Intrinsic Error Thresholds in Nearly Critical Toric Codes

Abstract

We study the protection of information in nearly critical topological quantum codes, constructed by perturbing topological stabilizer codes towards continuous quantum phase transitions. Our focus is on the transverse-field toric code subjected to local Pauli decoherence. Despite the strong quantum fluctuations of anyons when the transverse field is tuned infinitesimally close to the critical point, we show that a finite strength of Pauli decoherence remains necessary to irreversibly destroy information encoded in the ground-state manifold. Using a replica statistical physics mapping for the coherent information, we show that decoherence can be understood as introducing a two-dimensional inter-replica defect within a three-dimensional replica statistical physics model. A field theoretical analysis shows that this defect is perturbatively irrelevant to the bulk critical point, and cannot renormalize the transverse field strength, leading to a finite error threshold. We argue that a qualitatively similar conclusion can be drawn for a broad class of nearly critical topological codes, under a variety of decoherence channels.

Figures (2)

• Figure 1: Phases of information protection in the transverse-field toric code (TFTC) at field strength $h$, subjected to bit-flip decoherence of strength $p$. For all $h < h_{c}$, the TFTC exhibits a topological phase with four (nearly) degenerate ground states, which can be used to define a quantum code. The critical strength of bit-flip decoherence necessary to destroy the encoded information is finite throughout the phase, and remains finite even as $h \to h_{c}$. Red text denotes the phases of the 3$d$ replica statistical physics model which describes the Rényi coherent information; for $h < h_{c}$, information is destroyed when a 2$d$ defect surface exhibits an ordering transition.
• Figure 2: Depiction of the replica statistical physics model \ref{['eq:purity_statmech']}, which is used to compute the Rényi coherent information $I_{c}^{(n)}$ [Eq. \ref{['eq:renyi_coherent_info']}]. Each $n$th replica model consists of $n$ 3$d$ Ising models, represented using a high-temperature expansion as $n$ species of closed-loop anyon worldlines. The decoherence favors correlated worldline trajectories within the $\tau = 0$ surface. When pairs of worldlines proliferate along this surface, $I_{c}^{(n)}$ transitions from $2 - \mathcal{O}(e^{-L/\xi})$ to $0 + \mathcal{O}(e^{-L/\xi})$.