiltonian complexity, or circuit depth. By separating schedule function optimization from circuit digitization, FPC-QAOA enables accurate schedule approximations with minimal parameters while supporting arbitrarily deep digitized adiabatic evolutions, constrained only by NISQ hardware capabilities. This separation allows depth to scale without expanding the classical search space, mitigating overparameterization and optimization challenges typical of deep QAOA circuits, such as barren plateaus-like behaviors. We benchmark FPC-QAOA on random MaxCut instances and the Tail Assignment Problem, achieving performance comparable to or better than standard QAOA with nearly constant classical effort and significantly fewer quantum circuit evaluations. Experiments on the IBM Kingston superconducting processor with up to 50 qubits confirm robustness and hardware efficiency under realistic noise. These results position FPC-QAOA as a practical and scalable paradigm for variational quantum optimization on near-term quantum devices.
Abstract We introduce a novel quantum optimization paradigm: the Fixed-Parameter-Count Quantum Approximate Optimization Algorithm (FPC-QAOA). It is a scalable variational framework that maintains a constant number of trainable parameters regardless of the number of qubits, Hamiltonian complexity, or circuit depth. By separating schedule function optimization from circuit digitization, FPC-QAOA enables accurate schedule approximations with minimal parameters while supporting arbitrarily deep digitized adiabatic evolutions, constrained only by NISQ hardware capabilities. This separation allows depth to scale without expanding the classical search space, mitigating overparameterization and optimization challenges typical of deep QAOA circuits, such as barren plateaus-like behauvious. We benchmark FPC-QAOA on random MaxCut instances and the Tail Assignment Problem, achieving performance comparable to or better than standard QAOA with nearly constant classical effort and significantly fewer quantum circuit evaluations. Experiments on the IBM Kingston superconducting processor with up to 50 qubits confirm robustness and hardware efficiency under realistic noise. These results position FPC-QAOA as a practical and scalable paradigm for variational quantum optimization on near-term quantum devices.