Quantum Error Mitigation using Symmetry Expansion

TL;DR

This paper introduces a general framework called symmetry expansion for quantum error mitigation, which surpasses symmetry verification by reducing estimation bias and balancing sampling costs, demonstrated through simulations on the Fermi-Hubbard model.

Contribution

The paper develops a broad symmetry expansion framework that extends symmetry verification, enabling improved error mitigation with lower bias and manageable sampling costs.

Findings

Symmetry expansion schemes can reduce estimation bias 6 to 9 times compared to symmetry verification.

The sampling cost for bias reduction is only 2 to 6 times higher than symmetry verification.

The formalism applies to both inherent and engineered symmetries, including schemes like exponential error suppression.

Abstract

Even with the recent rapid developments in quantum hardware, noise remains the biggest challenge for the practical applications of any near-term quantum devices. Full quantum error correction cannot be implemented in these devices due to their limited scale. Therefore instead of relying on engineered code symmetry, symmetry verification was developed which uses the inherent symmetry within the physical problem we try to solve. In this article, we develop a general framework named symmetry expansion which provides a wide spectrum of symmetry-based error mitigation schemes beyond symmetry verification, enabling us to achieve different balances between the estimation bias and the sampling cost of the scheme. We show that certain symmetry expansion schemes can achieve a smaller estimation bias than symmetry verification through cancellation between the biases due to the detectable and undetectable noise components. A practical way to search for such a small-bias scheme is introduced. By numerically simulating the Fermi-Hubbard model for energy estimation, the small-bias symmetry expansion we found can achieve an estimation bias 6 to 9 times below what is achievable by symmetry verification when the average number of circuit errors is between 1 to 2. The corresponding sampling cost for random shot noise reduction is just 2 to 6 times higher than symmetry verification. Beyond symmetries inherent to the physical problem, our formalism is also applicable to engineered symmetries. For example, the recent scheme for exponential error suppression using multiple noisy copies of the quantum device is just a special case of symmetry expansion using the permutation symmetry among the copies.