Quantum computers promise dramatic advantages over their classical counterparts, but the answer to the most basic question "What is the source of the power in quantum computing?" has remained elusive. Here we prove a remarkable equivalence between the onset of contextuality and the possibility of universal quantum computation via magic state distillation. This is a conceptually satisfying link because contextuality provides one of the fundamental characterizations of uniquely quantum phenomena and, moreover, magic state distillation is the leading model for experimentally realizing fault-tolerant quantum computation. Furthermore, this connection suggests a unifying paradigm for the resources of quantum information: the nonlocality of quantum theory is a particular kind of contextuality and nonlocality is already known to be a critical resource for achieving advantages with quantum communication. In addition to clarifying these fundamental issues, this work advances the resource framework for quantum computation, which has a number of practical applications, such as characterizing the efficiency and trade-offs between distinct theoretical and experimental schemes for achieving robust quantum computation and bounding the overhead cost for the classical simulation of quantum algorithms.Quantum information enables dramatic new advantages for computation, such as Shor's factoring algorithm 1 and quantum simulation algorithms 2 . This naturally raises the fundamental question: what unique resources of the quantum world enable the advantages of quantum information? There have been many attempts to answer this question, with proposals including the hypothetical "quantum parallelism" 3 some associate with quantum superposition, the necessity of large amounts of entanglement 4 , and much ado about quantum discord 5 . Unfortunately none of these proposals have proven satisfactory 6-9 , and, in particular, none have helped resolve outstanding challenges confronting the field. For example, on the theoretical side, the most general classes of problems for which quantum algorithms might offer an exponential speed-up over classical algorithms are poorly understood. On the experimental side, there remain significant challenges to designing robust, large-scale quantum computers, and an important open problem is to determine the minimal physical requirements of a useful quantum computer 10,11 . A framework identifying relevant resources for quantum computation should help clarify these issues, for example, by identifying new simulation schemes for classes of quantum algorithms and by clarifying the trade-offs between the distinct physical requirements for achieving robust quantum computation. Here we establish that quantum contextuality, a generalization of nonlocality identified by Bell 14 and Kochen-Specker 15 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) [16][17][18] .Contextuality was first recognized as...

Recent results on the non-universality of fault-tolerant gate sets underline the critical role of resource states, such as magic states, to power scalable, universal quantum computation. Here we develop a resource theory, analogous to the theory of entanglement, that is relevant for fault-tolerant stabilizer computation. We introduce two quantitative measures-monotones-for the amount of non-stabilizer resource. As an application we give absolute bounds on the efficiency of magic state distillation. One of these monotones is the sum of the negative entries of the discrete Wigner representation of a quantum state, thereby resolving a long-standing open question of whether the degree of negativity in a quasi-probability representation is an operationally meaningful indicator of quantum behavior. New J. Phys. 16 (2014) 013009 V Veitch et al a common paradigm in quantum communication is two or more spatially separated parties communicating using classical communication and local quantum operations, considered 'cheap' resources, supplemented by 'expensive' resources that require global manipulation of quantum states, such as entanglement or quantum communication. This division of quantum operations into cheap and expensive parts motivates the development of a resource theory [27]. In the sense just explained, entanglement theory is the resource theory of quantum communication [2,3,29]. In this paper we develop a resource theory of quantum computation.The major obstacle to physical realizations of quantum computation is that real world devices suffer random noise when they execute quantum algorithms. Fault-tolerant quantum computation offers a framework to overcome this problem. Starting from a given error rate for the physical computation, logical encodings can be applied to create arbitrarily small effective error rates for the logically encoded computation. Transversal unitary gates, i.e. gates that do not spread errors within each code block, play a critical role in fault-tolerant quantum computation. Recent theoretical work has shown that a fault-tolerant scheme with a set of quantum gates that is both universal and transversal does not exist [11].Many-though not all-of the known fault-tolerant schemes are built around the stabilizer formalism. Stabilizer codes pick out a distinguished set of preparations, measurements and unitary transformations that have a fault-tolerant implementation; these are sometimes called 'stabilizer operations'. In this case the fault-tolerant operations are not only sub-universal but also actually efficiently classically simulable by the Gottesman-Knill theorem [17]. Thus to achieve universal quantum computation the stabilizer operations must be supplemented with some other fault-tolerant non-stabilizer resource.A celebrated scheme for overcoming this limitation is the magic state model of quantum computation [19,42] where the additional resource is a set of ancilla systems prepared in some (generally noisy) non-stabilizer quantum state. The idea is to consume non-stabilizer resource st...

A central problem in quantum information is to determine the minimal physical resources that are required for quantum computational speedup and, in particular, for fault-tolerant quantum computation. We establish a remarkable connection between the potential for quantum speed-up and the onset of negative values in a distinguished quasi-probability representation, a discrete analog of the Wigner function for quantum systems of odd dimension. This connection allows us to resolve an open question on the existence of bound states for magic-state distillation: we prove that there exist mixed states outside the convex hull of stabilizer states that cannot be distilled to non-stabilizer target states using stabilizer operations. We also provide an efficient simulation protocol for Clifford circuits that extends to a large class of mixed states, including bound universal states.

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