State-injection schemes of quantum computation in Spekkens' toy theory

TL;DR

This paper explores how Spekkens' toy theory can model certain quantum computational schemes, especially in qubits, by analyzing state-injection methods and their relation to contextuality, suggesting a unified framework for quantum computation.

Contribution

It characterizes Spekkens' subtheories equivalent to stabiliser quantum mechanics and demonstrates their sufficiency for universal quantum computation via state-injection.

Findings

Spekkens' subtheories can simulate stabiliser quantum mechanics in qubits.

All non-covariant Clifford gates can be injected within Spekkens' subtheories.

A novel computation scheme based on CCZ state injection is proposed.

Abstract

Spekkens' toy theory is a non-contextual hidden variable model with an epistemic restriction, a constraint on what the observer can know about the reality. It has been shown in [3] that for qudits of odd dimensions it is operationally equivalent to stabiliser quantum mechanics by making use of Gross' theory of discrete Wigner functions. This result does not hold in the case of qubits, because of the unavoidable negativity of any Wigner function representation of qubit stabiliser quantum mechanics. In this work we define and characterise the subtheories of Spekkens' theory that are operationally equivalent to subtheories of stabiliser quantum mechanics. We use these Spekkens' subtheories as a unifying framework for the known examples of state-injection schemes where contextuality is an injected resource to reach universal quantum computation. In addition, we prove that, in the case of qubits, stabiliser quantum mechanics can be reduced to a Spekkens' subtheory in the sense that all its objects that do not belong to the Spekkens' subtheory, namely non-covariant Clifford gates, can be injected. This shows that within Spekkens' subtheories we possess the toolbox to perform state-injection of every object outside of them and it suggests that there is no need to use bigger subtheories to reach universal quantum computation via state-injection. We conclude with a novel scheme of computation suggested by our approach which is based on the injection of CCZ states and we also relate different proofs of contextuality to different state injections of non-covariant gates.