Randomized Benchmarking beyond Groups

Randomized benchmarking (RB) is the gold standard for experimentally evaluating the quality of quantum operations. The current framework for RB is centered on groups and their representations but this can be problematic. For example, Clifford circuits need up to $O(n^2)$ gates and thus Clifford RB cannot scale to larger devices. Attempts to remedy this include new schemes such as linear cross-entropy benchmarking (XEB), cycle benchmarking, and nonuniform RB but they do not fall within the group-based RB framework. In this work, we formulate the universal randomized benchmarking (URB) framework, which does away with the group structure and also replaces the recovery-gate-plus-measurement component with a general “postprocessing” positive operator-valued measurement (POVM). Not only does this framework cover most of the existing benchmarking schemes but it also gives the language for and helps inspire the formulation of new schemes. We specifically consider a class of URB schemes called twirling schemes. For twirling schemes, the postprocessing POVM approximately factorizes into an intermediate channel, inverting maps, and a final measurement. This leads us to study the twirling map corresponding to the gate ensemble specified by the scheme. We prove that if this twirling map is strictly within unit distance of the Haar twirling map in induced diamond norm, the probability of measurement as a function of gate length is a single exponential decay up to small error terms. The core technical tool we use is the matrix perturbation theory of linear operators on quantum channels.

Popular Summary

Quantum computers must operate under difficult conditions. Superconducting qubits, for example, face disruption from ambient radioactivity and cosmic rays. Furthermore, quantum computers are expected to perform almost flawlessly, passing quality tests known as randomized benchmarking (RB) schemes. Mainstream RB schemes are centered on group structure. Here we offer a framework that obviates the group structure rule, allowing for quality tests that are more personalized for quantum computers.

We give a mathematical framework for RB schemes that do not require group structure, a sweeping generalization that captures almost every known scheme. For a certain, but surprisingly inclusive, class of such schemes, we prove that we can conveniently obtain an indicator of performance from the rate of an experimentally measured exponential decay. We give methods for how to show that an RB scheme personalized for a quantum computer or designed for a specialized task fulfills the requirements to be in this class.

Overturning the group structure regime allows for more customized quality tests. A follow up work, for example, develops the linear cross-entropy benchmarking test with Clifford circuits, which is designed for the quantum computers of tomorrow with more than a thousand qubits.

Figure 1

A visual illustration of the URB framework and the twirling schemes. (a) A generic URB scheme. Given a sequence length $m$, we randomly select $m$ IID samples from a distribution $\mu$. The experiment then applies $g_1,\dots ,g_m$ sequentially via an implementation map $\varphi$ on a fixed initial state ${\rho }_0$, followed by a postprocessing POVM $M$ that depends on the random elements $g_1,\dots ,g_m$. (b) Approximate factoring. When the postprocessing POVM admits an approximate factoring, it can be approximately decomposed into an intermediate channel $\mathcal{I}$, a sequence of inverting operations ${\varphi }^{\ast }(g_m),\dots ,{\varphi }^{\ast }(g_1)$, and a fixed measurement operator $M_0$. (c) A twirling map. The average joint operation of $\varphi (g)$ and ${\varphi }^{\ast }(g)$ sandwiching a channel can be formulated as a linear operator ${\mathrm{\Lambda }}_R$ on quantum channels. (d) A URB scheme in terms of twirling maps. Expressing the probability of measurement in terms of the twirling map ${\mathrm{\Lambda }}_R$ reduces the proof of the exponential decay to the study of its spectral properties.

Figure 2

A diagram showing the relationships between different classes of RB schemes, including our URB framework and class of twirling schemes. Character RB refers to the class of schemes in Ref. [29]. The space made by the URB and twirling classes leaves room for potentially new schemes.