Abstract
Analyzing pseudo-telepathy graph games, we propose a way to build contextuality scenarios exhibiting the quantum supremacy using graph states. We consider the combinatorial structures generating equivalent scenarios. We introduce a new tool called multipartiteness width to investigate which scenarios are harder to decompose and show that there exist graphs generating scenarios with a linear multipartiteness width.
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Notes
- 1.
The following three properties are equivalent: (i) \(D\subseteq \text {Even}(D)\); (ii) every vertex of G[D] has an even degree; (iii) G[D] is a union of Eulerian graphs. Notice that \(D\subseteq \text {Even}(D)\) does not imply that G[D] is Eulerian as it may not be connected.
- 2.
\(L_x\) is linear for the symmetric difference: \(L_x(D_1\oplus D_2)=L_x(D_1)\oplus L_x(D_2)\).
- 3.
The Foulis Randall product of scenarios [5] is the scenario \(H_A \otimes H_B\) with vertices \(V(H_A\otimes H_B)=V(H_A)\times V(H_B)\) and edges \(E(H_A\otimes H_B)=E_{A\rightarrow B}\cup E_{A\leftarrow B}\) where \(E_{A\rightarrow B}:=\{\cup _{a\in e_A} \{a\}\times f(a) : e_a\in E_A, f: e_A \rightarrow E_B\}\) and \(E_{A\leftarrow B}:=\{\cup _{b\in e_A} f(b)\times \{b\} : e_b\in E_b, f: E_B \rightarrow E_A\}\). In the multipartite case there are several ways to define products, however they all correspond to the same non-locality constraints [5]. Therefore one can just consider the minimal product \( ^{\min }\otimes _{i=1 }^n H_i\) which has vertices in the cartesian product \(V=\varPi V_i\) and edges \(\cup _{k\in [1,n]} E_k\) where \(E_k=\{(v_1\ldots ,v_n), v_i\in e_i \, \forall i\ne k,\, v_k\in f(\overrightarrow{v})\}\) for some edge \(e_i\in E(H_i)\) for every party \(i\ne k\) and a function \(\overrightarrow{v}\mapsto f(\overrightarrow{v})\) which assigns to every joint outcome \(\overrightarrow{v}=(v_1\ldots v_{k-1},v_{k+1},\ldots v_n)\) an edge \(f(\overrightarrow{v})\in E(H_k)\) (the \(k^{th}\) vertex is replaced by a function of the others).
- 4.
Note that for the questions x for which there exists no D involved in x, all the answers are allowed thus the constraints represented by the associated edge is a hyperedge of no-signaling scenario \(H_{Nsig}\).
- 5.
The probability distribution described in [8] corresponds to the quantum winning strategy on the graph state obtained from a cycle with 5 vertices.
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We would like to thank an anonymous reviewer for noticing a mistake in an earlier version and helpful comments.
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Anshu, A., Høyer, P., Mhalla, M., Perdrix, S. (2017). Contextuality in Multipartite Pseudo-Telepathy Graph Games. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_5
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