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

Spekkens' toy theory is a noncontextual hidden variable model with an epistemic restriction, a constraint on what the observer can know about the reality. In reproducing many features of quantum mechanics in an essentially classical model, it clarified our understanding of what behavior can be truly considered intrinsically quantum. In this work, we show that Spekkens' theory can be also used to help understanding aspects of quantum computation-in particular, an important subroutine in fault-tolerant quantum computation called state injection. State injection promotes fault-tolerant quantum circuits, which are usually limited to the classically efficiently simulatable stabilizer operations, to full universal quantum computation. We show that the limited set of faulttolerant operations used in standard state-injection circuits can be realized within Spekkens' theory, and that state injection leads to nonclassicality in the form of contextuality. To achieve this, we extend prior work connecting Spekkens' theory and stabilizer quantum mechanics, showing that subtheories of the latter can be represented within Spekkens' theory, in spite of the contextuality in qubit stabilizer quantum mechanics. The work shines light on the relationship between quantum computation and contextuality.