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Randomized Benchmarking of Quantum Gates
E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, D. J. Wineland
TL;DR
The paper addresses how to estimate quantum-gate errors without relying on accurate state preparation and measurement. It introduces randomized benchmarking, in which random gate sequences and exponential decay reveal computationally relevant error rates and can assess behavior over long computations.
Problem
Process tomography is limited by state-preparation, measurement, and gate errors, scales inefficiently with qubit number, and may not reveal error compounding in long sequences.
Method
Randomized benchmarking applies random Clifford-gate sequences of varying lengths and estimates average per-gate error from the increase in final-measurement errors.
Results
Exponential decay of sequence error probabilities yields the asymptotic average error probability per randomized computational gate.
Takeaways & Limitations
The method provides computationally relevant error estimates while accommodating imperfect preparation and measurement and testing error behavior in long sequences.
Abstract
from arXiv · showhide
A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a one-qubit error probability per randomized pi/2 pulse of 0.00482(17) in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.
I. INTRODUCTION
Scalable quantum computing requires gates with very low, context-independent errors, but process tomography is limited by calibration assumptions, exponential scaling, and short-sequence coverage. The paper proposes randomized benchmarking to estimate computationally relevant gate errors in long sequences.
- Motivation: Scalable quantum computing requires coherently controlling large numbers of qubits through many computational steps.The smallest useful applications described require hundreds of qubits and many millions of steps.
- Motivation: Error probabilities per unitary gate should be well below 10^-2, with a practical target below 10^-4.These targets motivate experimental methods capable of establishing very low error probabilities.
- Limitations of process tomography: Process tomography requires tomography gates with lower error than the gate being tested and scales exponentially with qubit number.It also does not establish that a gate performs equally well in every required context.
- Proposed method: Randomized benchmarking estimates error probability per gate in computational contexts using random unitary sequences and their inverses.Under stated noise assumptions, randomization depolarizes the noise and sequence fidelity decay represents its strength.
- Proposed method: The method applies random gate sequences of varying lengths and uses randomized measurements to assess computationally relevant errors.The gates are drawn from the Clifford group, which supports deterministic one-qubit Pauli measurements and targets errors relevant to stabilizer-based fault tolerance.
- Scope: The method provides only an overall average fidelity for gate noise unless refined to obtain more specific multiqubit noise information.The paper notes that related randomization strategies can provide more detailed information.
II. RANDOMIZED BENCHMARK OF ONE QUBIT
The one-qubit benchmark estimates computational-gate error by averaging randomized pulse sequences of different lengths and fitting their measurement-error growth. Its exponential decay model connects the fitted decay constant to average error, while sequence randomization also supports checks of specific computations.
- Experimental procedure: Each one-qubit experiment prepares |0⟩, applies alternating Pauli randomization and π/2 computational pulses, and performs a final measurement.The Pauli and computational pulses are chosen randomly from specified Clifford-group rotations.
- Sequence construction: The randomized sequence length l is defined as the number of π/2 pulses, while π pulses serve only to randomize errors.The combined π/2 and Pauli pulses are called randomized computational gates.
- Error model: For independent depolarizing errors, p_l = (1 − (1 − d_if)(1 − d)^l)/2, where d is the average depolarization probability per randomized computational gate.The preparation, initial Pauli pulse, and measurement contributions are combined in d_if.
- Error model: p_l decays exponentially to 1/2, and the decay constant yields d.The average computationally relevant error per step is related to the depolarization parameter by 1 − F_a = d/2.
- Sequence-level checks: The protocol can characterize specific randomized computations and fixed sequence instances, not only the overall average error.Sequences are generated independently, truncated at multiple lengths, completed with a final π/2 pulse, and randomized with Pauli pulses.
- Data reduction: The average error probability p_l is estimated from incorrect final measurements as a function of sequence length.The protocol averages estimates across computational sequences and Pauli randomizations, while also recording sequence-specific probabilities.
