Fig. 2: Adaptive KIK error mitigation.
From: Adaptive quantum error mitigation using pulse-based inverse evolutions
The estimate \({\left\langle A\right\rangle }_{{{{\rm{KIK}}}}}^{(M)}\) of a noiseless expectation value involves the execution of the circuits shown in (a) and (b). In particular, the survival probability μ is used to evaluate the coefficients \({a}_{{{{\rm{Adap}}}},m}^{(M)}[g(\mu )]\), for adaptive error mitigation (see main text for details). The green curve in (c, d) is the plot of λ−1/2 and it contains the eigenvalues of the operation that effectively suppresess the error channel (\({\left({{{{\mathcal{K}}}}}_{{{{\rm{I}}}}}{{{\mathcal{K}}}}\right)}^{-\frac{1}{2}}\) in Eq. (7)). The black dashed curves represent the polynomial approximations \(\mathop{\sum }\nolimits_{m = 0}^{M}{a}_{m}^{(M)}{\lambda }^{m}\) that appear in the integrand of (12), for third-order mitigation (M = 3). The better these approximations, the more accurate the corresponding error mitigation. This accuracy is related to the argument g(μ) in the optimal coefficients \({a}_{m}^{(3)}={a}_{{{{\rm{Adap}}}},m}^{(3)}[g(\mu )]\), which are obtained by minimizing (12) over the interval [g(μ),1]. Figures (c) and (d) correspond to g(μ) = μ2 and g(μ) = μ, respectively. In (c), λ−1/2 is very well approximated by \(\mathop{\sum }\nolimits_{m = 0}^{3}{a}_{{{{\rm{Adap}}}},m}^{(3)}[{\mu }^{2}]{\lambda }^{m}\) in the interval where the eigenvalues of \({\left({{{{\mathcal{K}}}}}_{I}{{{\mathcal{K}}}}\right)}^{-\frac{1}{2}}\) are distributed (jagged line in the background). In (d), the interval [μ,1] is too small to cover the full eigenvalue distribution and thus \(\mathop{\sum }\nolimits_{m = 0}^{3}{a}_{{{{\rm{Adap}}}},m}^{(3)}[\mu ]{\lambda }^{m}\) starts to deviate significantly from the green curve, as shown by the gray ellipse. The red curve corresponds to the Taylor polynomial \(\mathop{\sum }\nolimits_{m = 0}^{3}{a}_{{{{\rm{Adap}}}},m}^{(3)}[1]{\lambda }^{m}\) and is the less effective approximation, as seen in both (c) and (d).
