Investigating the limits of randomized benchmarking protocols

In this paper, we analyze the performance of randomized benchmarking protocols on gate sets under a variety of realistic error models that include systematic rotations, amplitude damping, leakage to higher levels, and $1/f$ noise. We find that, in almost all cases, benchmarking provides better than a factor-of-2 estimate of average error rate, suggesting that randomized benchmarking protocols are a valuable tool for verification and validation of quantum operations. In addition, we derive models for fidelity decay curves under certain types of non-Markovian noise models such as $1/f$ and leakage errors. We also show that, provided the standard error of the fidelity measurements is small, only a small number of trials are required for high-confidence estimation of gate errors.

Figure 1

Distance of maps from a depolarizing channel. For a range of error rates, unitary channels (red squares) were farthest from and amplitude-damping channels (blue circles) closest to depolarizing channels of the same error rate. Random maps (green triangles) typically fell between these extremes. The distance was measured using the diamond norm distance from a depolarizing channel, and $r$ is the average error rate.

Figure 2

SRB with fixed unitary Markovian noise for error rates $r={10}^{-4}$ (red squares), $r={10}^{-3}$ (green triangles), and $r={10}^{-2}$ (blue circles). (a) Convergence of the confidence $C$ (see text for details). The black line corresponds to the Hoeffding bound method explained in the text. (b) Convergence of the accuracy $\mu$. Note the rapid convergence of the estimate to within a factor of 2 of the average error rate by $K=100$.

Figure 3

The accuracy of several Markovian models with $K=10000$ sequences. Different random unitaries (blue circles), fixed random unitaries (black diamonds), generator-dependent unitaries (orange six-pointed stars), amplitude damping (green triangles), Gaussian unitaries (red squares), and slow drift (white five-pointed stars). All errors except generator-dependent unitary errors were estimated to within $25\%$ of $r$, with amplitude-damping noise determined significantly better. Generator-dependent unitary noise was estimated to within $50\%$ of $r$. The large horizontal spread at low error rates is due simply to the precision of the procedure used for generating random unitary channels of fixed error rate.

Figure 4

Simulated Ramsey experiments for $1/f$ noise. (a) The coherence $\sigma \left(t\right)=2\left|{\rho }_{12}\left(t\right)\right|$ is plotted versus time $t/t_g$ where $t_g$ is the duration of a gate. Each curve corresponds to a different noise power. The dashed lines interpolate the numerical data and every tenth data point is shown. (b) For each decay in (a) we extract $t_g$ as the time at which $\sigma =1/e$ and plot the corresponding average gate fidelity $F_g=[2+\sigma \left(t_g\right)]/3$ (blue points). The solid red curve is the gate fidelity for stochastic noise with exponential decay $\sigma \left(t\right)=e^{-t/T_2^{*}}$ of the coherences. The solid blue curve is the (analytic) gate fidelity for $1/f$ noise: $F_g=(2+e^{-{(t_g/T_2^{*})}^2})/3$.

Figure 5

Fidelity decay in the presence of $1/f$ noise ($A^{\prime }=2\times {10}^{-8}\to r\sim {10}^{-3}$) with application of random Clifford gates (blue circles, mean of four experiments, $K=2500$) and identity gates (red squares, $K=2500$). Also shown are fits to the FDM $F=0.486{\left(0.998\right)}^m+0.510$ (blue dotted line) and to the Gaussian model $F=0.495{\left(0.999\right)}^{m^2}+0.503$ (red dashed line). The standard error in the data is less than $4.8\times {10}^{-3}$ and fit parameters are rounded to three digits.

Figure 6

SRB results for several non-Markovian error models and error rates with $K=10000$ averaged sequences. Leakage, both random and fixed, was nearly always estimated to within a factor of 2, and there was no significant difference between the two types, except for the largest error rate, where random leakage had better accuracy, precision, and fit confidence. $1/f$ noise was estimated in all cases to within $25\%$ of $r$. Random leakage (blue circles), fixed leakage (green triangles), and $1/f$ (red squares).

Figure 7

Convergence in $K$ for standard RB with $1/f$ noise of (a) accuracy $\mu$ and (b) confidence interval $C$ on a fit of the fidelity decay to the exponential model. Noise strengths are $A^{\prime }=2.5\times {10}^{-9}$ $(T_2^{*}/t_g=93.95)$ (red squares), $A^{\prime }=2.0\times {10}^{-8}$ $(T_2^{*}/t_g=30.05)$ (green triangles), and $A^{\prime }=2.5\times {10}^{-7}$ $(T_2^{*}/t_g=8.35)$ (blue circles).

Figure 8

Fidelity decay in the presence of random leakage noise ($r={10}^{-3}$) with repeated application of random noisy Cliffords (blue circles, $K=10000$) and repeated application of a noisy identity gate, i.e., a fixed unitary qutrit gate (red line). Blue dotted line is $F=.666{(.998)}^m+.333$, the FDM fit to the RB data. Note the decay to $1/3$ (black dashed line). The standard errors in the data increase monotonically with $m$ from ${10}^{-6}$ to approximately ${10}^{-3}$, and fit parameters are rounded to three digits. The solid red line is the survival probability of the excited state $P\left(m\right)=|\langle 1|U^m|1\rangle {|}^2$ for repeated application of a fixed (randomly chosen) three-level unitary error $U$. If the excited state is nonstationary, $P\left(m\right)$ is an oscillatory function that is not periodic unless the stationary states accumulate commensurate phases.

Figure 9

The $90\%$ confidence intervals on the random leakage models are plotted against sample size $K$ for multiple values of $r={10}^{-4}$ (red squares), $r={10}^{-3}$ (green triangles), and $r={10}^{-2}$ (blue circles).

Figure 10

The diamond norm distance $D$ between $\overline{\mathcal{E}}$ and the depolarizing channel of error rate $r$ decreases as a function of $L$. $r={10}^{-4}$ (red squares), $r={10}^{-3}$ (green triangles), and $r={10}^{-2}$ (blue circles).

Figure 11

IRB in the presence of random unitary noise for $K=10000$. ${\widehat{r}}_{\mathrm{int}}$ is the estimated interleaved error rate, $r_{\mathrm{int}}$ is the true rate, and $\mu ={log}_{10}\left({\widehat{r}}_{\mathrm{int}}/r_{\mathrm{int}}\right)$. The average Clifford error is $r={10}^{-4}$ (red squares), $r={10}^{-3}$ (green triangles), $r={10}^{-2}$ (blue circles; blue triangles indicate negative IRB estimates).