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Optimal quantum control using randomized benchmarking
J. Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, I. -C. Hoi, E. Jeffrey, A. Megrant, J. Mutus, C. Neill, P. J. J. O`Malley, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. N. Cleland, John M. Martinis
TL;DR
As quantum-control errors become smaller, optimization requires a benchmarking signal that remains sensitive to fractional errors. The paper derives and demonstrates ORBIT, showing that halving error doubles its optimal sequence length and sensitivity while improving CZ-gate performance.
Problem
Quantum-control optimization needs benchmarking sensitivity that remains effective as gate errors become arbitrarily small.
Method
The paper derives randomized-benchmarking sequence-fidelity sensitivity to gate error and uses ORBIT to select an error-dependent sequence length for optimization.
Results
Halving the error from 0.001 to 0.0005 doubled the optimal sequence length and sensitivity, while CZ optimization reduced reference error per Clifford from 0.0361 to 0.0188.
Takeaways & Limitations
ORBIT retains sensitivity across progressively smaller errors, supporting its use for optimizing high-fidelity gates.
Abstract
from arXiv · showhide
We present a method for optimizing quantum control in experimental systems, using a subset of randomized benchmarking measurements to rapidly infer error. This is demonstrated to improve single- and two-qubit gates, minimize gate bleedthrough, where a gate mechanism can cause errors on subsequent gates, and identify control crosstalk in superconducting qubits. This method is able to correct parameters to where control errors no longer dominate, and is suitable for automated and closed-loop optimization of experimental systems.
Supplementary Information for ‘Optimal quantum control using randomized benchmarking’ · SCALING OF THE SENSITIVITY OF ORBIT WITH ERROR
The supplementary analysis shows that ORBIT’s sensitivity to fractional gate error is constant, allowing it to scale to arbitrarily small errors. The optimal sequence length changes with error, doubling when the error is halved.
- SCALING OF THE SENSITIVITY OF ORBIT WITH ERROR: ORBIT can in principle scale to arbitrarily small errors because its sensitivity to fractional error remains constant.This follows from the derived sensitivity of sequence fidelity to gate error.
- SCALING OF THE SENSITIVITY OF ORBIT WITH ERROR: For single-qubit benchmarking, sequence fidelity follows F = Ap^m + B with p = 1 − 2r, where r is the error per Clifford.The corresponding fidelity variation is dF/dr = −2Am(1−2r)^(m−1).
- SCALING OF THE SENSITIVITY OF ORBIT WITH ERROR: The optimal ORBIT sequence length is the characteristic decay m′ = −1/ln(1 − 2r).This is obtained by expressing fidelity as F = A exp(−m/m′) + B.
- SCALING OF THE SENSITIVITY OF ORBIT WITH ERROR: The same fractional sensitivity applies when improving a 99.0% gate to 99.9% or a 99.99% gate to 99.999%.Only the required value of m differs between these fidelity ranges.
- SCALING OF THE SENSITIVITY OF ORBIT WITH ERROR: 500 and 1000 are the maximum-sensitivity values of m′ for r = 0.001 and r = 0.0005, respectively, with A = 0.5.These examples illustrate the scaling predicted by the analysis.
- SCALING OF THE SENSITIVITY OF ORBIT WITH ERROR: When the error is halved, the optimal m and sensitivity both double.The examples r = 0.001 and r = 0.0005 exhibit this relationship.
CZ GATE FIDELITY BEFORE AND AFTER NELDER-MEAD OPTIMIZATION
The CZ gate was evaluated with reference and interleaved randomized benchmarking before and after Nelder–Mead optimization. A self-consistency calculation predicted expected errors per Clifford of 0.0318 before and 0.0185 after.
- CZ gate fidelity before and after Nelder–Mead optimization: Reference and interleaved randomized benchmarking data for the CZ gate are shown before and after improvement in Fig. S2.Figure S2a presents the pre-improvement data, while Fig. S2b presents the post-improvement data.
- CZ gate fidelity before and after Nelder–Mead optimization: The expected error per Clifford is rref,expected = 8.25rSQ + 1.5rCZ when gate errors are small and uncorrelated.Here, rSQ is the average single-qubit gate error and rCZ is the CZ gate error.
CONTROL CROSSTALK DATA
Supplementary data document randomized-benchmarking measurements for two-qubit gate optimization and control-crosstalk analysis. The data include sequence-fidelity results corresponding to main-text figures.
- Two-qubit gate optimization: Two-qubit randomized benchmarking measured reference, interleaved, and extracted CZ-gate errors before and after Nelder-Mead optimization.Before optimization, rref = 0.0361, rref+CZ = 0.0511, and rCZ = 0.0157; afterward, rref = 0.0188, rref+CZ = 0.0254, and rCZ = 0.0068.