Genuinely multipoint temporal quantum correlations and universal measurement-based quantum computing

Abstract

We introduce a constructive procedure that maps all spatial correlations of a broad class of $d$-level states of $N$ parties into temporal correlations between general $d$-outcome quantum measurements performed on a single $d$-level system. This allows us to present temporal phenomena analogous to genuinely multipartite nonlocal phenomena, such as Greenberger-Horne-Zeilinger correlations, which do not exist if only projective measurements on a single qubit are considered. The map is applied to certain lattice systems in order to replace one spatial dimension with a temporal one, without affecting measured correlations. We use this map to show how repeated application of a one-dimensional (1D) cluster gate leads to universal one-way quantum computing when supplemented with general two-outcome quantum measurements. In this way, we recover a temporal version of measurement-based quantum computing performed on a sequentially recreated 1D cluster.