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arXiv:1309.7650 · Physical Review A

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

Marcin Markiewicz, Anna Przysiężna, STEPHEN D. BRIERLEY, Tomasz Paterek

Published Jun 17, 2014• 11 citations
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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.

DOI
10.1103/physreva.89.062319

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We introduce a constructive procedure that maps all spatial correlations of a broad class of ddd-level states of NNN parties into temporal correlations between general ddd-outcome quantum measurements performed on a single ddd-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.
We introduce a constructive procedure that maps all spatial correlations of a broad class of states into temporal correlations between general quantum measurements. 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 qubits 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. In this way we prove that 1D cluster states are in fact universal for quantum computation when we allow generalised quantum measurements.
Abstract Multipartite quantum correlations are a powerful resource that underpins applications from quantum metrology to quantum computing. While most research has focused on spatial correlations, it is now becoming clear that a sequence of measurements on a single quantum system at different points in time reveals a similarly rich, yet fundamentally different structure of multipartite temporal correlations. Here we experimentally observe genuine multi-time correlations in a sequence of three generalized measurements on a single photon. These correlations, manifest by a simultaneous violation of all pairwise Bell inequalities, cannot be reproduced by any spatial quantum state of equal dimension. Our work lays the foundation for the exploration of temporal correlations arising in quantum networks for quantum information applications.
In this paper, we build a framework allowing for a systematic investigation of the fundamental issue: ' Which quantum states serve as universal resources for measurement-based (one-way) quantum computation? ' We start our study by re-examining what is exactly meant by 'universality' in quantum computation, and what the implications are for universal one-way quantum computation. Given the framework of a measurement-based quantum computer, where quantum information is processed by local operations only, we find that the most general universal one-way quantum computer is one which is capable of accepting arbitrary classical inputs and producing arbitrary quantum outputs—we refer to this property as CQ-universality. We then show that a systematic study of CQ-universality in one-way quantum computation is possible by identifying entanglement features that are required to be present in every universal resource. In particular, we find that a large class of entanglement measures must reach its supremum on every universal resource. These insights are used to identify several families of states as being not universal, such as one-dimensional (1D) cluster states, Greenberger–Horne–Zeilinger (GHZ) states, W states, and ground states of non-critical 1D spin systems. Our criteria are strengthened by considering the efficiency of a quantum computation, and we find that entanglement measures must obey a certain scaling law with the system size for all efficient universal resources. This again leads to examples of non-universal resources, such as, e.g. ground states of critical 1D spin systems. On the other hand, we provide several examples of efficient universal resources, namely graph states corresponding to hexagonal, triangular and Kagome lattices. Finally, we consider the more general notion of encoded CQ-universality, where quantum outputs are allowed to be produced in an encoded form. Again we provide entanglement-based criteria for encoded universality. Moreover, we present a general procedure to construct encoded universal resources.
We introduce a general scheme for sequential one-way quantum computation where static systems with long-living quantum coherence (memories) interact with moving systems that may possess very short coherence times. Both the generation of the cluster state needed for the computation and its consumption by measurements are carried out simultaneously. As a consequence, effective clusters of one spatial dimension fewer than in the standard approach are sufficient for computation. In particular, universal computation requires only a one-dimensional array of memories. The scheme applies to discrete-variable systems of any dimension as well as to continuous-variable ones, and both are treated equivalently under the light of local complementation of graphs. In this way our formalism introduces a general framework that encompasses and generalizes in a unified manner some previous system-dependent proposals. The procedure is intrinsically well suited for implementations with atom-photon interfaces.
Measurements on a single quantum system at different times reveal rich nonclassical correlations similar to those observed in spatially separated multipartite systems. Here we introduce a theory framework that unifies the description of temporal, spatial, and spatiotemporal resources for quantum correlations. We identify and experimentally demonstrate simple cases where an exact mapping between the domains is possible. We then identify correlation resources in arbitrary situations, where not all spatial quantum states correspond to a process and not all temporal measurements have a spatial analog. These results provide a starting point for the systematic exploration of multipoint temporal correlations as a powerful resource for quantum information processing.