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arXiv:quant-ph/0308115 · Journal of the Optical Society of America B

Measuring two-qubit gates

Andrew G. White, Alexei Gilchrist, Geoff J. Pryde, Jeremy L. O’Brien, Michael J. Bremner, Nathan K. Langford

Published Jan 26, 2007• 134 citations
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Abstract

Accurate characterization of two-qubit gates will be critical for any realization of quantum computation. We discuss a range of measurements for characterizing two-qubit gates. These measures are architecture-independent and span a range of complexity from simple measurement routines to full quantum-state and process tomography. Simple indicative measures, which flag but do not quantify gate operation in the quantum regime, include the fringe visibility, parity, Bell-state fidelity, and entanglement witnesses. Quantitative measures of gate output states include linear entropy and tangle; measures of, and error bounds to, whole-gate operation are provided by metrics such as process fidelity, process distance, and average gate fidelity. We discuss which measures are appropriate, given the stage of development of the gate, and highlight connections between them.

DOI
10.1364/josab.24.000172

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Accurate characterization of two-qubit gates will be critical for any realization of quantum computation. We discuss a range of measurements for characterizing two-qubit gates. These measures are architecture-independent and span a range of complexity from simple measurement routines to full quantum-state and process tomography. Simple indicative measures, which flag but do not quantify gate operation in the quantum regime, include the fringe visibility, parity, Bell-state fidelity, and entanglement witnesses. Quantitative measures of gate output states include linear entropy and tangle; measures of, and error bounds to, whole-gate operation are provided by metrics such as process fidelity, process distance, and average gate fidelity. We discuss which measures are appropriate, given the stage of development of the gate, and highlight connections between them.
In circuit-based quantum computing the available gate set typically consists of single-qubit gates acting on each individual qubit and at least one entangling gate between pairs of qubits.In certain physical architectures, however, some qubits may be "hidden" and lacking direct addressability through dedicated control and readout lines, for instance, because of limited on-chip routing capabilities, or because the number of control lines becomes a limiting factor for many-qubit systems.In this case, no single-qubit operations can be applied to the hidden qubits and their state cannot be measured directly.Instead, they may be controlled and read out only via single-qubit operations on connected "control" qubits and a suitable set of two-qubit gates.We first discuss the impact of such restricted control capabilities on the performance of specific qubit coupling networks.We then experimentally demonstrate full control and measurement capabilities in a superconducting two-qubit device with local single-qubit control and iSWAP and controlledphase two-qubit interactions enabled by a tunable coupler.We further introduce an iterative tune-up process required to completely characterize the gate set used for quantum process tomography and evaluate the resulting gate fidelities.
Quantum process tomography of each single-qubit quantum gate used in the IBM quantum processors is performed to compute gate error in order to check viability of complex quantum operations in the superconductivity-based quantum computers introduced by IBM and to compare the quality of these gates with the corresponding gates implemented using other technologies. Gate fidelities for all the allowed gates for every allowed position (i.e., by placing the gates in different qubit lines) have been obtained and reported for IBM QX4 architecture. Gate fidelities are observed to be lower than the corresponding values obtained in the other technologies, like NMR. Further, gate fidelities for all the single-qubit gates are obtained for IBM QX2 architecture by placing the gates in the third qubit line (q[2]q[2]q[2]). It's observed that the IBM QX4 architecture yields better gate fidelity compared to IBM QX2 in all cases except the case of Y⁡\operatorname{Y}Y gate as far as the gate fidelity corresponding to the third qubit line is concerned. In general, the analysis performed here leads to a conclusion that a considerable technological improvement would be inevitable to achieve the desired scalability required for the realization of complex quantum operations.
Remarkable experimental advances in quantum computing are exemplified by recent announcements of impressive average gate fidelities exceeding 99.9% for single-qubit gates and 99% for two-qubit gates. Although these high numbers engender optimism that fault-tolerant quantum computing is within reach, the connection of average gate fidelity with fault-tolerance requirements is not direct. Here we use reported average gate fidelity to determine an upper bound on the quantum-gate error rate, which is the appropriate metric for assessing progress towards fault-tolerant quantum computation, and we demonstrate that this bound is asymptotically tight for general noise. Although this bound is unlikely to be saturated by experimental noise, we demonstrate using explicit examples that the bound indicates a realistic deviation between the true error rate and the reported average fidelity. We introduce the Pauli distance as a measure of this deviation, and we show that knowledge of the Pauli distance enables tighter estimates of the error rate of quantum gates.
Abstract Universal quantum computation will require qubit technology based on a scalable platform1, together with quantum error correction protocols that place strict limits on the maximum infidelities for one- and two-qubit gate operations2, 3. Although various qubit systems have shown high fidelities at the one-qubit level4–10, the only solid-state qubits manufactured using standard lithographic techniques that have demonstrated two-qubit fidelities near the fault-tolerance threshold6 have been in superconductor systems. Silicon-based quantum dot qubits are also amenable to large-scale fabrication and can achieve high single-qubit gate fidelities (exceeding 99.9 per cent) using isotopically enriched silicon11, 12. Two-qubit gates have now been demonstrated in a number of systems13–15, but as yet an accurate assessment of their fidelities using Clifford-based randomized benchmarking, which uses sequences of randomly chosen gates to measure the error, has not been achieved. Here, for qubits encoded on the electron spin states of gate-defined quantum dots, we demonstrate Bell state tomography with fidelities ranging from 80 to 89 per cent, and two-qubit randomized benchmarking with an average Clifford gate fidelity of 94.7 per cent and an average controlled-rotation fidelity of 98 per cent. These fidelities are found to be limited by the relatively long gate times used here compared with the decoherence times of the qubits. Silicon qubit designs employing fast gate operations with high Rabi frequencies16, 17, together with advanced pulsing techniques18, should therefore enable much higher fidelities in the near future.
Accurate and precise control of large quantum systems is paramount to achieve practical advantages on quantum devices. Therefore, benchmarking the hardware errors in quantum computers has drawn significant attention lately. Existing benchmarks for digital quantum computers involve averaging the global fidelity over a large set of quantum circuits and are therefore unsuitable for specific multi-qubit gates used in analog quantum operations. Moreover, average global fidelity is not the optimal figure-of-merit for some of the applications specific to multi-qubit gates and analog devices , such as the study of many-body physics, which often use local observables. In this two-part paper, we develop a new figure-of-merit suitable for multi-qubit quantum gates based on the reduced Choi matrix of the operation. In the first part, we develop an efficient, scalable protocol to completely characterize the reduced Choi matrix. We identify two sources of sampling errors in measurements of the reduced Choi matrix and we show that there are fundamental limits to the rate of convergence of the sampling errors, analogous to the standard quantum limit and Heisenberg limit. A slow convergence rate of sampling errors would mean that we need a large number of experimental shots. We develop protocols using quantum information scrambling, which has been observed in disordered systems for e.g., to speed up the rate of convergence of the sampling error at state preparation Moreover, we develop protocols using squeezed and entangled initial states to enhance the convergence rate of the sampling error at measurement, which results in a metrologically enhanced reduced process tomography protocol.