Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography

Abstract

Quantum information processors promise fast algorithms for problems inaccessible to classical computers. But since qubits are noisy and error-prone, they will depend on fault-tolerant quantum error correction (FTQEC) to compute reliably. Quantum error correction can protect against general noise if—and only if—the error in each physical qubit operation is smaller than a certain threshold. The threshold for general errors is quantified by their diamond norm. Until now, qubits have been assessed primarily by randomized benchmarking, which reports a different error rate that is not sensitive to all errors, and cannot be compared directly to diamond norm thresholds. Here we use gate set tomography to completely characterize operations on a trapped-Yb+-ion qubit and demonstrate with greater than 95% confidence that they satisfy a rigorous threshold for FTQEC (diamond norm ≤6.7 × 10−4).

Experimental quantum verification in the presence of temporally correlated noise

“…26 The effect of gauge optimisation in the presence ofσ z errors and with use of the default gate set, however, is concerning as a key implied benefit of experimental GST is its ability to provide a rigorous upper bound on gate errors using a fully self-contained analysis package. Recent experimental work 10 on the topic claimed such upper bounds on gate errors using experimental GST and compared these to the fault-tolerance threshold with high reported confidence and tight uncertainties. The results above and observations made 24 suggest that there may be residual uncertainty in interpretation of such data due to the potential unresolved conflict between full gauge freedom and the nominal existence of a measurement basis constraining that freedom.…”

“…identical to those used in reference 10 and recommended as standard GST in the pyGSTi tutorials. In our numerical analysis, we extend the standard gate set from {G I , G x , G y } → {G I , G x , G y , −G x , −G y } while also expanding the germ set from 11 to 39 elements to maintain amplificational completeness (see Supplementary Material for details).…”

“…We define sequences to include fiducial operations and germs (see "Methods" and ref. 10 ), and calculate the corresponding walk lengths. Here we assume unit step size under application of either a constantσ z orσ x unitary error process (Fig.…”

“…11 Despite their differences, these protocols share the common theme that they were originally developed and mathematically formalised assuming that error processes are statistically independent and do not exhibit strong correlations in time. 1,2,10 Even in highly controlled laboratory environments there are a range of noise sources that, when applied to a qubit concurrent with logical gate operations, produce effective error models that diverge significantly from the assumptions underlying most QCVV protocols. For example, slow variations in ambient magnetic fields or drifts in amplifier gain can produce temporally correlated noise processes, often characterised through a power spectral density possessing large weight at low frequencies.…”

“…A variety of techniques have emerged including randomised benchmarking (RB), 1,2 purity benchmarking, 3 process tomography, [4][5][6][7] adaptive methods, 8 and gate-set tomography (GST). 9,10 Each protocol has relative strengths and weaknesses; for instance, RB has low experimental overhead but only provides average information about gate performance, while process tomography provides more information at the cost of unfavourable scaling in measurement overhead. 11 Despite their differences, these protocols share the common theme that they were originally developed and mathematically formalised assuming that error processes are statistically independent and do not exhibit strong correlations in time.…”