Fig. 3: Representation of the n-party Modular \({\mathsf{XOR}}\) games \({{{{\mathcal{G}}}}}_{p}\). | Nature Communications

Fig. 3: Representation of the n-party Modular \({\mathsf{XOR}}\) games \({{{{\mathcal{G}}}}}_{p}\).

From: Unconditional advantage of noisy qudit quantum circuits over biased threshold circuits in constant depth

Fig. 3

Each party Pi receives an input \({x}_{i}\in {{\mathbb{F}}}_{p}\) from a referee, forming a combined input string x with an ℓ1 − norm ∣x∣ = kp, where \(k\in {\mathbb{N}}\). Without further communication, each party responds with \({y}_{i}\in {{\mathbb{F}}}_{p}\), resulting in a collective output y = (y1, y2, …, yn). The players win if the output y has ℓ1-norm equal to the additive inverse of k modulo p, i.e., \(| y|=-k\,{{\mathrm{mod}}}\,\,p\). For context, \({{{{\mathcal{G}}}}}_{2}\) relates to the Mermin-Peres non-local game used in prior works14,17. The plot illustrates the upper bounds on the (winning) correlation of classical strategies for \({{{{\mathcal{G}}}}}_{2}\), \({{{{\mathcal{G}}}}}_{3}\), \({{{{\mathcal{G}}}}}_{5}\), and \({{{{\mathcal{G}}}}}_{7}\), whereas the optimal quantum strategy attains a maximum correlation of 1 for all \({{{{\mathcal{G}}}}}_{p}\).

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