Effect of Correlated Errors on Quantum Memory

TL;DR

This paper introduces a new model for correlated errors in quantum memory, demonstrating that quantum Tanner codes can achieve exponential retention time despite complex error correlations.

Contribution

It proposes a generalized correlation model for quantum errors and proves that quantum Tanner codes maintain exponential lifetime under broad non-Markovian, non-stationary error conditions.

Findings

Quantum Tanner codes achieve exponential retention time below a certain error threshold.

The new correlation model captures complex error dependencies beyond joint Hamiltonian models.

Decay rate of correlations does not determine code performance.

Abstract

Recent results on constant overhead LDPC code-based fault-tolerance against i.i.d. errors naturally lead to the question of fault-tolerance against errors with long-range correlations. Ideally, any correlation can be captured by a joint (system and bath) Hamiltonian. However, an arbitrary joint Hamiltonian is often intractable, and hence, the joint Hamiltonian model with pairwise terms was introduced and developed in a series of foundational works. However, the analysis of the new constant overhead codes in that error model appears to be quite challenging. In this paper, to model correlated errors in quantum memory, we introduce a correlation model which is a generalization of the well-known hidden random fields. This proposed model, which includes stationary and ergodic (non-Markov) error distributions, is shown to capture correlations not captured by the joint Hamiltonian model with pairwise terms. On the other hand, building on non-i.i.d. measure concentration, we show that for a broad class of non-Markov and (possibly) non-stationary error distributions, quantum Tanner codes ensure an exponential retention time (in the number of physical qubits), when the error rate is below a threshold. An implication of these results is that the rate of decay of the correlation with distance does not necessarily differentiate between good and bad correlation.