Optimization of Richardson extrapolation for quantum error mitigation

Quantum error mitigation is a key concept for the development of practical applications based on current noisy intermediate-scale quantum devices. One of the most promising methods is Richardson extrapolation to the zero noise limit. While its main idea is rather simple, the full potential of Richardson extrapolation has not been completely uncovered yet. We give an in-depth analysis of the relevant parameters of Richardson extrapolation and propose an optimized protocol for its implementation. This protocol allows for a precise control of the increase in statistical uncertainty and lays the foundation for a significant improvement of the mitigation performance achieved by increasing the number of nodes. Furthermore, we present a set of nodes that, on average, outperforms the linear, exponential, or Chebyshev nodes frequently used for Richardson extrapolation without requiring any additional resources.

Figure 1

Example for Richardson extrapolation with $n=3$ for a noisy expectation value which decays exponentially $E_{\lambda }=E^{*}{e}^{-\lambda }$ and has units according to $E^{*}$. The black diamond is the result without mitigation $R_0$ and the red square is the mitigated result $R_3$, both obtained with the same sampling budget. The colored interval indicates the standard deviation of the extrapolation using Lagrange polynomials. The error bars are the standard deviations of the samples $E_{{\lambda }_0}\left(x_j\right)$ adjusted according to Eqs. (8) and (11). The nodes lie equidistant.

Figure 2

Comparison of the different spacing methods for $n=5$ and $\mathrm{\Lambda }=10$. The scaling methods are tilted Chebyshev (T), extremal Chebyshev (C), exponential (E), and equidistant (L).

Figure 3

The ratio $\frac{(n+1)!}{C_n}$ for different values of $\mathrm{\Lambda }$ and $n$ for the spacing methods $x_j^T$, $x_j^C$, $x_j^E$, and $x_j^L$.

Figure 4

Schematic representation of the procedure for Richardson extrapolation.

Figure 5

Richardson extrapolation used on the Markovian noise model in Eq. (19). The bias is shown over the minimal noise strength ${\lambda }_0$ for $\mathrm{\Lambda }=8$ and 64 and $n=5$. The spacing methods are tilted Chebyshev (blue, solid), extremal Chebyshev (yellow, dashed), exponential (green, dotted), and equidistant (red, dot-dashed). The black solid line shows the result without mitigation $\left|\mathrm{Bias}\right[{\widehat{R}}_0\left]\right|$. The $y$ axis is in units of $E^{*}$. For the figures, we assume $E^{*}$ to be nonzero and positive.

Figure 6

Comparison of the bias of Richardson extrapolation on Markovian noise for fixed ${\lambda }_0=0.4$ and different $n$. The overhead in measurements is ${\mathrm{\Lambda }}^2=4^2,{32}^2$, and ${256}^2$, respectively. Note that the smaller the bias the better the QEM protocol works. The different spacing methods are tilted Chebyshev (blue, up-triangle), extremal Chebyshev (yellow, diamond), exponential (green, down-triangle), and equidistant (red, circle). The black square for $n=0$ is the bias without mitigation $\left|\mathrm{Bias}\right[{\widehat{R}}_0\left]\right|$. The $y$ axis is in units of $E^{*}$.

Figure 7

Comparison of the bias of Richardson extrapolation on the noise model in Eq. (20) with $\eta =0.1$ and ${\lambda }_0=0.4$ for different $n$. The overhead in measurements is ${\mathrm{\Lambda }}^2=4^2,{32}^2$, and ${256}^2$, respectively. Note that the smaller the bias the better the QEM protocol works. The different spacing methods are tilted Chebyshev (blue, up-triangle), extremal Chebyshev (yellow, diamond), exponential (green, down-triangle), and equidistant (red, circle). The black square for $n=0$ is the bias without mitigation $\left|\mathrm{Bias}\right[{\widehat{R}}_0\left]\right|$. The $y$ axis is in units of $E^{*}$.

Figure 8

The bias of mitigation for different levels of non-Markovianity. The noise is purely Markovian for $\eta =0$ and highly non-Markovian for $\eta =1$. There are three different values for $\mathrm{\Lambda }$—4 (purple), 32 (brown), and 256 (blue)—and two for $n$: 4 (solid) and 9 (dashed). The black solid line is the result without mitigation $\left|\mathrm{Bias}\right[{\widehat{R}}_0\left]\right|$. The dotted green line is the fake-node approach with $S\left(x\right)=x^2$ for $\mathrm{\Lambda }=4$ and $n=9$. For some $\eta$, there is a transition between over- and underestimation of the true expectation value $E^{*}$ which is why we see “dips,” where the bias briefly becomes zero. The $y$ axis is in units of $E^{*}$.

Figure 9

Comparison of the bias of Richardson extrapolation on the noise model in Eq. (20) with $\eta =0.9$ and ${\lambda }_0=0.4$ for different $n$. The overhead in measurements is ${\mathrm{\Lambda }}^2=4^2,{32}^2$, and ${256}^2$, respectively. The different spacing methods are tilted Chebyshev (blue, up-triangle), extremal Chebyshev (yellow, diamond), exponential (green, down-triangle), and equidistant (red, circle). The black square for $n=0$ is the bias without mitigation $\left|\mathrm{Bias}\right[{\widehat{R}}_0\left]\right|$. The $y$ axis is in units of $E^{*}$.