Spacetime Spins: Statistical mechanics for error correction with stabilizer circuits

TL;DR

This paper develops a statistical mechanics framework for analyzing quantum error correction in stabilizer circuits, enabling the comparison of static and dynamic codes, and providing tools for error rate estimation and threshold determination.

Contribution

It introduces a novel method to map stabilizer circuit error probabilities to classical spin models using spacetime subsystem codes, unifying static and dynamic quantum error correction analysis.

Findings

Analytical ranking of logical error rates and thresholds for repetition and toric codes.

Demonstration of the impact of transversal Clifford gates on logical error rates.

Monte Carlo simulations estimating maximum likelihood thresholds.

Abstract

A powerful method for analyzing quantum error-correcting codes is to map them onto classical statistical mechanics models. Such mappings have thus far mostly focused on static codes, possibly subject to repeated syndrome measurements. Recent progress in quantum error correction, however, has prompted new paradigms where codes emerge from stabilizer circuits in spacetime -- a unifying perspective encompassing syndrome extraction circuits of static codes, dynamically generated codes, and logical operations. We show how to construct statistical mechanical models for stabilizer circuits subject to independent Pauli errors, by mapping logical equivalence class probabilities of errors to partition functions using the spacetime subsystem code formalism. We also introduce a modular language of spin diagrams for constructing the spin Hamiltonians, which we describe in detail focusing on independent circuit-level X-Z error channels. With the repetition and toric codes as examples, we use our approach to analytically rank logical error rates and thresholds between code implementations with standard and dynamic syndrome extraction circuits, describe the effect of transversal logical Clifford gates on logical error rates, and perform Monte Carlo simulations to estimate maximum likelihood thresholds. Our framework offers a universal prescription to analyze, simulate, and compare the decoding properties of any stabilizer circuit, while revealing the innate connections between dynamical quantum systems and noise-resilient phases of matter.