[Paper Review] Error Mitigation for Deep Quantum Optimization Circuits by Leveraging Problem Symmetries

Ruslan Shaydulin, Alexey Galda|arXiv (Cornell University)|Jun 8, 2021

Quantum Computing Algorithms and Architecture46 references22 citations

TL;DR

This paper proposes an application-specific error mitigation technique for Quantum Approximate Optimization Algorithm (QAOA) circuits by projecting the quantum state into symmetry-restricted subspaces derived from classical symmetries of the objective function. By leveraging global bit-flip and qubit permutation symmetries, the method improves quantum state fidelity and increases the probability of sampling the optimal solution, achieving a 23% average fidelity improvement on IBM Quantum hardware with up to 5 qubits.

ABSTRACT

High error rates and limited fidelity of quantum gates in near-term quantum devices are the central obstacles to successful execution of the Quantum Approximate Optimization Algorithm (QAOA). In this paper we introduce an application-specific approach for mitigating the errors in QAOA evolution by leveraging the symmetries present in the classical objective function to be optimized. Specifically, the QAOA state is projected into the symmetry-restricted subspace, with projection being performed either at the end of the circuit or throughout the evolution. Our approach improves the fidelity of the QAOA state, thereby increasing both the accuracy of the sample estimate of the QAOA objective and the probability of sampling the binary string corresponding to that objective value. We demonstrate the efficacy of the proposed methods on QAOA applied to the MaxCut problem, although our methods are general and apply to any objective function with symmetries, as well as to the generalization of QAOA with alternative mixers. We experimentally verify the proposed methods on an IBM Quantum processor, utilizing up to 5 qubits. When leveraging a global bit-flip symmetry, our approach leads to a 23% average improvement in quantum state fidelity.

Motivation & Objective

• Address the challenge of high error rates and low gate fidelity in near-term quantum devices that limit QAOA performance.
• Overcome the performance plateau in QAOA objective values when increasing circuit depth due to noise and gate infidelities.
• Develop a low-overhead, application-specific error mitigation strategy tailored to the structure of optimization problems.
• Demonstrate the effectiveness of symmetry-based mitigation on real quantum hardware and noisy simulations.
• Extend symmetry verification techniques from quantum simulation to quantum optimization, particularly for QAOA with alternative mixers.

Proposed method

• Identify classical symmetries in the objective function (e.g., global bit-flip, qubit index permutation) that are preserved by the QAOA ansatz.
• Project the evolved quantum state into the +1 eigenspace of symmetry operators to suppress noise-induced errors.
• Perform projection either at the end of the circuit or throughout the evolution to maintain symmetry during unitary evolution.
• Use Pauli-Z based Hamiltonians to encode the objective function, enabling symmetry identification via Pauli string structure.
• Integrate the symmetry verification with existing error mitigation techniques, such as measurement error mitigation via calibration matrices.
• Implement and test the method on IBM Quantum processors (ibmq_bogota, ibmq_jakarta) using up to 5 qubits and noisy simulations with Qiskit.

Experimental results

Research questions

• RQ1 Can classical symmetries of the objective function be leveraged to improve QAOA state fidelity on noisy intermediate-scale quantum (NISQ) devices?
• RQ2 How does symmetry-based projection affect the expected objective value and probability of sampling the optimal solution in QAOA?
• RQ3 What is the relative overhead of symmetry verification, and how does it impact performance on current hardware with high error rates?
• RQ4 Can symmetry verification be combined effectively with other error mitigation techniques, such as measurement error mitigation?
• RQ5 Does the effectiveness of symmetry verification depend on the type of symmetry (e.g., bit-flip vs. qubit permutation) and circuit depth?

Key findings

• The proposed symmetry-based error mitigation improves quantum state fidelity by an average of 23% on IBM Quantum’s ibmq_bogota processor when leveraging global bit-flip symmetry.
• For qubit permutation symmetries, fidelity improvements are observed only in deeper circuits (e.g., p=3), indicating that relative overhead becomes favorable at higher depths.
• The method increases the probability of sampling the optimal binary string, thereby improving the success rate of QAOA on MaxCut problems.
• On noisy simulations, fidelity improvement from symmetry verification increases as the relative CNOT overhead decreases, supporting its viability for larger, deeper circuits.
• The approach is general and applicable to any objective function with symmetries, including QAOA with alternative mixers beyond the standard transverse field.
• Combining symmetry verification with measurement error mitigation leads to further performance gains, demonstrating compatibility with existing error mitigation stacks.