Single snapshot non-Markovianity of Pauli channels

TL;DR

Pauli channels shaped by twirling often do not admit a Markovian Pauli-Lindblad generator; by representing channels with Pauli pseudo-Lindblad generators that allow negative or complex rates, the authors show that non-Markovianity is typical in multi-qubit Pauli channels, even when the underlying physical noise is Markovian. They derive the generator from the Pauli transfer matrix via the Walsh-Hadamard transform, analyze simple, random, and physically motivated twirled noise, and validate the framework experimentally on superconducting qubits. The work also extends mitigation strategies to non-Markovian noise via generalized probabilistic error amplification and cancellation, with explicit overhead formulas and practical learning procedures. The results have direct implications for noise modeling and error mitigation in near-term quantum devices.

Abstract

Pauli channels are widely used to describe errors in quantum computers, particularly when noise is shaped via Pauli twirling. A common assumption is that such channels admit a Markovian generator, namely a Pauli-Lindblad model with non-negative rates, but the validity of this assumption has not been systematically examined. Here, using CP-indivisibility as our criterion for non-Markovianity, we study multi-qubit Pauli channels from a single snapshot of the dynamics. We find that while the generator always has the same structure as the standard Pauli-Lindblad model, the rates may be negative or complex. We show that random Pauli channels are almost always non-Markovian, with the probability of encountering a negative rate converging doubly exponentially to unity with the number of qubits. For physically motivated noise models shaped by Pauli twirling, including single-qubit over-rotations and two-qubit amplitude damping errors, we find that negative rates are generic, even when the underlying physical noise is Markovian. We generalize probabilistic error amplification and cancellation to non-Markovian generators, and quantify the sampling overhead introduced by negative and complex rates. Experiments on superconducting qubits confirm that allowing negative rates in the learned noise model yields more accurate predictions than restricting to non-negative rates.

Figures (6)

• Figure 1: Non-Markovianity of random Pauli channels. (a) The probability of having a negative Lindblad rate $p(\lambda_{\min}<0)$ as a function of infidelity $r$ grows with increasing system size $n$. Numerical simulations (circles) agree well with our analytical results in Eq. \ref{['eq:problambdamin']}. In the shaded grey region the rates for the smallest system size become complex and are not shown. (b) The minimum Lindblad rate $\lambda_{\min}$ at $r=0.09$ (dashed line in the top panel). The scatter points are samples from numerical simulations and the diamond markers indicate their mean values. Our analytical result for the mean in Eq. \ref{['eq:explambdamin']} agrees well with the numerical results. For larger $n$ the values concentrate around their mean. While negative values become more common at larger $n$, their magnitude gets smaller.
• Figure 2: Lindblad rates for a noisy Hadamard gate with timing error. Error in timing $\delta t$ results in over/under rotations. For $\abs{\Omega\delta t}<\pi/4$, where $\Omega$ is the strength of the Hamiltonian evolution, one of the rates $\lambda_Y$ is never positive. For larger errors $\pi/4<\abs{\Omega\delta t}<\pi/2$, all of the rates are complex (shaded region).
• Figure 3: Lindblad rates for a noisy $X_{\pi}$ gate with coherent amplitude and phase error. Phase errors can lead to off-axis errors $\epsilon$, whereas deviations from ideal amplitude $\Omega$ by $\delta \Omega$ cause over/under rotations errors. In the small noise regime, all the rates are real, with $\lambda_Z$ being negative (red region). For larger errors some of the rates become complex (black region).
• Figure 4: Lindblad rates for the two-qubit $ZZ_{\pi/2}(\pi/4)$ gate with amplitude damping error. (a) Lindblad rates (colors corresponding to the Pauli labels in the right panel) as a function of the damping rate $\kappa/J$, where $J$, the strength of the gate, sets the time scale. For all values of $\kappa/J$, one of the rates $\lambda_{ZZ}$ is nonpositive, and its magnitude grows with increasing $\kappa/J$. (b) Comparing different Lindblad rates at $\kappa/J=1$ corresponding to the dashed line in panel (a).
• Figure 5: Four-qubit experiment on ibm_pinguino1. (a) The structure of each layer with two parallel noisy CNOT gates $\tilde{\mathcal{G}}$, and the equivalent decomposition into a quasi-local noise channel $\mathcal{E}$ acting on the ideal gates $\mathcal{G}$. (b) The inferred Lindblad rates $(\lambda)$, including statistically significant negative values. (c) Comparison between experimentally measured expectation values ($f$) of nonlocal Pauli observables for 8 repetitions of the layer and model predictions for different noise models. Numbers in the legend indicate the average error between the model and raw data.