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Critical non-equilibrium phases from noisy topological memories

Amir-Reza Negari, Subhayan Sahu, Jan Behrends, Benjamin Béri, Timothy H. Hsieh

TL;DR

This work demonstrates an extended non-equilibrium critical phase in the surface code under heralded Pauli dephasing, identified by sub-exponential decay of the conditional mutual information $I(A{:}C|B)$ while correlations remain short-range. By mapping the classical dephased ensemble to the completely packed loop model with crossings (CPLC), the authors connect the critical regime to the Goldstone phase, predicting polylogarithmic decay of $I(A{:}C|B)$ and a diverging Markov length $\xi_{\mathrm M}$. They further analyze memory and decodability via punctured coherent information, showing that the Goldstone phase retains a partial, globally recoverable classical memory but not a quasi-local one, whereas a quasi-local decoder exists in the short-loop phase. The results introduce a concrete non-equilibrium critical memory in a topological code, provide a diagnostic to distinguish global versus quasi-local decoding, and suggest directions for fault-tolerant design and extensions to other LDPC codes and dynamical measurement scenarios.

Abstract

We demonstrate the existence of an extended non-equilibrium critical phase, characterized by sub-exponential decay of conditional mutual information (CMI), in the surface code subject to heralded random Pauli measurement channels. By mapping the resulting mixed state to the ensemble of completely packed loops on a square lattice, we relate the extended phase to the Goldstone phase of the loop model. In particular, CMI is controlled by the characteristic length scale of loops, and we use analytic results of the latter to establish polylogarithmic decay of CMI in the critical phase. We find that the critical phase retains partial logical information that can be recovered by a global decoder, but not by any quasi-local decoder. To demonstrate this, we introduce a diagnostic called punctured coherent information which provides a necessary condition for quasi-local decoding.

Critical non-equilibrium phases from noisy topological memories

TL;DR

This work demonstrates an extended non-equilibrium critical phase in the surface code under heralded Pauli dephasing, identified by sub-exponential decay of the conditional mutual information while correlations remain short-range. By mapping the classical dephased ensemble to the completely packed loop model with crossings (CPLC), the authors connect the critical regime to the Goldstone phase, predicting polylogarithmic decay of and a diverging Markov length . They further analyze memory and decodability via punctured coherent information, showing that the Goldstone phase retains a partial, globally recoverable classical memory but not a quasi-local one, whereas a quasi-local decoder exists in the short-loop phase. The results introduce a concrete non-equilibrium critical memory in a topological code, provide a diagnostic to distinguish global versus quasi-local decoding, and suggest directions for fault-tolerant design and extensions to other LDPC codes and dynamical measurement scenarios.

Abstract

We demonstrate the existence of an extended non-equilibrium critical phase, characterized by sub-exponential decay of conditional mutual information (CMI), in the surface code subject to heralded random Pauli measurement channels. By mapping the resulting mixed state to the ensemble of completely packed loops on a square lattice, we relate the extended phase to the Goldstone phase of the loop model. In particular, CMI is controlled by the characteristic length scale of loops, and we use analytic results of the latter to establish polylogarithmic decay of CMI in the critical phase. We find that the critical phase retains partial logical information that can be recovered by a global decoder, but not by any quasi-local decoder. To demonstrate this, we introduce a diagnostic called punctured coherent information which provides a necessary condition for quasi-local decoding.
Paper Structure (30 sections, 4 theorems, 58 equations, 14 figures)

This paper contains 30 sections, 4 theorems, 58 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Outline and summary of results
  3. Setup
  4. Surface code and Majorana representation
  5. Heralded Pauli dephasing channel and the ensemble
  6. Phase diagram of CMI and completely packed loop model
  7. Mapping to loop model
  8. CMI computation
  9. Critical phase as a memory
  10. Memory and the recovery
  11. Diagnostics of global recoverability
  12. Diagnostics of local recoverability
  13. No local recovery in the Goldstone phase
  14. Decoding transition: global vs local
  15. Goldstone phase: decoding is intrinsically global
  16. ...and 15 more sections

Key Result

theorem 1

Let $\{\rho^{(L)}_{RQ}\}_{L}$ be a family of reference--code states on $R\otimes Q$, where $Q=Q(L)$ is the physical register of a family of topological stabilizer codes with distance $d=d(L)$. Let $\rho^{(L)}_{RQ,\mathrm{noisy}}$ be the corresponding noisy reference--output state and let $\rho^{(L)} for some family of recovery channels $\{\mathcal{R}_L\}_{L}$ implementable by a quasi-local circuit

Figures (14)

  • Figure 1: The surface code subject to heralded complete Pauli dephasing has a phase diagram described by the completely packed loop model on a square lattice, and the parameters $0 \leq p,q \leq 1$ control the relative probabilities of $X, Y, Z$ Paulis in the dephasing channel. There are two topologically distinct short-loop phases which exhibit finite Markov length, and an extended Goldstone phase with system-spanning loops, which exhibit diverging Markov length. We also compute the persistence of a classical logical memory using the mutual information $I(R{:}Q)$ between a classical reference bit $R$ and the dephased code $Q$. Furthermore, we propose a punctured version of the mutual information $I(R{:}Q_{p^c})$ between the reference $R$ and part of the code $Q$ as a probe for local recoverability of the encoded information. Studying the interplay of phases, memory, and recoverability of encoded information in each of these phases, we find that the Goldstone phase is characterized by diverging Markov length, and partial classical memory that is globally but not locally recoverable. In contrast, in the short loop phase (shaded green), $X$-type logical information persists completely, and we show that it can be recovered using a quasi-local channel.
  • Figure 2: Left: A $d=5$ rotated surface code with $X$-type (black) and $Z$-type (white) plaquette stabilizers. Logical operators $\bar{Z}$ (red) and $\bar{X}$ (blue) run along boundaries. Right: The Majorana--qubit tensor-network representation, with 4 virtual Majorana modes (unfilled circles) attached to each physical qubit leg (filled circles). Neighboring virtual Majoranas are projected into even-parity dimers along edges (see text), while the 4 unpaired corner modes (red circles) encode the logical qubit degrees of freedom.
  • Figure 3: Mapping from the dephased surface code to loop model.
  • Figure 4: Phase diagram for the CPLC model Nahum_2013.
  • Figure 5: Loop configurations and geometries for the CMI computation.
  • ...and 9 more figures

Theorems & Definitions (7)

  • theorem 1: Punctured coherent/mutual information
  • proof : Proof sketch
  • corollary 1: Surface-code puncture invariance
  • lemma 1: Punctured recovery
  • lemma 2: Light-cone causality for channel circuits
  • proof
  • proof