Universal quantum computation and quantum error correction using discrete holonomies

TL;DR

This paper introduces a novel approach to quantum computation using discrete holonomies, combining measurement-based techniques with error correction, and bridges the gap between continuous and discrete holonomic quantum computation.

Contribution

It presents an explicit method for universal quantum computation via discrete holonomies using incomplete measurements, integrating quantum error correction naturally.

Findings

Discrete holonomies enable universal quantum gates.

Quantum error correction integrates seamlessly into the scheme.

Dense measurements recover continuous holonomies.

Abstract

Holonomic quantum computation exploits a quantum state's non-trivial, matrix-valued geometric phase (holonomy) to perform fault-tolerant computation. Holonomies arising from systems where the Hamiltonian traces a continuous path through parameter space have been well-researched. Discrete holonomies, on the other hand, where the state jumps from point to point in state space, have had little prior investigation. Using a sequence of incomplete projective measurements of the spin operator, we build an explicit approach to universal quantum computation. We show that quantum error correction codes integrate naturally in our scheme, providing a model for measurement-based quantum computation that combines the passive error resilience of holonomic quantum computation and active error correction techniques. In the limit of dense measurements we recover known continuous-path holonomies.