Optimal resource cost for error mitigation

One of the central problems for near-term quantum devices is to understand their ultimate potential and limitations. We address this problem in terms of quantum error mitigation by introducing a framework taking into account the full expressibility of near-term devices, in which the optimal resource cost for the probabilistic error cancellation method can be formalized. We provide a general methodology for evaluating the optimal cost by connecting it to a resource-theoretic quantifier defined with respect to the noisy operations that devices can implement. We employ our methods to estimate the optimal cost in mitigating a general class of noise, where we obtain an achievable cost that has a generic advantage over previous evaluations, as well as a fundamental lower bound applicable to a broad class of noisy implementable operations. We improve our bounds for several noise models, where we give the exact optimal costs for the depolarizing and dephasing noise, precisely characterizing the overhead cost while offering an operational meaning to the resource measure in terms of error mitigation. Our result particularly implies that the heuristic approach presented by Temme et al. [K. Temme, S. Bravyi, and J. M. Gambetta, Phys. Rev. Lett. 119, 180509 (2017)] is optimal even in our extended framework, putting fundamental limitations on the advantage provided by the extra degrees of freedom inherent in near-term devices for this noise model.

Figure 1

The ratio of the resource cost obtained by the systematic decomposition with discrete basis operations in Ref. [8] to the achievable cost obtained in Theorem 1 with respect to the noise strength, where the error channel is chosen as $(1-\epsilon )\mathrm{id}+\epsilon \mathcal{V}$ with $\mathcal{V}$ being a Haar random unitary acting on single-qubit systems (left) and two-qubit systems (right). The black dotted line corresponds to ${\gamma }_{\mathrm{disc}}/{\gamma }_{\mathrm{cont}}=1$. The number of samples used for each figure is ${10}^3$.