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

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.

Figure 1

Sequential generation of a multipartite MPS and its temporal counterpart. (a) Two-particle unitary gates $U_k$ (gray rectangles) are sequentially performed on particles (red circles) giving the state $\Psi$ (big red shape). In the end, projective measurements (blue shapes) are conducted on the particles. Dashed lines indicate how unitary gates are related to MPS matrices of Eq. (6). (b) Measurements can be shifted in time. (c) Every gate followed by a measurement can equivalently be represented as a POVM. (d) The circuit can be rearranged into a sequence of quantum channels performing POVM measurements. (e) We can reduce the required resources to two particles. After each measurement, one of them is reset and recycled into the remaining protocol.

Figure 2

2D examples of mappings from spatial to spatiotemporal correlations. (a) “Order 1” means that the state of the lattice is generated by applying the timelike (horizontal) gates before the respective next column of spacelike (vertical) gates. “Order 2” means that the spacelike gates are performed before the timelike gates connecting them. (b) Result of applying the map to the “Order 1” case. The spatial correlations of measurement outcomes of local projective measurements performed on each node of the lattice are mapped into temporal correlations of outcomes of local POVM measurements performed on a 1D lattice. (c) Result of applying the map to the “Order 2” case. The spatial correlations are mapped into temporal correlations of outcomes of POVMs on a 1D lattice, however in this case the POVM's have entangled ancillas. In both cases, at the last stage projective measurements are performed.

Figure 3

One-way quantum computing in time. In a standard spatial implementation, measurements are performed in a sequence along one dimension of a 2D cluster. This can be mapped into a spatiotemporal scenario as follows. First a vertical 1D cluster state (a blue-shaded rectangle) is prepared in the usual way. Next, these qubits undergo local POVM measurements, which effectively entangle the second column of qubits with the prepared 1D cluster, and the qubits of the 1D cluster are measured (which would never again take part in the computation, and therefore can be recycled). Since the contents of the gray rectangle are repeated throughout the computation, universal one-way quantum computing requires a single “gate” preparing 1D cluster state that after each preparation is followed by generalized measurements. In the last stage, local projective measurements need to be performed.