Contextuality as a Resource for Models of Quantum Computation with Qubits

Abstract

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...

Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates

“…A qutrit is the smallest system that exhibits inherent quantum features such as contextuality [28][29][30], which has been conjectured to be a resource for quantum computing [31][32][33][34][35]. NMR qudit quantum computing can be performed by using nuclei with spin s > 1 2 or can be modeled by two or * shrutidogra@iisermohali.ac.in † kavita@iisermohali.ac.in ‡ arvind@iisermohali.ac.in more coupled spin-1 2 nuclei [36,37].…”

Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen–Specker theorem: the Peres–Mermin square

“…Furthermore, failing to admit of a noncontextual model appears to be a resource. For instance, in the context of the state injection model for quantum computation [4,5], the failure of noncontextuality has been shown in some cases to be necessary for achieving universal quantum computation [6,7].…”

“…The definition asserts that an ontological model of an operational theory is noncontextual if two experimental procedures that are statistically indistinguishable at the operational level are statistically indistinguishable at the ontological level. They key point is that the notion of statistical indistinguishability at the operational level can be assessed in any operational theory 7 .…”

Quantum contextuality of a single neutrino under interactions with matter

“…At www.dropbox.com/sh/m678lxyn6fek2h6/AABM1Kg7wczX0JsRU-j2gcsHa?dl=0 one can find Mathematica code for their generation. In particular, following [44], after transforming the original, native problem equation (18)into the coherence-vector form equation (22) the numerical tool which is utilized is a Pythonbased exponentiation of finite dimensional matrices.…”

“…The experiments were carried out in the systems of trapped ions [15], photons [16], fermions [17], deuterium nucleus [18] or neutron [19]. Moreover, the quantum contextuality gains in the popularity also in the quantum computation's community [20][21][22]. All these aspects indicate that the contextuality has become the active area of research.…”