Estimating the Bias of $\mathrm{CX}$ Gates via Character Randomized Benchmarking

Recent work has demonstrated that high-threshold quantum error correction is possible for biased-noise qubits, provided that one can implement a controlled-sc not ($\mathrm{CX}$) gate that preserves the bias. Bias-preserving $\mathrm{CX}$ gates have been proposed for several biased-noise qubit platforms, most notably Kerr cats. However, experimentally measuring the noise bias is challenging, as it requires accurately estimating certain low-probability Pauli errors in the presence of much larger state-preparation-and-measurement (SPAM) errors. In this paper, we introduce bias-randomized benchmarking (BRB) as a technique for measuring bias in quantum gates. BRB, like all RB protocols, is highly accurate and immune to SPAM errors. Our first protocol, $\mathrm{CX}$-dihedral BRB, is a straightforward method to measure the bias of the entire $\mathrm{CX}$-dihedral group. Our second protocol, interleaved-bias randomized benchmarking (IBRB), is a generalization of interleaved RB tailored to the experimental constraints of biased-noise qubits; this is a more involved procedure that directly targets the bias of the $\mathrm{CX}$ gate alone. Our BRB procedures occupy a middle ground between classic RB protocols that only estimate the average fidelity and tomographic RB protocols that provide more detailed characterization of noise but require more measurements as well as experimental capabilities that are not necessarily available in biased-noise qubits.

Popular Summary

All qubits suffer from noise, which must be corrected to build a quantum computer. Recently, it has been realized that biased noise, where bit-flip errors are much less likely than dephasing errors, can be more easily corrected than unbiased noise. When correcting biased errors, it is important to use bias-preserving gates that do not introduce additional bit-flip errors. The main challenge for biased-noise error correction has been to realize a bias-preserving controlled-not ($\mathrm{CX}$) gate but in the past few years there have been multiple proposals for realizing bias-preserving $\mathrm{CX}$ gates. As these proposals are implemented, it will be important to be able to measure the bias of the $\mathrm{CX}$ gate.

Measuring the bias of a $\mathrm{CX}$ gate is challenging, as it requires estimating the probability of extremely unlikely bit-flip errors. Moreover, errors in preparing and measuring the qubits are often as common as errors in the $\mathrm{CX}$ gate. Randomized benchmarking (RB) is a standard technique to magnify the effects of gate errors and decouple gate errors from preparation and/or measurement errors. However, the usual RB methods rely on tools that are not available in biased-noise qubits.

In this paper, we introduce a new RB method that we dub bias-randomized benchmarking (BRB). BRB is tailored to the constraints of biased-noise qubits and allows us to determine the bias and fidelity of $\mathrm{CX}$ gates with high accuracy, even at very small bit-flip rates. As experimental groups begin to implement bias-preserving $\mathrm{CX}$ gates, BRB will be a key technique to accurately characterize these gates.

Figure 1

Simulated $\mathrm{CX}$-dihedral-bias-RB experiments. (a) An example of estimating $S_b(n)$, with the $S_b(n)$ curves plotted in orange and our estimates of $S_b(n)$ given by blue dots. To estimate sensitivity to statistical errors, we generate multiple fits of $S_b(n)$ using bootstrap resampling of our data, which are plotted in gray and generally overlap the exact $S_b(n)$ curves. (b) The probabilities $p_D$ and $p_{\mathrm{ND}}$ extracted from the fits for $50$ randomly generated error channels. The error bars denote the standard deviation over $50$ resamplings. (c) The estimated bias $\eta$ for each of these error channels. We see that even at very high bias, we can accurately estimate the value of $\eta$.

Figure 2

Simulated interleaved-bias-RB experiments. (a) An example of estimating $S_b(n)$, with the $S_b(n)$ curves plotted in orange and our estimates of $S_b(n)$ given by blue dots. Note that $S_b(n)$ is oscillatory for $b=0-,1-$, since the decay constants ${\lambda }_b$ and ${\kappa }_b$ have opposite sign. To robustly fit $S_b(n)$ in the presence of these oscillations does not require densely sampling data points, since we know the oscillatory period is always $2$; however, it does require taking data at both even and odd sequence lengths, as shown in the figure. To estimate sensitivity to statistical errors, we generate multiple fits of $S_b(n)$ using bootstrap resampling of our data, which are plotted in gray and generally overlap the exact $S_b(n)$ curves. (b) The probabilities $p_D$ and $p_{\mathrm{ND}}$ extracted from the fits for $50$ randomly generated error channels. The error bars denote the standard deviation over $50$ resamplings. (c) The estimated bias $\eta$ for each of these error channels. We see that even at very high bias, we can accurately estimate the value of $\eta$.

Figure 3

Randomizing a single ${\mathrm{CX}}_{c,t}$ gate in a circuit that measures the stabilizer of the XZZX code, where $c$ denotes the control qubit and $t$ the target. Rather than apply the bare ${\mathrm{CX}}_{c,t}$ gate, we randomly insert an element of ${\mathcal{Z}}_2$ before the gate and randomly apply either ${\mathrm{CX}}_{c,t}$ or $X_c{\mathrm{CX}}_{c,t}{X}_c$. This modification of the circuit results in an overall Pauli operator, which can be tracked in software.