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Quantum memory at nonzero temperature in a thermodynamically trivial system

Yifan Hong, Jinkang Guo, Andrew Lucas

TL;DR

The paper shows that passive quantum memory is possible without a nonzero-temperature thermodynamic phase transition by analyzing constant-rate classical and quantum LDPC codes with linear confinement. It demonstrates thermodynamic triviality (analytic free energy) alongside ergodicity-breaking slow Gibbs dynamics, yielding self-correction via local, memoryless dynamics. In the quantum case, hypergraph-product (HGP) codes inherit these properties: no phase transition but slow mixing time $t_{\text{mix}}\gtrsim e^{\alpha\sqrt{N}}$, with a Peierls-type argument governing cluster-based errors. It further proposes measurement-free quantum error correction (MFQEC) as a practical, finite-depth decoder leveraging Gibbs sampling, offering a scalable passive alternative to syndrome-based active decoding and suggesting experimental relevance for neutral-atom platforms.

Abstract

Passive error correction protects logical information forever in the thermodynamic limit by updating the system based only on local information and few-body interactions. A paradigmatic example is the classical two-dimensional Ising model: a Metropolis-style Gibbs sampler retains the sign of the initial magnetization (a logical bit) for thermodynamically long times in the low-temperature phase. Known models of passive quantum error correction similarly exhibit thermodynamic phase transitions to a low-temperature phase wherein logical qubits are protected by thermally stable topological order. Here, in contrast, we show that certain families of constant-rate classical and quantum low-density parity check codes have no thermodynamic phase transitions at nonzero temperature, but nonetheless exhibit ergodicity-breaking dynamical transitions: below a critical nonzero temperature, the mixing time of local Gibbs sampling diverges in the thermodynamic limit. Slow Gibbs sampling of such codes enables fault-tolerant passive quantum error correction using finite-depth circuits. This strategy is well suited to measurement-free quantum error correction and may present a desirable experimental alternative to conventional quantum error correction based on syndrome measurements and active feedback.

Quantum memory at nonzero temperature in a thermodynamically trivial system

TL;DR

The paper shows that passive quantum memory is possible without a nonzero-temperature thermodynamic phase transition by analyzing constant-rate classical and quantum LDPC codes with linear confinement. It demonstrates thermodynamic triviality (analytic free energy) alongside ergodicity-breaking slow Gibbs dynamics, yielding self-correction via local, memoryless dynamics. In the quantum case, hypergraph-product (HGP) codes inherit these properties: no phase transition but slow mixing time , with a Peierls-type argument governing cluster-based errors. It further proposes measurement-free quantum error correction (MFQEC) as a practical, finite-depth decoder leveraging Gibbs sampling, offering a scalable passive alternative to syndrome-based active decoding and suggesting experimental relevance for neutral-atom platforms.

Abstract

Passive error correction protects logical information forever in the thermodynamic limit by updating the system based only on local information and few-body interactions. A paradigmatic example is the classical two-dimensional Ising model: a Metropolis-style Gibbs sampler retains the sign of the initial magnetization (a logical bit) for thermodynamically long times in the low-temperature phase. Known models of passive quantum error correction similarly exhibit thermodynamic phase transitions to a low-temperature phase wherein logical qubits are protected by thermally stable topological order. Here, in contrast, we show that certain families of constant-rate classical and quantum low-density parity check codes have no thermodynamic phase transitions at nonzero temperature, but nonetheless exhibit ergodicity-breaking dynamical transitions: below a critical nonzero temperature, the mixing time of local Gibbs sampling diverges in the thermodynamic limit. Slow Gibbs sampling of such codes enables fault-tolerant passive quantum error correction using finite-depth circuits. This strategy is well suited to measurement-free quantum error correction and may present a desirable experimental alternative to conventional quantum error correction based on syndrome measurements and active feedback.
Paper Structure (22 sections, 21 theorems, 99 equations, 9 figures, 1 table)

This paper contains 22 sections, 21 theorems, 99 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classical codes
  3. Quantum codes
  4. Measurement-free quantum error correction
  5. Outlook
  6. Classical LDPC codes
  7. Linear parity-check codes
  8. Thermodynamics and Kramers-Wannier duality
  9. Lower bound on the classical mixing time
  10. Quantum LDPC codes
  11. CSS codes and the hypergraph product
  12. Thermodynamics
  13. Lower bound on the quantum mixing time
  14. Numerical simulations
  15. Codes built from tree graphs
  16. ...and 7 more sections

Key Result

Theorem 1.2

If $\Delta_B < \Delta_C$, then the rate of $\mathcal{C}_\mathcal{G}$ is at least where $\mathfrak{r}_{\rm des} \equiv 1-\Delta_B/\Delta_C$ is called the design rate.

Figures (9)

  • Figure 1: A snapshot of the energy landscape of a typical good classical LDPC code is depicted. Extensively deep minima separate codewords and low-energy configurations consisting of single flipped parity checks. Below a critical temperature, the Gibbs sampler becomes trapped inside its initial minimum and is unable to explore the full state space, preventing the system from reaching thermal equilibrium.
  • Figure 2: The hypergraph product construction is illustrated. (a) The Euclidean graph product is taken between the Tanner graph of a classical parity-check code and itself, producing four types of vertices. (b) The hypergraph product maps the four types of vertices in the product graph to qubits, $X$-checks and $Z$-checks, thus producing a quantum CSS code. The interactions in the hypergraph product code resemble those of its classical input code (in magenta).
  • Figure 3: (a) An example of a bottleneck state $\mathbf{x} \in E$ is depicted with the non-decodable cluster highlighted. (b) The partner state $\mathbf{y}(\mathbf{x}) \in A$ with this cluster removed. The Gibbs sampler will take a very long time to reach states containing non-decodable clusters at low temperature.
  • Figure 4: Top: The logical block error probability as a function of temperature is plotted. We set $\Delta t = 100$ and sample several codes from the $(4,5)$-LDPC ensemble, $(7,8)$-HGP ensemble and 2D surface code family. Bottom: The logical block error rate (normalized by $\Delta t$) as a function of temperature is plotted near the observed transition point. We examine one particular code from each of the aforementioned ensembles and vary the equilibration time. Error bands (shaded) denote the standard error.
  • Figure 5: The two steps of the mapping between $F^-_{\varphi} \in S_-$ and its complement $\bar{F}^-_{\varphi} \in S_+$ are depicted. (a) The first step involves flipping the branch of any -1 spin (shaded gray) on the fault path, thereby mapping $F^-_{\varphi}\rightarrow W^-_{\varphi}$. (b) The second step involves the inductive strategy (green circles) described in Lemma \ref{['lem:F_phi_0']}.
  • ...and 4 more figures

Theorems & Definitions (40)

  • Definition 1.1: Bipartite expansion Sipser_1996
  • Theorem 1.2: Constant rate Gallager_1962
  • proof
  • Lemma 1.3: Unique neighbor expansion Sipser_1996
  • proof
  • Corollary 1.4: Linear confinement
  • Theorem 1.5: Linear distance Sipser_1996
  • proof
  • Theorem 1.6: Random expansion; Theorem 8.7 of ModernCodingTheory
  • Lemma 1.7: Rate approaches design rate; Lemma 3.27 of ModernCodingTheory
  • ...and 30 more