Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code

Sergey Bravyi, Jeongwan Haah

TL;DR

This work analyzes the feasibility of self-correcting quantum memories in the 3D Cubic Code by coupling the memory to a thermal bath via Davies dynamics and decoding errors with a novel Renormalization Group decoder. It derives a rigorous, scalable lower bound on memory time $T_{mem}$ in regimes where the lattice size $L$ is below a critical value $L^*\sim e^{\beta/3}$, showing $T_{mem}\ge L^{c\beta-3}$ for some $c>0$, and demonstrates that the optimal memory time grows as $e^{\Theta(\beta^2)}$ for large inverse temperature $\beta$. A key technical ingredient is the logarithmic energy barrier guaranteed by the no-strings rule, enabling local decodability of low-barrier errors; the RG decoder further provides a universal constant threshold for topological stabilizer codes. Numerical simulations corroborate the analytic bounds, indicating that the bound is tight up to constant factors and supporting the potential for macroscopic memory times at suitable $L$ and $\beta$. Overall, the paper offers a concrete, scalable route to self-correcting quantum memories in 3D stabilizer codes and introduces a decoder with broad applicability and provable performance guarantees.

Abstract

A big open question in the quantum information theory concerns feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction if the memory is put in contact with a cold enough thermal bath. In this paper we derive a rigorous lower bound on the memory time $T_{mem}$ of the 3D Cubic Code model which was recently conjectured to have a self-correcting behavior. Assuming that dynamics of the memory system can be described by a Markovian master equation of Davies form, we prove that $T_{mem}\ge L^{cβ}$ for some constant $c>0$, where $L$ is the lattice size and $β$ is the inverse temperature of the bath. However, this bound applies only if the lattice size does not exceed certain critical value $L^*\sim e^{β/3}$. We also report a numerical Monte Carlo simulation of the studied memory indicating that our analytic bounds on $T_{mem}$ are tight up to constant coefficients. In order to model the readout step we introduce a new decoding algorithm which might be of independent interest. Our decoder can be implemented efficiently for any topological stabilizer code and has a constant error threshold under random uncorrelated errors.

Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code

TL;DR

This work analyzes the feasibility of self-correcting quantum memories in the 3D Cubic Code by coupling the memory to a thermal bath via Davies dynamics and decoding errors with a novel Renormalization Group decoder. It derives a rigorous, scalable lower bound on memory time in regimes where the lattice size is below a critical value , showing for some , and demonstrates that the optimal memory time grows as for large inverse temperature . A key technical ingredient is the logarithmic energy barrier guaranteed by the no-strings rule, enabling local decodability of low-barrier errors; the RG decoder further provides a universal constant threshold for topological stabilizer codes. Numerical simulations corroborate the analytic bounds, indicating that the bound is tight up to constant factors and supporting the potential for macroscopic memory times at suitable and . Overall, the paper offers a concrete, scalable route to self-correcting quantum memories in 3D stabilizer codes and introduces a decoder with broad applicability and provable performance guarantees.

Abstract

A big open question in the quantum information theory concerns feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction if the memory is put in contact with a cold enough thermal bath. In this paper we derive a rigorous lower bound on the memory time of the 3D Cubic Code model which was recently conjectured to have a self-correcting behavior. Assuming that dynamics of the memory system can be described by a Markovian master equation of Davies form, we prove that for some constant , where is the lattice size and is the inverse temperature of the bath. However, this bound applies only if the lattice size does not exceed certain critical value . We also report a numerical Monte Carlo simulation of the studied memory indicating that our analytic bounds on are tight up to constant coefficients. In order to model the readout step we introduce a new decoding algorithm which might be of independent interest. Our decoder can be implemented efficiently for any topological stabilizer code and has a constant error threshold under random uncorrelated errors.

Paper Structure

This paper contains 25 sections, 14 theorems, 54 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Memory Hamiltonian, thermal noise, and decoding
  3. 3D Cubic Code Hamiltonian
  4. Thermal noise
  5. Decoding
  6. Main results and sketch of the proof
  7. Previous work
  8. Analysis of thermal errors
  9. Energy barrier
  10. Lower bound on the memory time
  11. Choosing the optimal number of qubits
  12. Renormalization group decoder
  13. Metric convention and terminology
  14. Topological stabilizer codes
  15. Topological stabilizer codes
  16. ...and 10 more sections

Key Result

Theorem 1

There exists a decoder $\Phi_{ec}$ and a constant $c>0$ such that for for any inverse temperature $\beta>0$, any state $\rho(0)$ supported on the ground subspace of $H$, any evolution time $t\ge 0$, and any constant $0<a<1$ one has as long as $L\le e^{(1-a)\beta/3}$. The error correction algorithm used by the decoder has running time $poly(L)$.

Figures (6)

  • Figure 1: Stabilizer generators of the 3D Cubic Code. Here $X\equiv \sigma^x$ and $Z\equiv \sigma^x$ represent single-qubit Pauli operators, while $I$ is the identity operator. Double-letter indices represent two-qubit Pauli operators, for example, $IZ\equiv I\otimes Z$, $ZZ\equiv Z\otimes Z$, $II\equiv I\otimes I$ etc.
  • Figure 2: The memory time $T_{mem}$ vs. the system size $L$. In the upper inset is shown the exponent of the power law fit of $T_{mem}$ for the first a few system sizes. It is clear that $T_{mem} \propto L^{2.93 \beta -10.5}$ when $L < L^\star$, where $L^\star$ is the optimal system size where $T_{mem}$ reaches maximum. The data for $\beta = 4.3, 4.5, 4.7, 4.9, 5.1, 5.25$ are shown.
  • Figure 3: The maximum memory time $T_{mem}$ vs. the inverse temperature $\beta$. The memory time is maximized with respect to the system size. The logarithm of $T_{mem}$ clearly follows a quadratic relation with $\beta$ as oppose to a linear one.
  • Figure 4: Elementary syndromes created by $Z$ errors. The vertices which are on the dual lattice, represent the defects created by the error at the center. The elementary syndrome by $ZI$ is used to push the defects to the bottom and to the left. The syndromes by errors of weight three is used to push the defects to the bottom-left corner.
  • Figure 5: Construction of a hook of level 2 from the vacuum. The grid diagram represents the position and the number of defects in the ($x=z$)-plane. For each transition, an operator of weight 1 is applied. The total number of defects never exceeds 6. From a level-$0$ hook (the second diagram in the sequence), a level-$1$ hook (the last in the sequence) is constructed using extra 2 defects.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4: ChesiLossBravyiEtAl2009Thermodynamic
  • ...and 20 more