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Modified logarithmic Sobolev inequalities for CSS codes

Sebastian Stengele, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, Antonio Pérez-Hernández, Cambyse Rouzé, Simone Warzel

TL;DR

The paper proves modified logarithmic Sobolev inequalities (MLSI) for Davies quantum semigroups governing thermalization of translation-invariant CSS codes in D dimensions, under Dobrushin-Shlosman-type correlation decay. The authors develop explicit Davies conditional expectations, an approximate tensorization framework, and a multiscale analysis to derive uniform-in-size MLSI constants for the star and plaquette parts, whenever the DS-condition holds and jump rates are uniformly positive. This yields rapid thermalization at any positive temperature for the 2D toric code and the 3D star component, implying fast loss of quantum information and, consequently, strong lack of self-correction in these models. The results connect quantum MLSI to classical DS-conditioned correlation decay, enabling efficient Gibbs-state preparation and offering new insight into the dynamical stability of CSS codes under thermal noise.

Abstract

We consider the class of Davies quantum semigroups modelling thermalization for translation-invariant Calderbank-Shor-Steane (CSS) codes in D dimensions. We prove that conditions of Dobrushin-Shlosman-type on the quantum Gibbs state imply a modified logarithmic Sobolev inequality with a constant that is uniform in the system's size. This is accomplished by generalizing parts of the classical results on thermalization by Stroock, Zegarlinski, Martinelli, and Olivieri to the CSS quantum setting. The results in particular imply the rapid thermalization at any positive temperature of the toric code in 2D and the star part of the toric code in 3D, implying a rapid loss of stored quantum information for these models.

Modified logarithmic Sobolev inequalities for CSS codes

TL;DR

The paper proves modified logarithmic Sobolev inequalities (MLSI) for Davies quantum semigroups governing thermalization of translation-invariant CSS codes in D dimensions, under Dobrushin-Shlosman-type correlation decay. The authors develop explicit Davies conditional expectations, an approximate tensorization framework, and a multiscale analysis to derive uniform-in-size MLSI constants for the star and plaquette parts, whenever the DS-condition holds and jump rates are uniformly positive. This yields rapid thermalization at any positive temperature for the 2D toric code and the 3D star component, implying fast loss of quantum information and, consequently, strong lack of self-correction in these models. The results connect quantum MLSI to classical DS-conditioned correlation decay, enabling efficient Gibbs-state preparation and offering new insight into the dynamical stability of CSS codes under thermal noise.

Abstract

We consider the class of Davies quantum semigroups modelling thermalization for translation-invariant Calderbank-Shor-Steane (CSS) codes in D dimensions. We prove that conditions of Dobrushin-Shlosman-type on the quantum Gibbs state imply a modified logarithmic Sobolev inequality with a constant that is uniform in the system's size. This is accomplished by generalizing parts of the classical results on thermalization by Stroock, Zegarlinski, Martinelli, and Olivieri to the CSS quantum setting. The results in particular imply the rapid thermalization at any positive temperature of the toric code in 2D and the star part of the toric code in 3D, implying a rapid loss of stored quantum information for these models.

Paper Structure

This paper contains 29 sections, 30 theorems, 184 equations, 7 figures.

Table of Contents

  1. Introduction and results
  2. Families of CSS Hamiltonians
  3. Davies Lindbladians and thermal equilibria
  4. MLSI and rapid mixing towards equilibrium
  5. MLSI from uniform correlation decay
  6. Comparison to existing literature
  7. Fast mixing via gap estimates.
  8. Preparation of Gibbs states.
  9. Comments on the proof
  10. Equilibrium
  11. Properties of the Gibbs state
  12. Dobrushin-Shlosman condition
  13. DS-condition at any positive temperature
  14. Davies Lindbladians
  15. Self-adjointness and stationary states
  16. ...and 14 more sections

Key Result

Theorem 1.1

For any CSS code the partition function factorizes, $Z_\Lambda^\boxplus = Z_\Lambda^\medwhitestar \ Z_\Lambda^\square \ 2^{-|\Lambda|}$, with $|\Lambda |$ the number of qubits within $\Lambda$, and hence the Gibbs state factorizes as well

Figures (7)

  • Figure 1: Examples of CSS codes. Stars are colored orange, plaquettes blue. The commutation relation \ref{['eq:commrule']} applies to all. (a) The stars (top) and plaquettes (bottom) of the rotated surface code. (b) The rotated surface code on $5\times 5$ qubits. The red vertical line is a logical $X$-operator, and the red horizontal line is a logical $Z$-operator. (c) The stars (top) and plaquette (bottom) of the $2$D toric code, which are then combined in a $2$D regular lattice. (d) The stars of the $2$D hexagonal color code are the faces of the hexagonal lattice. Qubits are placed on the vertices. The plaquettes are, up to switching $X\leftrightarrow Z$, identical to the stars. (e) The star (left) and plaquettes (right) of the $3$D Toric code. There are three orientations of the plaquette. Each plaquette is connected to $4$ qubits, each star to $6$. (f) The star (left) and plaquette (right) of a $2$D tessellation code on the hexagonal lattice.
  • Figure 2: Unit cells and boundaries. Top left: a single unit cell of the rotated surface code containing four qubits, two stars (orange) and two plaquettes (blue). Bottom left: A single unit cell of the $2$D hexagonal color code containing two qubits and one star. Right: a small rectangle with boundary conditions. The dashed unit cells lie at the boundary and are modified. The shaded interactions are truncated or fully removed. The crossed-out qubits are not interacting and can be ignored.
  • Figure 3: Geometry in the DS-condition. Left: Two rectangles $U\uplus V$ and $V\uplus W$ intersecting in $V$. The example shows an alignment of $U$ and $W$ along the first coordinate direction. Their union $U \uplus V\uplus W$ is again a rectangle. The height and width of $W$ are bounded by the square of the width of $V$. Right: The same setup for periodic boundaries. Now $V=V_1\uplus V_2$ is a disjoint union of two rectangles. The quantity $l_1$ appears in the proof of the MLSI in \ref{['sec:mainresult']}.
  • Figure 4: The geometry for \ref{['def:DSIIId']}. Two nested regions $R\subset R'\subseteq \Lambda$ and a spin $v$ in the complement $(R')^c$. In \ref{['def:DSIIId']}, flipping the spin at $v$ changes the (conditional) Gibbs measure in $R$ by a term decaying exponentially in the distance between $v$ and $R$.
  • Figure 5: A subset of the Ising model and the corresponding partition of the interactions. For illustration purposes, the Ising interactions are taken as the star part of a CSS code. Plaquette interactions are not discussed. Left: Qubits (black dots) and interactions (lines) of the $2$D Ising model. A subset of qubits $R$ (red box) and the interactions $\mathop{\mathrm{\mathbb{S}}}\nolimits_R$ (blue, thick lines) with support intersecting $R$. Right: Only the qubits in $R$ and their interactions. The interactions are partitioned by support in $R$ (orange sets). Most sets of this partition contain only a single interaction, only sets at the boundary contain multiple interactions.
  • ...and 2 more figures

Theorems & Definitions (66)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • ...and 56 more