A central question in quantum computation is to identify the resources that are responsible for quantum speed-up. Quantum contextuality has been recently shown to be a resource for quantum computation with magic states for odd-prime dimensional qudits and two-dimensional systems with real wave functions. The phenomenon of state-independent contextuality poses a priori an obstruction to characterizing the case of regular qubits, the fundamental building block of quantum computation. Here, we establish contextuality of magic states as a necessary resource for a large class of quantum computation schemes on qubits. We illustrate our result with a concrete scheme related to measurement-based quantum computation. DOI: 10.1103/PhysRevLett.119.120505 The model of quantum computation by state injection (QCSI) [1] is a leading paradigm of fault-tolerance quantum computation. Therein, quantum gates are restricted to belong to a small set of classically simulable gates, called Clifford gates [2], that admit simple fault-tolerant implementations [3]. Universal quantum computation is achieved via injection of magic states [1], which are the source of quantum computational power of the model.A central question in QCSI is to characterize the physical properties that magic states need to exhibit in order to serve as universal resources. In this regard, quantum contextuality has recently been established as a necessary resource for QCSI. This was first achieved for quopit systems [4,5], where the local Hilbert space dimension is an odd prime power, and subsequently for local dimension two with the case of rebits [6]. In the latter, the density matrix is constrained to be real at all times.In this Letter we ask "Can contextuality be established as a computational resource for QCSI on qubits?" This is not a straightforward extension of the quopit case because the multiqubit setting is complicated by the presence of stateindependent contextuality among Pauli observables [7,8]. Consequently, every quantum state of n ≥ 2 qubits is contextual with respect to Pauli measurements, including the completely mixed one [5]. It is thus clear that contextuality of magic states alone cannot be a computational resource for every QCSI scheme on qubits.Yet, there exist qubit QCSI schemes for which contextuality of magic states is a resource, and we identify them in this Letter. Specifically, we consider qubit QCSI schemes M O that satisfy the following two constraints: (C1) Resource character. There exists a quantum state that does not exhibit contextuality with respect to measurements available in M O . (C2) Tomographic completeness. For any state ρ, the expectation value of any Pauli observable can be inferred via the allowed operations of the scheme.The motivation for these constraints is the following. Condition (C1) constitutes a minimal principle that unifies, simplifies and extends the quopit [5] and rebit [6] settings. While seemingly a weak constraint, it excludes the possibility of Mermin-type state-independent contextuality [7,8] amo...

We provide the first example of a symmetry protected quantum phase that has universal computational power. This two-dimensional phase is protected by one-dimensional line-like symmetries that can be understood in terms of local symmetries of a tensor network. These local symmetries imply that every ground state in the phase is a universal resource for measurement based quantum computation.arXiv:1803.00095v2 [quant-ph]

We describe schemes of quantum computation with magic states on qubits for which contextuality and negativity of the Wigner function are necessary resources possessed by the magic states. These schemes satisfy a constraint. Namely, the non-negativity of Wigner functions must be preserved under all available measurement operations. Furthermore, we identify stringent consistency conditions on such computational schemes, revealing the general structure by which negativity of Wigner functions, hardness of classical simulation of the computation, and contextuality are connected.

We provide a cohomological framework for contextuality of quantum mechanics that is suited to describing contextuality as a resource in measurement-based quantum computation. This framework applies to the parity proofs first discussed by Mermin, as well as a different type of contextuality proofs based on symmetry transformations. The topological arguments presented can be used in the state-dependent and the state-independent case.

We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et al., New J. Phys. 14, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides amplitude estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to robustness of magic.