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arXiv:2407.10396 · New Journal of Physics

Randomised benchmarking for universal qudit gates

David Amaro-Alcalá, Barry C. Sanders, Hubert de Guise

Published Jul 1, 2024• 7 citations
arXivHTMLPDF

Abstract

Abstract We aim to establish a scalable scheme for characterising diagonal non-Clifford gates for single- and multi-qudit systems; d is a prime-power integer. By employing cyclic operators and a qudit T gate, we generalise the dihedral benchmarking scheme for single- and multi-qudit circuits. Our results establish a path for experimentally benchmarking qudit systems and are of theoretical and experimental interest because our scheme is optimal insofar as it does not require preparation of the full qudit Clifford gate set to characterise a non-Clifford gate. Moreover, combined with Clifford randomised benchmarking, our scheme is useful to characterise the generators of a universal gate set.

DOI
10.1088/1367-2630/ad6635

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Abstract We aim to establish a scalable scheme for characterising diagonal non-Clifford gates for single- and multi-qudit systems; d is a prime-power integer. By employing cyclic operators and a qudit T gate, we generalise the dihedral benchmarking scheme for single- and multi-qudit circuits. Our results establish a path for experimentally benchmarking qudit systems and are of theoretical and experimental interest because our scheme is optimal insofar as it does not require preparation of the full qudit Clifford gate set to characterise a non-Clifford gate. Moreover, combined with Clifford randomised benchmarking, our scheme is useful to characterise the generators of a universal gate set.
We introduce a scheme to characterise a qudit T gate that has different noise than a set of Clifford gates. We developed our scheme through representation theory and ring theory to generalise non-Clifford interleaved benchmarking to qudit systems. By restricting to the qubit case, we recover the dihedral benchmarking scheme. Our characterisation scheme provides experimental physicists a practical method for characterising universal qudit gate sets and advances randomised benchmarking research by providing the characterisation of a complete qudit library.
Randomised benchmarking is a widely used experimental technique to characterise the average error of quantum operations. Benchmarking procedures that scale to enable the characterisation of n-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomised benchmarking procedure over n-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabiliser codes. We present efficient methods for representing and composing group elements, sampling them uniformly and synthesising corresponding poly(n)-sized circuits. The procedure provides experimental access to two independent parameters that together characterise the average gate fidelity of a group element. A scalable procedure to determine the error of quantum operations has been developed by researchers in the United States. Andrew Cross and colleagues at the IBM T.J. Watson Research Center have expanded previously used protocols to a broader range of quantum circuits, thereby enabling error benchmarking for all common logical operations used in leading quantum computing protocols. The standard technique for error characterization in these devices is randomized benchmarking, which provides benefits such as insensitivity to measurement errors, but on the other hand only functions in a limited number of circumstances. In their theoretical study, the researchers demonstrate how randomized benchmarking approaches can be expanded to a much broader range of logical operations. The procedure can be readily implemented experimentally, and provides access to important experimental noise parameters in quantum computing.
We introduce a characterization scheme for a universal qutrit gate set. Motivated by the rising interest in qutrit systems, we apply our criteria to establish that our hyperdihedral group underpins a scheme to characterize the performance of a qutrit T gate. Our resulting qutrit scheme is feasible, as it requires resources and data analysis techniques similar to resources employed for qutrit Clifford randomized benchmarking. Combining our T gate benchmarking procedure for qutrits with known qutrit Clifford-gate benchmarking enables complete characterization of a universal qutrit gate set.
We describe a practical experimental protocol for robustly characterizing the error rates of non-Clifford gates associated with dihedral groups, including small single-qubit rotations. Our dihedral benchmarking protocol is a generalization of randomized benchmarking that relaxes the usual unitary 2-design condition. Combining this protocol with existing randomized benchmarking schemes enables practical universal gate sets for quantum information processing to be characterized in a way that is robust against state-preparation and measurement errors. In particular, our protocol enables direct benchmarking of the $\ensuremath{\pi}/8$ gate even under the gate-dependent error model that is expected in leading approaches to fault-tolerant quantum computation.
Randomized benchmarking (RB) protocols are the most widely used methods for assessing the performance of quantum gates.However, the existing RB methods either do not scale to many qubits or cannot benchmark a universal gate set.Here, we introduce and demonstrate a technique for scalable RB of many universal and continuously parametrized gate sets, using a class of circuits called randomized mirror circuits.Our technique can be applied to a gate set containing an entangling Clifford gate and the set of arbitrary single-qubit gates, as well as gate sets containing controlled rotations about the Pauli axes.We use our technique to benchmark universal gate sets on four qubits of the Advanced Quantum Testbed, including a gate set containing a controlled-S gate and its inverse, and we investigate how the observed error rate is impacted by the inclusion of non-Clifford gates.Finally, we demonstrate that our technique scales to many qubits with experiments on a 27-qubit IBM Q processor.We use our technique to quantify the impact of crosstalk on this 27-qubit device, and we find that it contributes approximately 2=3 of the total error per gate in random many-qubit circuit layers.
Abstract We introduce unitary-gate randomized benchmarking (URB) for qudit gates by extending single- and multi-qubit URB to single- and multi-qudit gates. Specifically, we develop a qudit URB procedure that exploits unitary 2-designs. Furthermore, we show that our URB procedure is not simply extracted from the multi-qubit case by equating qudit URB to URB of the symmetric multi-qubit subspace. Our qudit URB is elucidated by using pseudocode, which facilitates incorporating into benchmarking applications.