Blind-spots of Randomized Benchmarking Under Temporal Correlations

TL;DR

This work analyzes randomized benchmarking (RB) under temporally correlated, non-Markovian noise, including classical memory and memory arising from quantum environments. Using the process-matrix formalism, it derives analytic expressions for the average sequence fidelity (ASF) and shows that classical memory typically yields a sum of exponentials rather than a single decay, while monotonicity of the ASF distinguishes quantum memory effects. It identifies conditions under which RB becomes blind to temporal correlations (complete blindness) and provides operational criteria to witness such correlations, along with implications for worst-case (diamond-norm) errors, which can behave differently from average RB metrics. The findings highlight the need for complementary diagnostics to characterize non-Markovian noise in quantum hardware and offer guidance for interpreting RB data in the presence of classical memory and for optimizing strategies to mitigate or exploit memory effects in fault-tolerant design.

Abstract

Randomized benchmarking (RB) is a widely adopted protocol for estimating the average gate fidelity in quantum hardware. However, its standard formulation relies on the assumption of temporally uncorrelated noise, an assumption often violated in current devices. In this work, we derive analytic expressions for the average sequence fidelity (ASF) in the presence of temporally correlated (non-Markovian) noise with classical memory, including cases where such correlations originate from interactions with a quantum environment. We show how the ASF can be interpreted to extract meaningful benchmarking parameters under such noise and identify classes of interaction Hamiltonians that render temporal correlations completely invisible to RB. We further provide operational criteria for witnessing temporal correlations due to quantum memory through RB experiments. Importantly, while classical correlations may remain undetectable in the ASF data, they can nonetheless significantly affect worst-case errors quantified by the diamond norm, a metric central to fault tolerant quantum computing. In particular, we demonstrate that temporal correlations may suppress worst-case errors highlighting that temporal correlations may not always have detrimental effects on gate performance.

Figures (8)

• Figure 1: Schematic diagram of process matrix $W$ with $n$ intervention labs $\{L_1,L_2,\cdots,L_n\}$. The process matrix captures the most general correlations compatible with the local validity of quantum mechanics in the labs
• Figure 2: Schematic of two time-step multi-time process with labs $L_1,L_2$ and process matrix (in grey) capturing multi-time correlations due to system-environment interactions. Top figure shows a general multi-time process with initial state and system-environment unitaries. The middle figure shows a Markovian process where the initial state is a product state and environment traced out and prepared at each time-step. The bottom figure represents a process with classical memory where the environment feed-forwards classical information.
• Figure 3: RB protocol in the presence of non-Markovian noise: The process matrix construction provides a natural framework to study non-Markovian effects in RB. Each noisy gate $\{\mathcal{C}_1, \mathcal{C}_2, \cdots,\mathcal{C}_{m+1}\}$ can be divided into ideal Clifford gates $\{G_1,G_2,\cdots,G_{m+1}\}$ which act only on the system and the noise is captured by the process matrix comb (in grey).
• Figure 4: Modified RB protocol where the initial state and the final measurement are randomized using Clifford operations denoted by $\mathcal{V}, \mathcal{V}^{\dagger}$.
• Figure 5: Comparing the performance of ESPRIT with single exponential fit for the simulated data with exponents 0.9 and 0.99. The RMSE (root mean square error) and adjusted $R^2$ values for ESPRIT fit is 0.0068 and 0.9908 respectively, while the same parameters take respective values 0.0116 and 0.9735 for single exponential fitting.