Randomized benchmarking techniques have been an essential tool for assessing the performance of contemporary quantum devices. The goal of this tutorial is to provide a pedagogical, self-contained, introduction to randomized benchmarking. With this intention, every chapter is also supplemented with an accompanying Python notebook, illustrating the essential steps of each protocol. In addition, we also introduce more recent trends in the field that bridge shadow tomography with randomized benchmarking, namely through the gate-set shadow protocol.
Randomized benchmarking techniques have been an essential tool for assessing the performance of contemporary quantum devices. The goal of this tutorial is to provide a pedagogical, self-contained, introduction to randomized benchmarking. With this intention, every chapter is also supplemented with an accompanying Python notebook, illustrating the essential steps of each protocol. In addition, we also introduce more recent trends in the field that bridge shadow tomography with randomized benchmarking, namely through the gate-set shadow protocol.
As quantum technology matures, the efficient benchmarking of quantum devices remains a key challenge. Although sample-efficient, information-theoretic benchmarking techniques have recently been proposed, there is still a gap in adapting these techniques to contemporary experiments. In this work, we re-formulate five of the most common randomized benchmarking techniques in the modern language of the gate-set shadow tomography. This reformulation brings along several concrete advantages over conventional formulations of randomized benchmarking. For standard and interleaved randomized benchmarking, we can reduce the required gate-set size and, using median-of-means estimators, also reduce the required experimental sample size. For simultaneous and correlated randomized benchmarking, we can additionally reconstruct the Pauli-terms of correlated noise channels using additional post-processing of only a single experimental dataset. We also present a minimal approach to extract leakage errors. Our work provides a clear avenue for comprehensive, reliable, and convenient benchmarking of quantum devices, with all methods formulated under a single umbrella technique that can be easily adapted to a range of experimental quantities and gate sets.
Randomized benchmarking is a powerful technique to efficiently estimate the performance and reliability of quantum gates, circuits, and devices. Here, we propose to perform randomized benchmarking in a coherent way, where superpositions of different random sequences rather than independent samples are used. We show that this leads to a uniform and simple protocol having significant advantages with respect to gates that can be benchmarked, and in terms of efficiency and scalability. We show that, e.g., universal gate sets, the set of n-qudit Pauli operators or more general sets including arbitrary unitaries, as well as a particular n-qudit Clifford gate using only the Pauli set, can be efficiently benchmarked. The price to pay is an additional complexity to add control to the involved quantum operations. However, we demonstrate that this can be done by using auxiliary degrees of freedom that are naturally available in basically any physical realization, and are independent of the gates to be tested.
Benchmarking noise in quantum gates is crucial for developing dependable quantum computers, as noise directly affects gate fidelity and computational accuracy. Randomized benchmarking (RB) protocols, which evaluate performance through random gate sequences, provide a reliable way to quantify average error rates due to their scalability and robustness in averaging out complex, device-specific noise. In this talk, we will discuss several variants of randomized benchmarking protocols including an overview of our recent work in this area.
Randomized measurements are increasingly appreciated as powerful tools to estimate properties of quantum systems, e.g., in the characterization of hybrid classical-quantum computation. On many platforms they constitute natively accessible measurements, serving as the building block of prominent schemes like shadow estimation. In the real world, however, the implementation of the random gates at the core of these schemes is susceptible to various sources of noise and imperfections, strongly limiting the applicability of protocols. To attenuate the impact of this shortcoming, in this work we introduce an error-mitigated method of randomized measurements, giving rise to a robust shadow estimation procedure. On the practical side, we show that error mitigation and shadow estimation can be carried out using the same session of quantum experiments, hence ensuring that we can address and mitigate the noise affecting the randomization measurements. Mathematically, we develop a picture derived from Fourier-transforms to connect randomized benchmarking and shadow estimation. We prove rigorous performance guarantees and show the functioning using comprehensive numerics. More conceptually, we demonstrate that, if properly used, easily accessible data from randomized benchmarking schemes already provide such valuable diagnostic information to inform about the noise dynamics and to assist in quantum learning procedures.
We present a pedagogical, hands-on tutorial on \emph{replica tensor-network} techniques for random quantum circuits. At its core, the method recasts circuit-averaged observables acting on multiple copies of the system as the contraction of a classical tensor network, equivalently the partition function of a statistical-mechanics model whose effective spins live in the commutant of the gate ensemble. The framework is general: changing the observable or the initial state modifies only the replica boundary conditions, while changing the ensemble modifies the bulk tensors. Focusing on quantum-information diagnostics, from metrics of wavefunction spreadings to entanglement quantifiers, we illustrate the approach in both clean and noisy random unitary circuits. We then briefly explain how the methodology extends to other ensembles, such as orthogonal or Clifford circuits. The lecture notes are accompanied by \texttt{ReplicaTN}, a self-contained C++/Python library and pedagogical notebooks.
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.