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Efficient measurement of quantum gate error by interleaved randomized benchmarking
Easwar Magesan, Jay M. Gambetta, B. R. Johnson, Colm A. Ryan, Jerry M. Chow, Seth T. Merkel, Marcus P. da Silva, George A. Keefe, Mary B. Rothwell, Thomas A. Ohki, Mark B. Ketchen, M. Steffen
TL;DR
The paper addresses scalable estimation of individual quantum-gate errors while accounting for state-preparation and measurement errors. It interleaves random Clifford gates with a gate of interest and applies the protocol to a superconducting qubit, obtaining bounded estimates that compare favorably with quantum process tomography.
Problem
The paper seeks a scalable way to estimate the error rates of individual quantum computational gates while accounting for state-preparation and measurement errors.
Method
The protocol interleaves a gate of interest with random Clifford gates, measures sequence survival probabilities, and fits their decay to estimate gate performance.
Results
The protocol produced bounded error estimates for Xπ/2 and Yπ/2 gates, including an estimated error rate of 0.003 ± 0.003 with bounds [0, 0.016].
Takeaways & Limitations
Interleaved benchmarking provides a scalable estimate independent of SPAM errors and a more reliable Clifford-gate performance estimate than quantum process tomography in this experiment.
Takeaways & Limitations
The first-order fidelity-decay model is valid only when the average variation of gate-dependent noise is sufficiently small, specifically γ^2 ≪ 2/[m(m + 1)].
Abstract
from arXiv · showhide
We describe a scalable experimental protocol for obtaining estimates of the error rate of individual quantum computational gates. This protocol, in which random Clifford gates are interleaved between a gate of interest, provides a bounded estimate of the average error of the gate under test so long as the average variation of the noise affecting the full set of Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits. We apply this protocol to a superconducting qubit system and find gate errors that compare favorably with the gate errors extracted via quantum process tomography.
Interleaving benchmarking protocol.—
The protocol benchmarks an individual Clifford gate by comparing standard randomized benchmarking with sequences that interleave the gate of interest. Survival probabilities are averaged and fitted to estimate the gate’s depolarizing parameter, from which bounded error estimates are obtained.
- Standard randomized benchmarking: Standard randomized benchmarking samples random Clifford sequences that compose to the identity and estimates average error from fidelity decay.The final inverse Clifford is computed efficiently using the Gottesman–Knill theorem.
- Standard randomized benchmarking: Survival probability includes state-preparation and measurement errors through the prepared state ρψ and measurement operator Eψ.Averaging over K random sequences produces the sequence fidelity Fseq(m, ψ).
- Interleaved benchmarking: Interleaved sequences alternate uniformly random Clifford gates with the target gate C, then append an inverse of the combined sequence.Sequence length is defined by the number of random gates.
- Interleaved benchmarking: The interleaved sequence fidelity is fitted to zeroth- or first-order decay models to obtain the target gate’s depolarizing parameter pC.The same survival-probability measurement and averaging procedure is used for the interleaved sequences.
- Error estimation: The target gate error is estimated from the standard and interleaved depolarizing parameters, with bounds supplied for general imperfect random gates.The protocol is exact for perfect random gates or when the average error is depolarizing; otherwise the bounds account for imperfect randomization.
Experimental implementation.—
The experiment implements interleaved randomized benchmarking for single-qubit gates and compares its estimates with quantum process tomography. The protocol uses randomized Clifford sequences, models fidelity decay, and reports gate-error estimates and bounds for Xπ/2 and Yπ/2 rotations.
- Experimental implementation.—: The interleaved sequences benchmark Xπ/2 and Yπ/2 gates by inserting each target rotation between random Clifford gates.Their lower fidelities reflect an effective doubling in the number of pulses in the interleaved sequences.
- Experimental implementation.—: 0.003 ± 0.003 is the best estimated error rate for the tested Xπ/2 and Yπ/2 gates, with bounds [0, 0.016].The estimate is obtained from the standard and interleaved depolarizing parameters using the protocol’s error expression.
- Experimental implementation.—: The protocol was tested against intentional Xπ/2 pulse miscalibrations and reliably tracked the anticipated pulse infidelity.Table I compares applied over-rotation, predicted error, experimentally extracted error, and bounds from Eq. (5).
- Experimental implementation.—: 32 random sequences of lengths 2–96 were used to measure standard and interleaved randomized-benchmarking fidelities.Standard RB gave p = 0.984 ± 0.004; interleaved Xπ/2 and Yπ/2 gave pC = 0.978 ± 0.005 and 0.979 ± 0.001.
- Experimental implementation.—: Quantum process tomography produced higher reported gate errors than interleaved benchmarking, which the authors attribute to state-preparation and measurement errors.The QPT errors were 0.011+0.011 −0.009 and 0.020+0.009 −0.008, respectively.
- Experimental implementation.—: The derivation uses the Clifford group’s unitary 2-design property to replace channel twirls with depolarizing channels.The first-order fitting model remains valid when the average variation of gate-dependent noise is sufficiently small, γ^2 ≪ 2/[m(m + 1)].
- Experimental implementation.—: The paper presents a scalable protocol for benchmarking individual quantum gates that accounts for state-preparation and measurement errors.The gate error is exact for perfect random gates or depolarizing average noise, and bounded when average noise variation is small.
Conclusion.—
The authors acknowledge discussions with Antonio Córcoles, John Smolin, and Joseph Emerson, along with support from NSERC, CIFAR, the Ontario government, and IARPA.
- Conclusion.—: The authors acknowledge discussions with Antonio Córcoles, John Smolin, and Joseph Emerson.
- Conclusion.—: EM acknowledges support from NSERC, CIFAR, and the Ontario government.
- Conclusion.—: The work acknowledges IARPA support under contract W911NF-10-1-0324.