Fig. 4: Numerical results for the biases of the inaccurate error model with under-mitigated and over-mitigated cases.
From: Noisy probabilistic error cancellation and generalized physical implementability
a, b Colored filling regions denote the bias of expectations for different Pauli operators, and solid curves denote the median of the bias of the operator expectations. Dashed curves denote the distance between the mitigated error \({\hat{{{{\mathcal{E}}}}}}^{-1}\circ {{{\mathcal{E}}}}\) and \({{{\mathcal{I}}}}\) characterized by the implementability function \({p}_{{{{\mathcal{Q}}}}}({{{\mathcal{I}}}}-{\hat{{{{\mathcal{E}}}}}}^{-1}\circ {{{\mathcal{E}}}})\), which upper bounds the bias of the operator expectations in Eq. (26). Dotted curves denote the CPTP upper bound of \({p}_{{{{\mathcal{Q}}}}}({{{\mathcal{I}}}}-{\hat{{{{\mathcal{E}}}}}}^{-1}\circ {{{\mathcal{E}}}})\), as shown in Eqs. (45) and (46) for the under-mitigated and over-mitigated errors, respectively. The mitigated error channel \({{{{\mathcal{E}}}}}_{i}^{-1}\circ {{{{\mathcal{E}}}}}_{i}\) for each layer is assumed to be the same, \({{{{\mathcal{E}}}}}_{i}={{{{\mathcal{E}}}}}_{0}\), and randomly sampled from the Pauli-Lindblad error model in Eq. (21), with a fixed single-layer error rate \(\Delta {\lambda }^{{{{\rm{under}}}}}=0.05\Delta {\lambda }_{i}^{{{{\rm{under}}}}}\ge 0\) for the under-mitigated error channel (a), and \(\Delta {\lambda }_{i}^{{{{\rm{over}}}}}=-\Delta {\lambda }_{i}^{{{{\rm{under}}}}}\) for over-mitigated error channel (b). c Dotted dashed curves denote the implementability function \({p}_{{{{\mathcal{Q}}}}}({\hat{{{{\mathcal{E}}}}}}^{-1}\circ {{{\mathcal{E}}}})\) for the under-mitigated and over-mitigated errors. Since the under-mitigated error \({\hat{{{{\mathcal{E}}}}}}^{-1}\circ {{{\mathcal{E}}}}\) is CPTP, the bias is lower than the CPTP upper bound as shown in (a). In contrast, the over-mitigated error channel \({\hat{{{{\mathcal{E}}}}}}^{-1}\circ {{{\mathcal{E}}}}\) is not CPTP, the bias increases exponentially as the layer number L grows, which is bounded by the non-CPTP upper bound in Eq. (46) as shown in (b). For details of simulation, see Supplementary Note VI.
