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Fault-Tolerant Logical Gates in the IBM Quantum Experience
Robin Harper, Steven T. Flammia
TL;DR
It was unclear whether error-detecting codes could improve encoded computation on noisy small-scale devices, where correlated noise might undermine expected gains. The paper benchmarks fault-tolerant gates in the [4] [2] code and finds a roughly tenfold reduction in two-qubit infidelity, while noting that whole-circuit performance remains above the fault-tolerance threshold.
Problem
Whether a small error-detecting code can conclusively improve encoded computation on noisy devices remains uncertain, including whether correlated noise defeats fidelity gains.
Source: pp. 1, 3Method
Results
Takeaways & Limitations
Virtualizing the noisy CNOT and detecting errors each contribute to the observed reduction, yielding an overall tenfold decrease in fault-tolerant gate-set infidelity.
Source: p. 4The device is not convincingly below the fault-tolerance threshold for complete circuits because the benchmarking approach is insensitive to SPAM errors.
Source: p. 2
Abstract
from arXiv · showhide
Quantum computers will require encoding of quantum information to protect them from noise. Fault-tolerant quantum computing architectures illustrate how this might be done but have not yet shown a conclusive practical advantage. Here we demonstrate that a small but useful error detecting code improves the fidelity of the fault-tolerant gates implemented in the code space as compared to the fidelity of physically equivalent gates implemented on physical qubits. By running a randomized benchmarking protocol in the logical code space of the [4,2,2] code, we observe an order of magnitude improvement in the infidelity of the gates, with the two-qubit infidelity dropping from 5.8(2)% to 0.60(3)%. Our results are consistent with fault-tolerance theory and conclusively demonstrate the benefit of carrying out computation in a code space that can detect errors. Although the fault-tolerant gates offer an impressive improvement in fidelity, the computation as a whole is not below the fault-tolerance threshold because of noise associated with state preparation and measurement on this device.
Supplementary material
The supplementary material details how the [4] [2] code and randomized benchmarking were implemented on IBM-QX5, including practical choices for state preparation and error post-selection. It also explains fitting approximations and experimental constraints that qualify some comparisons.
Implementation details
- QASM supported end-of-run error detection and post-selection, but the available interface offered no obvious way to perform error correction during a run.Successful runs had substantially improved fidelity, despite fewer runs remaining after discarding detected errors.Source: Supplementary material · p. 5
Experimental protocol
- Each experiment used 30 gate lengths, with every sequence measured 1,024 times; nine repetitions produced 36 sequences per length for each of four run types.Uncertainties were estimated by non-parametric bootstrapping, while reported infidelities were per real Clifford because QASM compiled group elements into shorter gates.Source: Supplementary material · pp. 5, 7, 8
Logical |00⟩ state
- Fault-tolerant logical |00⟩ preparation was omitted: QX5 connectivity adds 8 single-qubit and 5 two-qubit gates, while RB absorbs preparation errors into SPAM.The experiments therefore used only the first four gates of the preparation circuit.Source: Supplementary material · p. 6, Figure 3 · p. 6
Real RB
- The Realizable Group R(2) has 576 elements and averages just over four basic gates per element, compared with just over seven for the full two-qubit Clifford group.R(2) is half the size of the real Clifford group and 20 times smaller than the full Clifford group; the supplement verifies its use for Real RB.Source: Supplementary material · p. 6
Fitting the curves
- The conditional survival fit treats the probability of returning to the code space as approximately constant, although exact fit equations remain future work.Measured non-return rates rose from 0.3 in Real runs and 0.4 in Phased runs toward 0.5; the authors report this approximation did not alter the nearly order-of-magnitude improvement.Source: Supplementary material · p. 7
Comparison runs
- Physical-qubit benchmarking used qubits 12 and 5; a separate pair, qubits 12 and 13, yielded 5.4(2)% versus 5.8(2)% for the main pair.Runs were interleaved and batched near one another in time because device fidelity varied with recalibration and usage.Source: Supplementary material · pp. 5, 8