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.
Highlights
The ultimate goal of implementing a fully faulttolerant quantum error correction scheme for quantum algorithms with provable speed-up may still be years away
When we look at all symmetry expansion schemes including symmetry verifications, we see that the small-bias scheme we found in Section 7.2 can achieve the lowest bias
Different symmetry expansions correspond to different sampling weight distributions among the symmetry operators, with symmetry verification corresponding to the uniform weight distribution over the full symmetry group
Summary
The ultimate goal of implementing a fully faulttolerant quantum error correction scheme for quantum algorithms with provable speed-up may still be years away. Symmetry verification is one such error-mitigation technique that projects the noisy output quantum state back into the symmetry subspace defined by the physical problem we try to solve [12, 13]. Its core idea relies on the permutation symmetry among the copies, but the way it is implemented does not involve projecting the noisy state into the correct symmetry subspace, and does not fall within the symmetry verification framework. We will provide a general framework named symmetry expansion that encompasses a much wider range of symmetry-based error mitigation schemes beyond symmetry verification. We will discuss ways to search within this wide range of symmetry expansion schemes and identify one that can outperform symmetry verification, just as virtual distilla-.
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