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Efficient quantum Gibbs sampling of stabilizer codes using hybrid computation

Ivan H. C. Shum, Angela Capel

TL;DR

This work tackles the problem of efficiently sampling Gibbs states for stabilizer-code Hamiltonians, specifically the rotated surface code and toric code, using hybrid quantum-classical approaches with local gates. It builds a Heisenberg-picture framework that decouples the Hamiltonians into $X$- and $Z$-subsystems via sequences of CX gates, enabling most sampling to be performed classically after diagonalization. It achieves Gibbs-state preparation depths of about $O(L/2)$ for the rotated surface code and $O(L)$ for the toric code, and also shows a non-local scheme yielding logarithmic-depth sampling for a periodic $1$D Ising model, along with improvements over prior $O(L)$/$O(L^2)$/$O(L^3)$ bounds. The results provide locality-friendly routes to thermal-state preparation of stabilizer codes, with potential implications for quantum simulation and fault-tolerant quantum computation.

Abstract

We present hybrid Gibbs sampling algorithms for the stabilizer code Hamiltonians of the rotated surface code and the toric code with only local quantum algorithms, using $\sim L/2$ quantum circuit depth to prepare the Gibbs state of the rotated surface code Hamiltonian, and $L$ quantum circuit depth to prepare the Gibbs state of the toric code Hamiltonian, being $L$ the side of the side of the square lattice. We further show that if we allow for non-local gates, the Gibbs state of the periodic 1D Ising model can be prepared in logarithmic depth and linearly many simultaneous measurements.

Efficient quantum Gibbs sampling of stabilizer codes using hybrid computation

TL;DR

This work tackles the problem of efficiently sampling Gibbs states for stabilizer-code Hamiltonians, specifically the rotated surface code and toric code, using hybrid quantum-classical approaches with local gates. It builds a Heisenberg-picture framework that decouples the Hamiltonians into - and -subsystems via sequences of CX gates, enabling most sampling to be performed classically after diagonalization. It achieves Gibbs-state preparation depths of about for the rotated surface code and for the toric code, and also shows a non-local scheme yielding logarithmic-depth sampling for a periodic D Ising model, along with improvements over prior // bounds. The results provide locality-friendly routes to thermal-state preparation of stabilizer codes, with potential implications for quantum simulation and fault-tolerant quantum computation.

Abstract

We present hybrid Gibbs sampling algorithms for the stabilizer code Hamiltonians of the rotated surface code and the toric code with only local quantum algorithms, using quantum circuit depth to prepare the Gibbs state of the rotated surface code Hamiltonian, and quantum circuit depth to prepare the Gibbs state of the toric code Hamiltonian, being the side of the side of the square lattice. We further show that if we allow for non-local gates, the Gibbs state of the periodic 1D Ising model can be prepared in logarithmic depth and linearly many simultaneous measurements.

Paper Structure

This paper contains 8 sections, 3 theorems, 19 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Notation, convention, explanation of algorithms
  3. Hybrid Gibbs sampling the rotated surface code
  4. Hybrid Gibbs sampling the toric code
  5. Quantum Gibbs sampling the Ising loop
  6. Quantum Gibbs sampling the toric code
  7. Acknowledgments
  8. Classical Gibbs sampling algorithms

Key Result

Theorem 1

There is an $\mathcal{O}(L)$ depth local mapping from the both the 2D toric code and the 2D rotated surface code, to two decoupled systems that can be identified as 1D Ising models.

Figures (18)

  • Figure 1: Flow of algorithm
  • Figure 2: Basic operations after conjugation of $CX$ gates. We denote the sum of a real linear combination of stabilizers by red/blue cross/circles, lines and polygons, with red corresponding to purely $X$ interactions and blue corresponding to purely $Z$ interactions. An arrow is used to signify a $CX$ gate conjugation, pointly from the source to the target.
  • Figure 3: $4\times 3$ rotated surface code; the even sites host pink/$X$ operators, while the odd sites host blue/$Z$ operators
  • Figure 4: First step of algorithm when $L=6$
  • Figure 5: Resulting Hamiltonian after the first step of quantum circuit evolution
  • ...and 13 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3