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Error correction with brickwork Clifford circuits

Twan Kroll, Jonas Helsen

Abstract

We prove that random 1D Clifford brickwork circuits form (in expectation) good approximate quantum error correction codes in logarithmic depth. Our proof makes use of the statistical mechanics techniques for random circuits developed by Dalzell et al. [PRX Quantum 3, 010333], adapted extensively to our own purpose. We also consider exact error correction, where we give matching upper and lower bounds for the required depth in which random 1D Clifford brickwork circuits become error correcting.

Error correction with brickwork Clifford circuits

Abstract

We prove that random 1D Clifford brickwork circuits form (in expectation) good approximate quantum error correction codes in logarithmic depth. Our proof makes use of the statistical mechanics techniques for random circuits developed by Dalzell et al. [PRX Quantum 3, 010333], adapted extensively to our own purpose. We also consider exact error correction, where we give matching upper and lower bounds for the required depth in which random 1D Clifford brickwork circuits become error correcting.
Paper Structure (11 sections, 15 theorems, 118 equations, 2 figures)

This paper contains 11 sections, 15 theorems, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. (Approximate) error correction
  5. Noise models
  6. Designs
  7. Brickwork circuits
  8. Logarithmic depth approximate error correction
  9. Exact Error Correction
  10. Additional technical material
  11. Missing proofs of Section \ref{['sec:EC']}

Key Result

Theorem 1

For appropriate constants $0<r<1$ and $c>\frac{1}{|\log r|}$ with $D(n)=c\log(n)$, the expected Choi error can be bounded for appropriate error thresholds as follows:

Figures (2)

  • Figure 1: Trajectories in the stat-mech model. Paths labeled blue annihilate while the orange paths survive. Each path carries with it a weight that must be controlled in order to prove a bound on the expected Choi error (i.e. entanglement infidelity). Note that the surviving domain walls partition the qubits into "independent" intervals. This insight is key to achieving theorem \ref{['thm:informal']}. For the sake of readability of this figure, we apply all gates in a single layer in the same time step, while in the proof of theorem \ref{['thm:informal']}, to avoid ambiguities, we will apply them one-by-one in a specified order (in effect moving only one domain wall at a time).
  • Figure 2: A diagram showing the architecture when the rate is $\frac{2}{5}$ and there are 2 blocks in depth $6$, i.e., $n=10$. Each gate is a uniformly sampled $2$-qubit Clifford gate.

Theorems & Definitions (31)

  • Theorem 1: Informal
  • Theorem 2: Informal
  • Definition 1
  • Proposition 1
  • Definition 2
  • Lemma 1: Second Order Moment
  • Lemma 2: Partition function
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 21 more