Rare Events and Griffiths Phases in Topological Quantum Error Correction

TL;DR

The paper addresses how extended rare-error events, modeled as spatio-temporal disorder, impact the thresholds of topological quantum codes. It maps decoding to disordered statistical-mechanics models—the 2D RBIM for the repetition code and the 3D RPGT for the toric code—and analyzes the role of largest rare regions via Griffiths physics, including McCoy-Wu analogies and dualities. The main findings are that linear rare regions create a Griffiths phase with stretched-exponential scaling of logical failure in the repetition code, while planar rare regions in the toric code cause an immediate loss of the decoding threshold; these results hold across analytical arguments and finite-size numerics using MWPM decoding. The work highlights the importance of mitigating long-lasting error bursts in quantum hardware and suggests universal scaling behavior that could guide error-m mitigation strategies in future QEC implementations.

Abstract

The performance of quantum error correcting (QEC) codes are often studied under the assumption of spatio-temporally uniform error rates. On the other hand, experimental implementations almost always produce heterogeneous error rates, in either space or time, as a result of effects such as imperfect fabrication and/or cosmic rays. It is therefore important to understand if and how their presence can affect the performance of QEC in qualitative ways. In this work, we study effects of non-uniform error rates in the representative examples of the 1D repetition code and the 2D toric code, focusing on when they have extended spatio-temporal correlations; these may arise, for instance, from rare events (such as cosmic rays) that temporarily elevate error rates over the entire code patch. These effects can be described in the corresponding statistical mechanics models for decoding, where long-range correlations in the error rates lead to extended rare regions of weaker coupling. For the 1D repetition code where the rare regions are linear, we find two distinct decodable phases: a conventional ordered phase in which logical failure rates decay exponentially with the code distance, and a rare-region dominated Griffiths phase in which failure rates are parametrically larger and decay as a stretched exponential. In particular, the latter phase is present when the error rates in the rare regions are above the bulk threshold. For the 2D toric code where the rare regions are planar, we find no decodable Griffiths phase: rare events which boost error rates above the bulk threshold lead to an asymptotic loss of threshold and failure to decode. Unpacking the failure mechanism implies that techniques for suppressing extended sequences of repeated rare events (which, without intervention, will be statistically present with high probability) will be crucial for QEC with the toric code.

Figures (17)

• Figure 1: Statistical mechanics models: (a) a 2D RBIM corresponding to the decoding problem of the repetition code and (b) a 3D RPGT corrsponding to the toric code. The horizontal $x/y$ directions are the spatial directions of the codes, and the vertical $z$ direction is time. Both models have time varying bit flip errors. The red shaded regions are rare regions where the error rate is higher, i.e. where the vertical bond/plaquette coupling strength is weaker. Bit flip and measurement errors are shown via thick red links/dark shaded plaquettes. A higher density of them is located in the rare regions. The largest rare region is of height $\sim \mathcal{O}(\log L)$. (c) and (d) are the stat mech and decoding phase diagrams of the models in (a) and (b) respectively as a function of $p_{\rm rare}$ (see main text).
• Figure 2: The relevant defects for the models shown in Fig. \ref{['fig:cartoons']}(a,b). Panel (a) depicts the repetition code where the defect is a line of flipped vertical bonds corresponding to a domain wall. Panel (b) depicts the toric code where defect is a line of flipped plaquettes, also called a flux tube. Both defects are shown in rare regions where they are more likely to occur. Logical errors correspond to large defects, of sizes comparable to the linear system size $L$
• Figure 3: Threshold of the repetition code with time varying qubit error rates. For a length $L$ repetition code, we take $L$ rounds of measurements. The threshold is at $p_{\rm rare} \approx 0.2$ and this is the point $p_{\rm D}$ in the phase diagram \ref{['fig:cartoons']}(c). In the inset, the median failure rate at $p_{\rm rare} = 0.08, 0.15$ (green, blue respectively). Though difficult to properly resolve the difference, when $p_{\rm rare} < p_0^{2D} \approx 0.1$, the failure rate is consistent with exponential decay and when $p_0^{2D} < p_{\rm rare} < p_{\rm D}$, the failure rate is consistent with stretched exponential decay in $L$.
• Figure 4: Stretched exponential power and defect cost scaling varies continuously throughout the phase. The inset shows the empirical value of $z$ obtained from the slopes of the lines in the panel as a function of $p_{\rm rare}$. Note that empirically, $z$ does not exactly equal $0$ for $p_{\rm rare} < p_0^{2D}$ (see Appendix \ref{['sec:clean']})
• Figure 5: Threshold of a length $L$ repetition code with $\log_2 L$ measurements taken, with single error rate $p$. The defect cost only vanishes at a point $p_{\rm D}$ which is above $p_0^{2D}$. The inset shows a vertical cut through the plot at a point beneath the threshold where we observe the failure rate decaying as a stretched exponential in $L$.