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

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. In reproducing many features of quantum mechanics in an essentially classical model, it clarified our understanding of what behaviour 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 fault tolerant operations used in standard state injection circuits can be realised within Spekkens' theory, and that state-injection leads to non-classicality in the form of contextuality. To achieve this, we extend prior work connecting Spekkens' theory and stabilizer quantum mechanics, showing that sub-theories of the latter can be represented within Spekkens' theory, in spite of the contextuality in qubit stabilizer quantum mechanics. The work shines new light on the relationship between quantum computation and contextuality.

Contextuality as a resource for measurement-based quantum computation beyond qubits

“…Many features of quantum theory have been proposed as the origin of this supposed quantum computational power: entanglement, superposition, and exponential scaling of Hilbert spaces to name a few. Recently, contextuality has been investigated in a variety of scenarios as a potential resource for quantum computation [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Contextuality can be thought of as the impossibility of assigning pre-determined outcomes to all potential measurements of a quantum system, independent of their measurement context [15][16][17][18].…”

Uniqueness of all fundamental noncontextuality inequalities

“…Recently, contextuality has also been applied more directly to quantum key distribution [12,13] and variants of randomness certification [14] and self-testing [15]. Further, it has been uncovered to be the resource powering the measurement based model and a class of fault tolerant models of quantum computation [16,17], among others [1,[16][17][18][19][20][21][22][23].…”

A Classical Formulation of Quantum Theory?

“…Spekkens’ model has previously been provided with a contextual extension, but without fully capturing quantum theory [ 21 ]. It has also been shown to hold strong similarities to stabilizer physics [ 22 , 23 , 24 ], thereby providing a link to a subtheory of quantum mechanics that plays an important role in quantum computing. Spekkens’ model is naturally set in phase space, which provides the backbone for our work.…”