We describe a universal scheme of quantum computation by state injection on rebits (states with real density matrices). For this scheme, we establish contextuality and Wigner function negativity as computational resources, extending results of [M. Howard et al., Nature 510, 351-355 (2014)] to two-level systems. For this purpose, we define a Wigner function suited to systems of n rebits, and prove a corresponding discrete Hudson's theorem. We introduce contextuality witnesses for rebit states, and discuss the compatibility of our result with state-independent contextuality.

We propose a new strategy to decode color codes, which is based on the projection of the error onto three surface codes. This provides a method to transform every decoding algorithm of surface codes into a decoding algorithm of color codes. Applying this idea to a family of hexagonal color codes, with the perfect matching decoding algorithm for the three corresponding surface codes, we find a phase error threshold of approximately 8.7%. Finally, our approach enables us to establish a general lower bound on the error threshold of a family of color codes depending on the threshold of the three corresponding surface codes. These results are based on a chain complex interpretation of surface codes and color codes.

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...