Hamiltonian Formulation of Quantum Error Correction and Correlated Noise: The Effects Of Syndrome Extraction in the Long Time Limit

TL;DR

This paper develops a Hamiltonian-based framework to analyze long-term effects of correlated noise on quantum error correction, deriving conditions under which fault-tolerance thresholds are maintained or broken.

Contribution

It introduces a Hamiltonian formulation for realistic correlated noise models and establishes a dimensional criterion for the validity of fault-tolerance thresholds.

Findings

Fast-decaying correlations support fault-tolerance thresholds.

Slow-decaying correlations can invalidate traditional fault-tolerance proofs.

Explicit calculations are provided for spin-boson and frustration models.

Abstract

We analyze the long time behavior of a quantum computer running a quantum error correction (QEC) code in the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models, and assuming that QEC is indeed possible, we find formal expressions for the probability of a faulty path and the residual decoherence encoded in the reduced density matrix. Systems with non-zero gate times (``long gates'') are included in our analysis by using an upper bound on the noise. In order to introduce the local error probability for a qubit, we assume that propagation of signals through the environment is slower than the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalized spin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlations decay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem of fault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, the traditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities to criteria that occur in the theory of quantum phase transitions.