Mitigating Quantum Errors via Truncated Neumann Series

Abstract

Quantum gates and measurements on quantum hardware are inevitably subject to hardware imperfections that lead to quantum errors. Mitigating such unavoidable errors is crucial to explore the power of quantum hardware better. In this paper, we propose a unified framework that can mitigate quantum gate and measurement errors in computing quantum expectation values utilizing the truncated Neumann series. The essential idea is to cancel the effect of quantum error by approximating its inverse via linearly combining quantum errors of different orders produced by sequential applications of the quantum devices with carefully chosen coefficients. Remarkably, the estimation error decays exponentially in the truncated order, and the incurred error mitigation overhead is independent of the system size, as long as the noise resistance of the quantum device is moderate. We numerically test this framework for different quantum errors and find that the computation accuracy is substantially improved. Our framework possesses several vital advantages: it mitigates quantum gate and measurement errors in a unified manner, it neither assumes any error structure nor requires the tomography procedure to completely characterize the quantum errors, and most importantly, it is scalable. These advantages empower our quantum error mitigation framework to be efficient and practical and extend the ability of near-term quantum devices to deliver quantum applications.

Information recoverability of noisy quantum states

“…While the shadow destructivity and the retrieving cost capture different aspects of shadow information recoverability, a single unified measure that captures both aspects may be worth further studies. For future work, it would be interesting to see how the techniques presented in this work can be combined with quantum error correction [46][47][48] or quantum error mitigation [37,[41][42][43][44]. We also expect that our ideas could be applied to near-term quantum tasks or applications on noisy quantum devices [49].…”

An overview of quantum error mitigation formulas

“…Truncated Neumann series Recently, Wang et al [30] proposed error mitigation with truncated Neumann series. In this protocol, a formula resembling that in Eq.…”

“…( 5) can be found in Appendix B of Ref. [30]. Suppose 𝒰 = ℰ ∘ [U], where ℰ is the overall noise effect and [U] prepares the error-free state…”

Information recoverability of noisy quantum states

“…Now recall the retrieving cost, which quantifies the resources required to recover a certain piece of shadow information. The corresponding retrieving protocol can be considered as a method to mitigate errors, endowing the retrieving cost with a practical meaning in quantum error mitigation [23][24][25][26][27][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. In probabilistic error cancellation (PEC) (see, e.g., [23][24][25][26][27]), the sampling cost is used to depict the resource to be consumed, which has the same meaning as retrieving cost.…”

“…Meanwhile it establishes an optimal way of extracting noiseless classical information from noisy states, which could be useful for near-term quantum information processing tasks. For future work, it would be interesting to see how the techniques presented in this work can be combined with quantum error correction [44][45][46] or quantum mitigation [35,[39][40][41][42]. We also expect that our ideas could be further applied to near-term quantum tasks or applications on noisy quantum devices [47].…”