Abstract In quantum computation the target fidelity of the qubit gates is very high, with the admissible error being in the range from 10−3 to 10−4 and even less, depending on the protocol. The direct experimental determination of such a tiny error is very challenging by standard quantum-process tomography. Instead, the method of randomized benchmarking, which uses a random sequence of Clifford gates, has become a standard tool for determination of the average gate error as the decay constant in the exponentially decaying fidelity. In this paper, the task for determining a tiny error is addressed by sequentially repeating the same gate multiple times, which leads to the coherent amplification of the error, until it reaches large enough values to be measured reliably. If the transition probability is p = 1 − ϵ with ϵ ≪ 1 in the single process, then classical intuition dictates that the probability after N passes should be P N ≈ 1 − N ϵ . However, this classical expectation is misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) symmetry of the qubit propagator, resulting in explicit analytic relations that link the N-pass propagator to the single-pass one in terms of Chebyshev polynomials. In particular, the relations suggest that in some special cases the N-pass transition probability degrades as P N = 1 − N 2 ϵ , i.e. dramatically faster than the classical probability estimate. In the general case, however, the relation between the single-pass and N-pass propagators is much more involved. Recipes are proposed for unambiguous determination of the gate errors in the general case for both Clifford and non-Clifford gates.
Abstract In quantum computation the target fidelity of the qubit gates is very high, with the admissible error being in the range from 10−3 to 10−4 and even less, depending on the protocol. The direct experimental determination of such a tiny error is very challenging by standard quantum-process tomography. Instead, the method of randomized benchmarking, which uses a random sequence of Clifford gates, has become a standard tool for determination of the average gate error as the decay constant in the exponentially decaying fidelity. In this paper, the task for determining a tiny error is addressed by sequentially repeating the same gate multiple times, which leads to the coherent amplification of the error, until it reaches large enough values to be measured reliably. If the transition probability is p = 1 − ϵ with ϵ ≪ 1 in the single process, then classical intuition dictates that the probability after N passes should be P N ≈ 1 − N ϵ . However, this classical expectation is misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) symmetry of the qubit propagator, resulting in explicit analytic relations that link the N-pass propagator to the single-pass one in terms of Chebyshev polynomials. In particular, the relations suggest that in some special cases the N-pass transition probability degrades as P N = 1 − N 2 ϵ , i.e. dramatically faster than the classical probability estimate. In the general case, however, the relation between the single-pass and N-pass propagators is much more involved. Recipes are proposed for unambiguous determination of the gate errors in the general case for both Clifford and non-Clifford gates.
In quantum computation the target fidelity of the qubit gates is very high, with the admissible error being in the range from 10−3 to 10−4 and even less, depending on the protocol. The direct experimental determination of such an extremely small error is very challenging by standard quantum-process tomography. Instead, the method of randomized benchmarking, which uses a random sequence of Clifford gates, has become a standard tool for determination of the average gate error as the decay constant in the exponentially decaying fidelity. In this paper, the task for determining a tiny error is addressed by sequentially repeating the \emph{same} gate multiple times, which leads to the coherent amplification of the error, until it reaches large enough values to be measured reliably. If the transition probability is p=1−ε with ε≪1 in the single process, then classical intuition dictates that the probability after N passes should be PN≈1−Nε. However, this classical expectation is misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) symmetry of the qubit propagator, resulting in explicit analytic relations that link the N-pass propagator to the single-pass one in terms of Chebyshev polynomials. In particular, the relations suggest that in some special cases the N-pass transition probability degrades as PN=1−N2ε, i.e. dramatically faster than the classical probability estimate. In the general case, however, the relation between the single-pass and N-pass propagators is much more involved. Recipes are proposed for unambiguous determination of the gate errors in the general case, and for both Clifford and non-Clifford gates.
Cross-entropy benchmarking (XEB) with single-qubit reference sequences is widely used to characterize multi-qubit gates in large-scale quantum processors, despite the lack of a rigorous theoretical justification. Here we show that the commonly employed additive single-qubit errors approximation underlying this approach breaks down and leads to a systematic overestimation of gate fidelities. We derive an analytical expression for the joint decay of simultaneous single-qubit reference sequences, identify when interleaved circuits generate sufficient randomization for the standard depolarizing approximation, and obtain a refined expression for the interleaved gate fidelity estimation. Experiments on a superconducting quantum processor validate the theory and demonstrate that fidelities obtained using XEB with single-qubit references agree with those extracted from standard interleaved randomized benchmarking (IRB), while achieving higher precision due to reduced reference-sequence errors. Our results establish theoretical foundation for interleaved benchmarking protocols employing single-qubit reference sequences and show that, with appropriate post-processing, they provide a simple and accurate alternative to multi-qubit Clifford reference circuits.
Abstract Implementing arbitrary single-qubit gates with near perfect fidelity is among the most fundamental requirements in gate-based quantum information processing. In this work, we fabricate a transmon qubit with long coherence times and demonstrate single-qubit gates with the average gate error below 10 −4 , i.e. (7.42 ± 0.04) × 10 −5 by randomized benchmarking (RB). To understand the error sources, we experimentally obtain an error budget, consisting of the decoherence errors lower bounded by (4.62 ± 0.04) × 10 −5 and the leakage rate per gate of (1.16 ± 0.04) × 10 −5 . Moreover, we reconstruct the process matrices for the single-qubit gates by the gate set tomography (GST), with which we simulate RB sequences and obtain single-qubit fidelities consistent with experimental results. We also observe non-Markovian behavior in the experiment of long-sequence GST, which may provide guidance for further calibration. The demonstration extends the upper limit that the average fidelity of single-qubit gates can reach in a transmon-qubit system, and thus can be an essential step towards practical and reliable quantum computation in the near future.
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 benchmarking of quantum gates is crucial for understanding and enhancing the performance of quantum hardware. A standard method for this is interleaved benchmarking, a technique which estimates the error on an interleaved target gate by comparing cumulative error rates of randomized sequences implemented with the interleaved gate and without it. In this work, we show both numerically and experimentally that the standard approach of interleaved randomized benchmarking (IRB), which uses the multi-qubit Clifford group for randomization, can produce highly inaccurate and even physically impossible estimates for the error on the interleaved gate in the presence of coherent errors. Fortunately we also show that interleaved benchmarking performed with cycle benchmarking, which randomizes with single qubit Pauli gates, provides dramatically reduced systematic uncertainty relative to standard IRB, and further provides as host of additional benefits including data reusability. We support our conclusions with a theoretical framework for bounding systematic errors, extensive numerical results comparing a range of interleaved protocols under fixed resource costs, and experimental demonstrations on three quantum computing platforms.
Raman qubits, represented by two ground or metastable quantum states coupled via an intermediate state, hold some advantages over directly coupled qubits, most notably much longer radiative lifetimes, shorter gate duration, and lower radiation intensity due to using electric-dipole allowed optical transitions. They are also relatively simple to implement and control, making them an attractive option for building quantum gates for quantum computers. In this work, we present a simple and fast tomographic method to measure the errors of Raman qubit gates possessing the Morris-Shore dynamic symmetry. The latter occurs when the qubit states are on two-photon resonance and the driving fields have the same time dependence. The method is based on repeating the same gate multiple times, which amplifies the small coherent errors to sufficiently large values, which can be measured with high accuracy and precision. Then the (small) gate errors can be determined from the amplified errors by using the analytical connections between them.