Towards a general framework of Randomized Benchmarking incorporating non-Markovian Noise

Abstract

The rapid progress in the development of quantum devices is in large part due to the availability of a wide range of characterization techniques allowing to probe, test and adjust them. Nevertheless, these methods often make use of approximations that hold in rather simplistic circumstances. In particular, assuming that error mechanisms stay constant in time and have no dependence in the past, is something that will be impossible to do as quantum processors continue scaling up in depth and size. We establish a theoretical framework for the Randomized Benchmarking protocol encompassing temporally-correlated, so-called non-Markovian noise, at the gate level, for any gate set belonging to a wide class of finite groups. We obtain a general expression for the Average Sequence Fidelity (ASF) and propose a way to obtain average gate fidelities of full non-Markovian noise processes. Moreover, we obtain conditions that are fulfilled when an ASF displays authentic non-Markovian deviations. Finally, we show that even though gate-dependence does not translate into a perturbative term within the ASF, as in the Markovian case, the non-Markovian sequence fidelity nevertheless remains stable under small gate-dependent perturbations.

Operational Markovianization in randomized benchmarking

“…It has been shown in [25] that the generalization of equation ( 1) to the non-Markovian gate-independent [46] noise case, has the (generally non-exponential) form…”

“…More generally than Markovian quality factors, they encode both SPAM and fidelity-like information of the noise across a whole RB process, reducing to A, B and p of equation (1) in the Markovian limit. In general, equation ( 3) is difficult to work with even in the time-stationary [47] noise regime, as opposed to the Markovian case, although it has been previously studied in [24,25].…”

“…where the sequence of gates and noise maps can be depicted as in figure 8; the average over the gate-set, E G , can then be computed analytically at least either when G is the multi-qubit Clifford group (generally any unitary 2-design) [24] or when it is a finite group [25] (extensions beyond finite groups or groups in general, have also been proposed, at least within the Markovian case [14,83]).…”

“…Explicitly, when G is a finite group admitting a multiplicity-free representation [84], the analytical ASF for a RB experiment with m gates drawn uniformly from G, under gate-independent non-Markovian noise, takes the form ( [25])…”

“…These kinds of correlations and their effects are generally known as non-Markovianity [16][17][18] and can be fully described in the quantum realm through multi-time generalizations of quantum channels [19][20][21]. A signature aspect of non-Markovian noise in RB is the non-trivial non-exponential decay of the data rendered by it [22][23][24][25][26], which makes the information about the noise and its correlations remarkably hard to analyze. Moreover, the key feature of RB of robustness against SPAM errors ceases to hold if any of these are correlated.…”