Modeling error correction with Lindblad dynamics and approximate channels

TL;DR

We study error correction for a quantum code under realistic Lindblad noise, incorporating both $1$-qubit and $2$-qubit interactions within the $5$-qubit code under a code-capacity framework. We compare a full Lindblad dynamics, a composite-channel approximation, a multi-qubit Pauli approximation, and 1Q approximations to determine when they reproduce the code's performance. We find that the composite-channel approximation remains accurate across relevant parameters and timescales until noncommuting terms accumulate, while the Pauli approximation can be accurate only at short times and may underestimate logical failures at longer times, with 1Q approximations generally unreliable in the presence of 2Q noise. The results provide guidance for realistically modeling quantum error correction and for designing decoders that leverage connectivity and dominant noise terms to improve logical fidelity in near-term devices.

Abstract

We analyze the performance of a quantum error correction code subject to physically motivated noise modeled by a Lindblad master equation. We consider dissipative and coherent single-qubit terms and two-qubit crosstalk, studying how different approximations of the noise capture the success rate of a code. Focusing on the five-qubit code and adapting it to partially correct two-qubit errors in relevant parameter regimes according to the noise model, we find that a composite-channel approximation where every noise term is considered separately captures the behavior in many physical cases up to long timescales, eventually failing due to the effect of noncommuting terms. In contrast, we find that single-qubit approximations do not properly capture the error correction dynamics with two-qubit noise, even for short times. A Pauli approximation going beyond a single-qubit channel is sensitive to the details of the noise, state, and decoder, and succeeds at short timescales relative to the noise strength, beyond which it fails. Furthermore, we point out a mechanism for a Pauli model failure where it underestimates the failure rate of a code even with frequent syndrome projection and correction cycles. These results shed light on the performance of error correction in the presence of realistic noise and can advance the ongoing efforts towards useful quantum error correction.

Figures (17)

• Figure 1: (a) A schematic depiction of the dynamical simulation of the underlying noise. The initial logical (pure) state is evolved in small increments of time, governed by the full Lindbaldian and turns into a mixed state. (b) A schematic depiction of the composite-channel approximation, obtained by solving each term in the Lindbladain separately, grouped into 1Q channels and 2Q channels corresponding to the 1Q terms and 2Q terms in the Lindbladian. We then apply the 2Q channel on every pair of connected qubits in the device, and the 1Q channels on every qubit. For example, in this sketch the qubit marked in red is connected to $n=2$ other qubits. (c) A sketch of a 1Q approximation, obtained by replacing the 2Q channels by 1Q channels. As discussed in the text, such an approximation is not reliable unless the 2Q noise can be neglected.
• Figure 2: The failure probability after error recovery -- $\eta$ of Eq. \ref{['eq:eta']} -- vs. $t$, the duration of the noise channel, and the noise parameters in Eqs. \ref{['eq:EagleParams1']}-\ref{['eq:EagleParams2']}, with dominant crosstalk noise. (a) An average over the initial states in Eq. \ref{['eq:states to average']}, in the short-time regime, showing that all approximations are identical. (b) The initial state is $|0\rangle_L$, shown up to the intermediate timescale, when the Pauli approximation breaks, and also the composite-channel approximation loses accuracy (shown for two possible orderings, see the text for details).
• Figure 3: The difference between the logical error when the FC ZZ decoder is used vs. the standard decoder, with ZZ crosstalk noise acting with all-to-all connectivity. Here $h=0$, $T_2=2T_1$, and the initial state $|+\rangle_L$ is evolved for $t=0.5\,\mu$s using the dynamical simulation. Positive values (colored red) imply that the modified decoder is beneficial, and this can be seen to hold for a wide range of parameters in the regime of dominant crosstalk.
• Figure 4: The failure probability after error recovery -- $\eta$ of Eq. \ref{['eq:eta']} -- vs. $t$, for $t$ the time of the noise channel, and the noise parameters in Eqs. \ref{['eq:EagleParams1']}-\ref{['eq:EagleParams2']}, with dominant crosstalk noise, and the modified FC ZZ decoder. An average over the initial states in Eq. \ref{['eq:states to average']}, in the short-time regime, shows that the FC ZZ decoder performs much better than the regular decoder, even though the Pauli approximation begins to diverge from the dynamical simulation already at $t\approx 0.7 \mu$s.
• Figure 5: The failure probability $\eta$ vs. $|\zeta|t$, for $t$ the time of the noise channel, with only ZZ crosstalk noise, of magnitude $|\zeta|$ [Eq. \ref{['eq:2QParams']}]. The composite-channel approximation is exact and identical to the full dynamical solution in this case. The curves are an average over the six logical Pauli eigenstates [Eq. \ref{['eq:states to average']}] in the short-time limit, with (a) the regular decoder, and (b) the FC ZZ decoder (the inset showing the same data using log-log axes). We can see that the latter decoder outperforms the former (by about an order of magnitude), as expected. The Pauli approximation holds until $|\zeta| t\approx 0.2$ for the regular decoder, while it doesn't hold for any value of $|\zeta| t$ for the FC ZZ decoder. See the text for a detailed discussion.