III. TRAPPED-ION-QUBIT IMPLEMENTATION
The experiment implemented randomized benchmarking on a trapped 9Be+ ion qubit, using Raman transitions and phase-controlled rotations. Hardware and FPGA constraints limited sequences to about 100 computational pulses.
- The qubit used two ground-state hyperfine levels of a trapped 9Be+ ion, identified as computational states |0⟩ and |1⟩.
- State preparation used optical pumping after laser cooling, while state-dependent fluorescence distinguished the two qubit states.
- Raman transitions implemented x- and y-axis rotations, while programmed beam-phase changes implemented z-axis rotations.
- The FPGA supported about 100 computational pulses, and the longest experimental sequence contained 96 computational gates.
IV. EXPERIMENTAL RESULTS
The experiment sampled randomized gate sequences at multiple lengths and repeated Pauli randomizations, with recalibration and measurement statistics used to assess error probabilities.
- Four random computational sequences were truncated to 17 lengths ranging from 2 to 96 gates.
- Each truncated sequence received 8 Pauli randomizations and was applied 8160 times in four randomized experimental groups.
- Pulse durations, resonant frequencies, and Stark shifts were automatically recalibrated at regular intervals.
- Independent experiments quantifying different error types were consistent with the randomized benchmarking data.
V. THEORETICAL CONSIDERATIONS
The theoretical analysis defines randomized benchmarking through asymptotic exponential decay and identifies assumptions governing how gate errors can be interpreted. It also distinguishes average error notions and notes that randomized benchmarking primarily gives an overall noise fidelity.
- Randomized benchmarking estimates an asymptotic average error probability per randomized computational gate from the eventual exponential decay of measured error probabilities.
- The analysis assumes memoryless errors, independent errors on parallel disjoint-qubit gates, stationary errors, and no leakage outside the computational subsystem.
- Figure 1 plots final-state fidelity on a logarithmic scale across randomized sequence lengths and shows scatter exceeding standard errors, suggesting significant coherent-error contributions.
- The asymptotic average error probability need not equal the average error probability, although the authors conjecture useful bounds relating them.
- Pauli randomization converts errors to random Pauli operators to first order, while computational-gate randomization averages errors over the Clifford group.
VI. BENCHMARKING MUTLIPLE QUBITS
Randomized benchmarking can extend to multiple qubits by incorporating multiqubit gates and using Pauli-based final measurements. The one-qubit data show exponential fidelity decay corresponding to a per-gate error estimate, while multiqubit readout uses parity.
- Benchmarking multiple qubits: The method is intended to verify that many multiqubit gates can be applied consistently without worsening error behavior during computation.
- Experimental benchmark: 0.00482(17) is the error probability per randomized computational gate implied by the fitted exponential fidelity decay.
- Benchmarking multiple qubits: Randomized benchmarking can be extended to two or more qubits by adding multiqubit gates and relying on rapid mixing rather than sampling every Clifford element.
- Benchmarking multiple qubits: For multiple qubits, a stabilizing Pauli product is converted to σz measurements, whose parity provides a deterministic no-error answer.
APPENDIX A: DIRECT ERROR CHARACTERIZATIONS
Direct error-characterization experiments quantify major pulse-error sources and provide consistency checks for randomized benchmarking. The estimates identify phase decoherence and amplitude fluctuations as substantially larger contributions than spontaneous emission.
- The direct error-characterization experiments measure initial gate errors and serve as consistency checks for randomized benchmarking data.Known error sources include phase errors, amplitude errors, and spontaneous emission.
- 0.0037(1) is the estimated contribution of unrefocusable phase decoherence to the error probability per step.The estimate comes from fitting the initial part of a refocused Ramsey decay curve.
- 0.0090(7) is the inferred error contribution per step from Ramsey measurements without refocusing.This larger value is attributed to refocusing effects from Pauli randomization in the benchmarking sequences.
- 0.006(3) is the estimated amplitude-fluctuation contribution per step from the initial oscillations of a Rabi-flopping curve.This estimate is consistent with the randomized-experiment error probability, while also including phase fluctuations during computation pulses.