Fig. 3: Numerical results for the biases of the noisy cancellations with separate and direct methods.
From: Noisy probabilistic error cancellation and generalized physical implementability
a, b, d, and e, Colored filling regions denote the biases of expectations for different Pauli operators, and solid curves denote the median of the biases of the operator expectations. Dashed curves denote the distance between the imperfectly canceled error \({{{{\mathcal{E}}}}}_{\lambda }^{-1}\circ {{{\mathcal{E}}}}\) and \({{{\mathcal{I}}}}\) in the implementability function \({p}_{{{{\mathcal{Q}}}}}({{{\mathcal{I}}}}-{{{{\mathcal{E}}}}}_{\lambda }^{-1}\circ {{{\mathcal{E}}}})\), which upper bounds the bias of the operator expectations by Eq. (26). Dotted curves denote the CPTP upper bound of \({p}_{{{{\mathcal{Q}}}}}({{{\mathcal{I}}}}-{{{{\mathcal{E}}}}}_{\lambda }^{-1}\circ {{{\mathcal{E}}}})\), see Eqs. (33) and (34) for the direct and separate cancellation methods, respectively. The error \({{{{\mathcal{E}}}}}_{j}\) is assumed to be the same for each layer, \({{{{\mathcal{E}}}}}_{j}={{{{\mathcal{E}}}}}_{0}\), and the error \({{{{\mathcal{E}}}}}_{0}\) as well as the noises \({{{{\mathcal{N}}}}}_{i}\) on Pauli gates \({{{{\mathcal{P}}}}}_{i}\) are randomly sampled from the Pauli-Lindblad error model (21), with a fixed single-layer error rate λ = ∑iλi. c and f, Dotted dashed curves denote the implementability function \({p}_{{{{\mathcal{Q}}}}}({{{{\mathcal{E}}}}}_{\lambda }^{-1}\circ {{{\mathcal{E}}}})\) of the imperfectly canceled error for direct and separate cancellation methods. a--c, For a small error rate λ = 0.05, since the imperfectly canceled errors \({{{{\mathcal{E}}}}}_{\lambda }^{-1}\circ {{{\mathcal{E}}}}\) (c) are CPTP for circuits with a layer number less than 20, the biases of both the separate (a) and direct (b) cancellation methods are limited by the CPTP upper bound, Eqs. (33) and (34), and the bias of the direct cancellation is smaller than the separate cancellation method. d–f For a large error rate λ = 0.5, since the imperfectly canceled error \({{{{\mathcal{E}}}}}_{\lambda }^{-1}\circ {{{\mathcal{E}}}}\) for the direct cancellation method (f) will eventually not be CPTP with the cumulation of errors \({{{\mathcal{E}}}}={{{{\mathcal{E}}}}}_{0}^{L}\). The bias increases exponentially and surpasses the CPTP bound (e), as the layer number L grows. However, the error \({{{{\mathcal{E}}}}}_{\lambda }^{-1}\circ {{{\mathcal{E}}}}\) for the separate cancellation method (f) is still CPTP, since each the error of individual layer \({{{{\mathcal{E}}}}}_{0\lambda }^{-1}\circ {{{{\mathcal{E}}}}}_{0}\) is CPTP. The bias of separate cancellation is still bounded by the CPTP upper bound (f). For details of simulation, see Supplementary Note VI.
