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On thermal stability of topological qubit in Kitaev's 4D model

R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki

TL;DR

The paper examines thermal stability of Kitaev's four-dimensional model as a quantum memory under finite temperature with Markovian weak coupling, demonstrating that topological qubits built from observables $X$ and $Z$ can have relaxation times that grow exponentially with system size below a critical temperature $T_{\mathrm{crit}}$.Using Davies generators, it reduces the quantum dynamics of protected observables to a classical $Z_2$ gauge Ising model, and shows that Gibbs states concentrate on short-loop configurations, enabling stable topological observables constructed from homology classes.A concrete polynomial-time measurement protocol is given for the topological observables, and the work highlights both the potential for thermally robust quantum memory and the remaining challenges for self-correcting memory, encoding, and readout.Overall, the results provide a rigorous link between finite-temperature stability, topological protection, and thermodynamic limit considerations in Kitaev-type models, with implications for quantum memory design and statistical mechanics.

Abstract

We analyse stability of the four-dimensional Kitaev model - a candidate for scalable quantum memory - in finite temperature within the weak coupling Markovian limit. It is shown that, below a critical temperature, certain topological qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their construction involves polynomial in system's size algorithm which uses as an input the results of measurements performed on all individual spins. We also discuss the drawbacks of such candidate for quantum memory and mention the implications of the stability of qubit for statistical mechanics.

On thermal stability of topological qubit in Kitaev's 4D model

TL;DR

The paper examines thermal stability of Kitaev's four-dimensional model as a quantum memory under finite temperature with Markovian weak coupling, demonstrating that topological qubits built from observables $X$ and $Z$ can have relaxation times that grow exponentially with system size below a critical temperature $T_{\mathrm{crit}}$.Using Davies generators, it reduces the quantum dynamics of protected observables to a classical $Z_2$ gauge Ising model, and shows that Gibbs states concentrate on short-loop configurations, enabling stable topological observables constructed from homology classes.A concrete polynomial-time measurement protocol is given for the topological observables, and the work highlights both the potential for thermally robust quantum memory and the remaining challenges for self-correcting memory, encoding, and readout.Overall, the results provide a rigorous link between finite-temperature stability, topological protection, and thermodynamic limit considerations in Kitaev-type models, with implications for quantum memory design and statistical mechanics.

Abstract

We analyse stability of the four-dimensional Kitaev model - a candidate for scalable quantum memory - in finite temperature within the weak coupling Markovian limit. It is shown that, below a critical temperature, certain topological qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their construction involves polynomial in system's size algorithm which uses as an input the results of measurements performed on all individual spins. We also discuss the drawbacks of such candidate for quantum memory and mention the implications of the stability of qubit for statistical mechanics.

Paper Structure

This paper contains 18 sections, 6 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Markovian approximation in weak coupling limit
  3. Autocorrelation functions, decay rate and fidelity
  4. Upper bound for decay rate
  5. From quantum to classical in Kitaev-type models
  6. Observables $X$ and $Z$
  7. $3D$ Kitaev model
  8. $4D$ Kitaev model
  9. Stability conditions for Kitaev model
  10. Bound for decay rate for dressed observables
  11. Gibbs state is concentrated on configurations without long loops
  12. Stability of Kitaev 4D model
  13. Stability of topological observables
  14. Observables depending only on homology classes
  15. Construction of stable topological observables
  16. ...and 3 more sections

Key Result

Proposition 1

With the above notation, suppose now that the Gibbs state is of the form where $\rho_{anc}$ is a state on ancilla. We then have where $\epsilon$ is upper bound for the rates $-\langle X,{\cal L} (X)\rangle_\beta$ and $-\langle Z,{\cal L}(Z)\rangle_\beta$.

Figures (1)

  • Figure 1: Efficient algorithm for determining surface closing the loop.

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 2
  • Proposition 3
  • ...and 1 more