Effect of noise correlations on randomized benchmarking

Abstract

Among the most popular and well studied quantum characterization, verification and validation techniques is randomized benchmarking (RB), an important statistical tool used to characterize the performance of physical logic operations useful in quantum information processing. In this work we provide a detailed mathematical treatment of the effect of temporal noise correlations on the outcomes of RB protocols. We provide a fully analytic framework capturing the accumulation of error in RB expressed in terms of a three-dimensional random walk in "Pauli space." Using this framework we derive the probability density function describing RB outcomes (averaged over noise) for both Markovian and correlated errors, which we show is generally described by a gamma distribution with shape and scale parameters depending on the correlation structure. Long temporal correlations impart large nonvanishing variance and skew in the distribution towards high-fidelity outcomesconsistent with existing experimental data -highlighting potential finite-sampling pitfalls and the divergence of the mean RB outcome from worst-case errors in the presence of noise correlations. We use the Filtertransfer function formalism to reveal the underlying reason for these differences in terms of effective coherent averaging of correlated errors in certain random sequences. We conclude by commenting on the impact of these calculations on the utility of single-metric approaches to quantum characterization, verification, and validation.

Experimental quantum verification in the presence of temporally correlated noise

“…The differences in the distribution of measured survival probabilities over sequences under these two noise models reproduces the central predictions of ref. 17 We compare the characteristics of the distributions themselves against analytic predictions for both slowly and rapidly varying noise, beginning with the measured expectation, E I ð Þ, and variance, V I ð Þ (Fig. 2b, c), finding good agreement by taking only the applied noise strength as an input into a theoretical model (see Supplementary Materials).…”

“…Recent theoretical studies have demonstrated that measurements on RB sequences in the presence of temporal noise correlations, can produce a divergence between average and worst-case reported trace fidelities. 17,20 Thus we find that measurement outcomes for different RB sequences are characterised by distributions with distinctly different shapes depending on the temporal correlations in the noise. The standard practice of combining all measurements to extract an RB error rate, p RB , from the decay of the mean over all J-gate sequences as a function of J, results in a global ensemble average and does not take advantage of this information (formally, as the noise we implement exhibits temporal correlations, the value of p RB one extracts may not be meaningful as a measure of average Clifford gate error).…”

“…17 . Error accumulation over a given Clifford sequence maps to a "random walk" in a three-dimensional vector-space representing the action of sequential error unitaries in the operator space spanned by the Pauli operators,σ fx;y;zg (Fig.…”

“…In the case of slowly varying noise, and to first-order approximation, the net error can be separated into two independent parts, kRk 2 ¼ δ 2 kṼk 2 , where δ is the value of the noise and kṼk is the net unit-step walk specific to a particular sequence. 17 However, in the case of rapidly varying noise these two terms are no longer separable and the net error must be calculated as the convolution of the noise value at each timestep and each individual step in the random walk,…”

“…15 Understanding that such an approach is untenable when attempting to rigorously compare QCVV results to metrics relevant to quantum error correction has recently led to an expansion of theoretical activity in this space. 3,[16][17][18][19][20] In this work our objectives are to experimentally characterise and explain the impact of temporally correlated noise processes on the outputs of QCVV protocols, and to identify potential modifications enabling users to improve the utility of the information returned. We perform QCVV experiments using a single trapped 171 Yb + ion as a long-lived, high-stability qubit.…”