A Universal Operator Theoretic Framework for Quantum Fault Tolerance

TL;DR

This paper presents a comprehensive operator theoretic framework for quantum fault tolerance, enabling more precise error threshold calculations and unifying various error correction schemes under a single formalism.

Contribution

It introduces a universal operator theoretic approach based on the Quantum Computer Condition, unifying existing error correction protocols and improving error threshold estimations.

Findings

More accurate error thresholds demonstrated formally and numerically

All known coding schemes are special cases of the QCC

New operator theoretic form of entanglement-assisted quantum error correction

Abstract

In this paper we introduce a universal operator theoretic framework for quantum fault tolerance. This incorporates a top-down approach that implements a system-level criterion based on specification of the full system dynamics, applied at every level of error correction concatenation. This leads to more accurate determinations of error thresholds than could previously be obtained. This is demonstrated both formally and with an explicit numerical example. The basis for our approach is the Quantum Computer Condition (QCC), an inequality governing the evolution of a quantum computer. We show that all known coding schemes are actually special cases of the QCC. We demonstrate this by introducing a new, operator theoretic form of entanglement assisted quantum error correction, which incorporates as special cases all known error correcting protocols, and is itself a special case of the QCC.