General Framework for Randomized Benchmarking

Randomized benchmarking refers to a collection of protocols that in the past decade have become central methods for characterizing quantum gates. These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors. Over the years many versions have been developed, however a comprehensive theoretical treatment of randomized benchmarking has been missing. In this work, we develop a rigorous framework of randomized benchmarking general enough to encompass virtually all known protocols as well as novel, more flexible extensions. Overcoming previous limitations on error models and gate sets, this framework allows us, for the first time, to formulate realistic conditions under which we can rigorously guarantee that the output of any randomized benchmarking experiment is well described by a linear combination of matrix exponential decays. We complement this with a detailed analysis of the fitting problem associated with randomized benchmarking data. We introduce modern signal processing techniques to randomized benchmarking, prove analytical sample complexity bounds, and numerically evaluate performance and limitations. In order to reduce the resource demands of this fitting problem, we introduce novel, scalable postprocessing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of randomized benchmarking protocols. These postprocessing techniques overcome shortcomings in efficiency of several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking. Finally, we discuss, in full generality, how and when randomized benchmarking decay rates can be used to infer quality measures like the average fidelity. On the technical side, our work substantially extends the recently developed Fourier-theoretic perspective on randomized benchmarking by making use of the perturbation theory of invariant subspaces, as well as ideas from signal processing.

Popular Summary

Quantum computers promise a revolution in computing power, however we first need to build them! A central problem in building quantum computers is that quantum logic operations are highly error-prone, due to interactions with the environment, or just misspecified control sequences. This means we need good methods to characterize such errors (in order to fix them later). The gold standard of such methods is called randomized benchmarking, which has been used and expanded extensively in the last decade. However the mathematics of randomized benchmarking were not totally understood, which meant that we could not necessarily prove that randomized benchmarking always works the way it should. Moreover, over the years dozens of different variations of the protocol were developed and it was not clear how they all related to one another.

In this work we give a comprehensive overview of randomized benchmarking, focusing in particular on its mathematical fundamentals. We rigorously prove that randomized benchmarking works as intended under a wide range of circumstances, using advanced mathematical tools. Relatedly we also unify all existing randomized benchmarking protocols under a single umbrella, and provide a clean classification of all the different types of randomized benchmarking. With this effort we hope to provide some clarity in the rapidly growing field of quantum characterization and benchmarking, which makes it easier for experimentalists to have trust in their experiments and for theorists to develop novel procedures for the characterization of quantum gates. Thus we ultimately provide a small but vital piece of the enormous puzzle that is building quantum computers.

Figure 1

The basic structure of RB. The RB data-collection phase iterates the following steps: after (1) the preparation of an initial state ${\rho }_0$, (2) a sequence of $m$ random gates $g_i$ is applied, (3) followed by the gate inverting the sequence $g_{\mathrm{inv}}$ up to an end gate ${\mathrm{g}}_{\mathrm{end}}$ and (4) a final POVM measurement. In the data-processing phase the measurement data, for many random sequences and different sequence lengths is postprocessed in a classical computer to extract decay parameters quantifying the imperfections in the implemented gates.

Figure 2

An overview of RB schemes, indicating how they fit within our typology (see Sec. 5d) of RB schemes and what theorem covers the behavior of their output data (see Sec. 6). An $\ast$ indicates that the protocol has a nontrivial postprocessing scheme, while $\ast \ast$ indicates that the protocol in its original specification has no inversion gate. We discuss how this is equal to uniform RB (with inversion) together with a postprocessing step in Sec. 8.

Figure 3

The condition number of the Vandermonde matrix for different Hankel matrix dimensions ($\propto$ RB sequence length) for three different sets of poles. The dashed lines indicate the asymptotic expression of Lemma 13. Note that the minimum for the green line is due to the scaling of the maximal eigenvalue of $W_{M-L}$. We observe (not depicted here) that the minimum singular value of the Hankel matrix, as appearing in Theorem 11, is monotonically increasing in $L$ for all three sets of poles.

Figure 4

Here we show the dependency of the conditioning number of the Vandermonde matrix on the spacing of two poles $z_0,z_1$, for infinite sequence length. We see that the conditioning depends drastically on the distance between the two poles, but not on the absolute location of the poles on the real line. The orange line at ${10}^2$ is added for the purpose of comparison.

Figure 5

The dependency of the condition number in the limit of infinite sequence length on the number of poles for different families of poles. These families are defined in Table 1.

Figure 6

Mean Hausdorff distance between the real set of poles and the reconstructed set of poles (via ESPRIT) for different families of poles (as defined in Table 1) and Hankel dimension $L$ ($\propto$ maximal RB sequence length $M$) versus the number of samples used per expectation value estimation. Each data point is averaged over $100$ repetitions. For all families we see that the reconstruction essentially fails until a sampling threshold is reached, after this threshold the accuracy of the estimation increases rapidly with increased number of samples. This threshold increases strongly with the number of poles in the family across all families and also depends on the maximal sequence length. This latter dependence is mediated by the actual locations of the poles in the complex plane, which is as expected.

Figure 7

Mean Hausdorff distance of the reconstruction (via ESPRIT) for poles $z=F_2(4)=(0.9,0.968,0.99,0.997)$ for different number of samples and Hankel dimension $L$ ($\propto$ maximal RB sequence length $M$). Each data point is averaged over $100$ repetitions. We see again that reconstruction essentially fails, until a threshold is reached both in the number of samples and in maximal sequence length after which the accuracy of reconstruction increases with increasing number of samples and in $L$.

Figure 8

RB (data-collection phase)

Figure 9

An estimator for $k_{\lambda }(m)$