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Rapid progress in quantum technology has transformed quantum computing and quantum information science from theoretical possibilities into tangible engineering challenges. Breakthroughs in quantum algorithms, quantum simulations, and quantum error correction are bringing useful quantum computation closer to fruition. These remarkable achievements have been facilitated by advances in quantum characterization, verification, and validation (QCVV). QCVV methods and protocols enable scientists and engineers to scrutinize, understand, and enhance the performance of quantum information-processing devices. In this Tutorial, we review the fundamental principles underpinning QCVV, and introduce a diverse array of QCVV tools used by quantum researchers. We define and explain QCVV's core models and concepts -- quantum states, measurements, and processes -- and illustrate how these building blocks are leveraged to examine a target system or operation. We survey and introduce protocols ranging from simple qubit characterization to advanced benchmarking methods. Along the way, we provide illustrated examples and detailed descriptions of the protocols, highlight the advantages and disadvantages of each, and discuss their potential scalability to future large-scale quantum computers. This Tutorial serves as a guidebook for researchers unfamiliar with the benchmarking and characterization of quantum computers, and also as a detailed reference for experienced practitioners.

The rapid pace of development in quantum computing technology has sparked a proliferation of benchmarks for assessing the performance of quantum computing hardware and software. Good benchmarks empower scientists, engineers, programmers, and users to understand a computing system's power, but bad benchmarks can misdirect research and inhibit progress. In this Perspective, we survey the science of quantum computer benchmarking. We discuss the role of benchmarks and benchmarking, and how good benchmarks can drive and measure progress towards the long-term goal of useful quantum computations, i.e., "quantum utility". We explain how different kinds of benchmark quantify the performance of different parts of a quantum computer, we survey existing benchmarks, critically discuss recent trends in benchmarking, and highlight important open research questions in this field.

We present an algorithm that decomposes any $n$-qubit Clifford operator into a circuit consisting of three subcircuits containing only CNOT or CPHASE gates with layers of one-qubit gates before and after each of these subcircuits. As with other asymptotically optimal Clifford compilation algorithms, the resulting circuit contains $O(n^2/\log n)$ two-qubit gates. The derivation of our algorithm only requires the symplectic representation of Clifford gates, basic row and column matrix manipulations, and some known properties of general matrices over 0 and 1., Comment: 5 pages

Scalable quantum processors require high-fidelity universal quantum logic operations in a manufacturable physical platform. Donors in silicon provide atomic size, excellent quantum coherence and compatibility with standard semiconductor processing, but no entanglement between donor-bound electron spins has been demonstrated to date. Here we present the experimental demonstration and tomography of universal 1- and 2-qubit gates in a system of two weakly exchange-coupled electrons, bound to single phosphorus donors introduced in silicon by ion implantation. We surprisingly observe that the exchange interaction has no effect on the qubit coherence. We quantify the fidelity of the quantum operations using gate set tomography (GST), and we use the universal gate set to create entangled Bell states of the electrons spins, with fidelity ~ 93%, and concurrence 0.91 +/- 0.08. These results form the necessary basis for scaling up donor-based quantum computers.

Gate set tomography (GST) provides precise, self-consistent estimates of the noise channels for all of a quantum processor's logic gates. But GST experiments are large, involving many distinct quantum circuits. This has prevented their use on systems larger than two qubits. Here, we show how to streamline GST experiment designs by removing almost all redundancy, creating smaller and more scalable experiments without losing precision. We do this by analyzing the "germ" subroutines at the heart of GST circuits, identifying exactly what gate set parameters they are sensitive to, and leveraging this information to remove circuits that duplicate other circuits' sensitivities. We apply this technique to two-qubit GST experiments, generating streamlined experiment designs that contain only slightly more circuits than the theoretical minimum bounds, but still achieve Heisenberg-like scaling in precision (as demonstrated via simulation and a theoretical analysis using Fisher information). In practical use, the new experiment designs can match the precision of previous GST experiments with significantly fewer circuits. We discuss the prospects and feasibility of extending GST to three-qubit systems using our techniques., Comment: 11 pages, 6 figures, to be published in proceedings of 2023 IEEE International Conference on Quantum Computing and Engineering (QCE). V2: Minor edits to acknowledgements

Gate set tomography (GST) is a self-consistent and highly accurate method for the tomographic reconstruction of a quantum information processor's quantum logic operations, including gates, state preparations, and measurements. However, GST's experimental cost grows exponentially with qubit number. For characterizing even just two qubits, a standard GST experiment may have tens of thousands of circuits, making it prohibitively expensive for platforms. We show that, because GST experiments are massively overcomplete, many circuits can be discarded. This dramatically reduces GST's experimental cost while still maintaining GST's Heisenberg-like scaling in accuracy. We show how to exploit the structure of GST circuits to determine which ones are superfluous. We confirm the efficacy of the resulting experiment designs both through numerical simulations and via the Fisher information for said designs. We also explore the impact of these techniques on the prospects of three-qubit GST., Comment: 46 pages, 13 figures. V2: Minor edits to acknowledgments

Closed-loop control algorithms for real-time calibration of quantum processors require efficient filters that can estimate physical error parameters based on streams of measured quantum circuit outcomes. Development of such filters is complicated by the highly nonlinear relationship relationship between observed circuit outcomes and the magnitudes of elementary errors. In this work, we apply the extended Kalman filter to data from quantum gate set tomography to provide a streaming estimator of the both the system error model and its uncertainties. Our numerical examples indicate extended Kalman filtering can achieve similar performance to maximum likelihood estimation, but with dramatically lower computational cost. With our method, a standard laptop can process one- and two-qubit circuit outcomes and update gate set error model at rates comparable with current experimental execution., Comment: Revised to version that appeared in the conference