Noise in quantum hardware remains the biggest roadblock for the implementation of quantum computers. To fight the noise in the practical application of near-term quantum computers, instead of relying on quantum error correction which requires large qubit overhead, we turn to quantum error mitigation, in which we make use of extra measurements. Error extrapolation is an error mitigation technique that has been successfully implemented experimentally. Numerical simulation and heuristic arguments have indicated that exponential curves are effective for extrapolation in the large circuit limit with an expected circuit error count around unity. In this Article, we extend this to multi-exponential error extrapolation and provide more rigorous proof for its effectiveness under Pauli noise. This is further validated via our numerical simulations, showing orders of magnitude improvements in the estimation accuracy over single-exponential extrapolation. Moreover, we develop methods to combine error extrapolation with two other error mitigation techniques: quasi-probability and symmetry verification, through exploiting features of these individual techniques. As shown in our simulation, our combined method can achieve low estimation bias with a sampling cost multiple times smaller than quasi-probability while without needing to be able to adjust the hardware error rate as required in canonical error extrapolation.

量子硬件中的噪声仍然是实现量子计算机的最大障碍。为了在近期量子计算机的实际应用中与噪声抗争，我们不再依赖于需要大量量子位开销的量子误差校正，而转向了量子误差缓解技术，其中我们使用了额外的测量方法。错误外推是一种已通过实验成功实施的错误缓解技术。数值模拟和启发式论证表明，指数曲线可有效地在较大的电路极限中进行外推，且预期电路误差计数约为1。在本文中，我们将其扩展到多指数误差外推法，并为其在保利噪声下的有效性提供了更严格的证据。我们的数值模拟进一步验证了这一点，显示了单指数外推的估计精度提高了几个数量级。此外，我们开发了将错误外推与其他两种错误缓解技术相结合的方法：通过利用这些单独技术的功能，来实现准概率和对称性验证。如我们的仿真所示，我们的组合方法可以实现低估计偏差，且采样成本比准概率小数倍，而无需能够按照规范误差外推法调整硬件误差率。通过利用这些单独技术的功能。如我们的仿真所示，我们的组合方法可以实现低估计偏差，且采样成本比准概率小数倍，而无需能够按照规范误差外推法调整硬件误差率。通过利用这些单独技术的功能。如我们的仿真所示，我们的组合方法可以实现低估计偏差，且采样成本比准概率小数倍，而无需能够按照规范误差外推法调整硬件误差率。