Contextuality Supplies the Magic for Quantum Computation

We know that quantum mechanics enables the performance of computational and cryptographic tasks that are impossible (or impracticable) using only classical physics. It seems natural to examine manifestations of inherently quantum behaviour as potential sources of these capabilities. Here [1] we establish that quantum contextuality, a generalization of nonlocality identified by Bell [2] and Kochen-Specker [3] almost 50 years ago, is a critical resource for quantum speed-up within the leading model for fault-tolerant quantum computation, known as magic state distillation (MSD) [4], [5]. We consider the framework of fault-tolerant stabilizer quantum computation, which provides the most promising route to achieving robust universal quantum computation thanks to the discovery of high-threshold codes in two-dimensional geometries. In this framework, only a subset of quantum operations - namely, stabilizer operations - can be achieved via a fault-tolerant encoding. These operations define a closed subtheory (i.e. sets of states, transformations and measurements) of quantum theory - the stabilizer subtheory - which is not universal and in fact admits an efficient classical simulation. The stabilizer subtheory can be promoted to universal quantum computation through MSD which relies on a large number of ancillary resource states. Using d-level (where d is an odd prime) quantum systems - "qudits" - as the fundamental unit of information leads to mathematical and conceptual simplifications [6] (as well as improved efficiency and thresholds for the MSD subroutine). The term stabilizer refers to the fact that the states arising in our subtheory are simultaneous eigenstates of elements of the finite Heisenberg-Weyl (or, equivalently, the generalized Pauli-) group [7], [8], [9].If the input states to an MSD subroutine are unsuitable, then the overall computation remains classically efficiently simulable. We show that quantum contextuality plays a critical role in characterizing t- e suitability of quantum states for MSD. Our approach builds on recent work [10] that has established a remarkable connection between contextuality and graph-theory. We use this combinatorial framework to identify non-contextuality inequalities such that the onset of state-dependent contextuality, using stabilizer measurements, coincides exactly with the possibility of universal quantum computing via MSD.