Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes

TL;DR

This paper introduces the theory of quasi-exact fault-tolerant quantum computation using quasi codes, especially valence-bond-solid codes, bridging the gap between noisy and fully fault-tolerant quantum computing.

Contribution

It develops the theoretical framework for quasi-exact quantum error correction and demonstrates its application with valence-bond-solid codes as concrete examples.

Findings

Defined quasi error-correction theory using quantum instruments

Introduced notions of quasi universality, quasi code distances, and quasi thresholds

Identified valence-bond-solid codes as a class of quasi codes

Abstract

In this work, we develop the theory of quasi-exact fault-tolerant quantum (QEQ) computation, which uses qubits encoded into quasi-exact quantum error-correction codes ("quasi codes"). By definition, a quasi code is a parametric approximate code that can become exact by tuning its parameters. The model of QEQ computation lies in between the two well-known ones: the usual noisy quantum computation without error correction and the usual fault-tolerant quantum computation, but closer to the later. Many notions of exact quantum codes need to be adjusted for the quasi setting. Here we develop quasi error-correction theory using quantum instrument, the notions of quasi universality, quasi code distances, and quasi thresholds, etc. We find a wide class of quasi codes which are called valence-bond-solid codes, and we use them as concrete examples to demonstrate QEQ computation.