Contextuality as a computational resource

In 2014, Howard et al. showed that the contextuality possessed solely by the magic state is a necessary resource for UQC in state-injection schemes on qudits of odd prime dimensions [54]. More precisely, they proved that a noisy input state must violate Cabello-Severini-Winter inequality (equation (3.3)) in order to be distilled into a magic state. Therefore UQC via magic states implies that the magic state manifests contextuality. The free part of the computation is composed by stabilizer circuits and the inequality is built considering a scenario where the vertices of the graph represent stabilizer rank-1 projectors.

This result triggered a whole field of research focused on studying the role of contextuality in quantum computation. The main question that remains open is to find a result analogue to Howard’s for qubits. As already discussed, multi-qubit SQM shows state-independent contex- tuality, which means that the result of Howard et al. cannot be extended to the qubit case. However, it is possible to restrict the free part of the computation to subtheories of SQM and obtain analogous results to the qudit case. In 2015, Delfosse et al considered state-injection schemes on rebits, where the free part is composed by the CSS circuits, presenting no state- independent contextuality. More precisely, we define the CSS subtheory, that we denote with

Generalized

non-contextuality measurementsSharp Preparations Transformations Outcome determinism + = _{non-Contextuality}Kochen-Specker State-dependent State-independent Unsharp measurements

- Broken by: GHZ paradox. - Yes Qudit SQM. - No Qubit SQM. - CSW inequalities.

- Broken by: P-M square. - Yes Qudit SQM. - No Qubit SQM. No QM. No QM. No QM. No QM.

Figure 3.3: Schematic representation of the relations between different notions of non-contextuality. The notion of non-contextuality due to Spekkens [34] generalises the notion due to Kochen-Specker [33], separating it from the notion of outcome determinism and extending it to unsharp measurements, preparations and transformations. Non-contextual ontological models of quantum mechanics are impossible for both notions (and also for just preparation and transformation non-contextuality, but not for measurement non-contextuality, as proven in [34]). There exist examples of Kochen-Specker contextuality that, given a set of pro- jective measurements, arise only for certain states. In this case we talk about state-dependent contextuality (the usual example of this is the GHZ paradox). Violations of Cabello-Severini- Winter (CSW) inequality quantify this contextuality. When the contextuality arguments hold for any quantum state (e.g. in the Peres-Mermin (P-M) square) we talk of state-independent contextuality. It results that qubit SQM is Kochen-Specker contextual, while odd dimensional qudit SQM is not, due to the intrinsic difference in the structure of the Pauli groups in the two cases.

(Sr,Tr,Mr), where Sr,Tr,Mr are the sets of allowed quantum states, transformations and

measurements respectively, as follows. The set _Sr, a subset of the stabilizer states, is com-

posed by CSS states [87], i.e. stabilizer states|ψi , whose corresponding stabilizer group S(|ψi) decomposes into an X and a Z part; i.e. S(_{|ψi) = S}X(|ψi) ∪ SZ(|ψi), where all elements of

are the eigenstates of the allowed observables belonging to the set Mr,

Mr ={X(x), Z(p)|x, p ∈ Zn2}. (3.6)

The set of allowed transformations is composed by the CSS preserving gates, subset of the Clifford group Cn,2, Tr={g ∈ Cn,2|g |ψi ∈ Sr,∀ |ψi ∈ Sr} = * _n O i=1 Hi, CN OT (i, j), Xi, Zi + , (3.7)

where i, j ∈ {1, 2, . . . , n} and i 6= j. The universal quantum computation is reached by injecting two particular magic states to the free subtheory of CSS rebits just described [55].

The proof that contextuality is a necessary resource for UQC injected with the magic states is based on the construction of a contextuality witness similar to the Cabello-Severini-Winter inequality, in the sense that they both consist of linear operators for which the range of ex- pectation values allowed by quantum mechanics is strictly greater than the one allowed for non-contextual ontological models. Moreover, they also developed a Wigner function that, analogously to Gross Wigner function for odd qudit stabilizer states, is non-negative if and only if it represents CSS states. It is defined as follows:

Ar(λ) = 1 2n X T (λ0_)∈A (−1)[λ,λ0]_{W (λˆ} 0 ), (3.8)

where ˆW (λ) = Z(p)X(x), λ = (x, p), andA = { ˆW (λ)|x · p = 0 mod 2}. The set A is the set of inferred observables. “Inferred” means that these observables may not be directly measurable, but they can be inferred by multiple measurements. For example, in the case of two qubits, the set MrandA are Mr ={II, IX, IZ, XI, ZI, XX, ZZ}, and A = {II, IX, IZ, XI, ZI, XX, ZZ, XZ,

ZX, Y Y}, i.e. the set of all rebits observables. Delfosse’s Wigner function shares most of the properties that Gross’ Wigner function possesses, as defined in the previous chapter. How- ever, it is not factorizable (equation (2.26)), i.e. it is composed by phase-point operators of n qubits that are not given by the tensor products of the ones for the single qubit, e.g.

Ar((0, 0), (0, 0))6= Ar(0, 0)⊗ Ar(0, 0).

There is a strict relation between the notions of contextuality and negativity of the Wigner functions (and more general quasi-probability representations). More precisely, as proven in [123], the existence of a non-negative quasi-probability distribution representing a subtheory is equivalent to the existence of a non-contextual ontological model for it. Therefore, finding a non-negative Wigner function for a subtheory guarantees that it is non-contextual. The con- verse is not true, as it may be that quasi-probability representations different from the Wigner functions are non-negative even if the Wigner functions unavoidably show some negativity. When restricted to Pauli measurements, in the case of qudits of odd dimensions, Delfosse et al. proved that the negativity of the Wigner functions and Kochen-Specker contextuality are equivalent notions [124]. This result does not hold for a single qudit, for which there are states that fit in a non-contextual ontological model, even if no non-negative Wigner functions exist. When considering more generic subtheories of multi-qubit SQM that represent the free part of the computation, the obstacle to be avoided in order to find results showing contextuality as a resource possessed solely by the magic state is again the presence of state-independent contextuality. In 2017, Bermejo-Vega et al. showed that, with the condition that the free part of the computation does not show state-independent contextuality, the contextuality of the magic state is a necessary resource for universal quantum computation. The proof they developed is quite different from the previous ones based on the construction of a contextuality witness. It relies on a characterization of non-contextual ontological models of state-injection schemes on qubits, which shows a contradiction if universality of the scheme and the non- contextuality of the magic state are assumed. This result is also treated in the framework of Raussendorf et al., which is based on Wigner functions [57]. We now describe this framework, that will be crucial to state the results of this chapter.