Abstract
In this chapter, we discuss bottom-up theories of continuous-variable cluster states and one-way quantum computation. We first review formalisms of cluster states using state vectors, stabilizers, and nullifiers. Second, by applying the extended van Loock-Furusawa criteria, we show a sufficient condition of entanglement for an arbitrary cluster state. Third, we show that an approximation of any cluster state (Gaussian cluster states) with an arbitrary graph can be generated by using an appropriate network of beam splitters. As examples of Gaussian cluster states, we show beam splitter networks for the generation of \(n\)-mode linear and star cluster states with an arbitrary \(n\). Fourth, we discuss input-coupling schemes. We introduce the squeezer-based input-coupling scheme, which is simpler than the teleportation-based input-coupling scheme. Fifth, We show several cluster reshaping schemes: erasing, wire shortening with a homodyne measurement, as well as cluster connections with zero, one, or two homodyne measurements. Sixth, we discuss Gaussian operations in one-way quantum computation. As the last issue, we introduce the “delta notation” of one-way quantum computation, where the error derived from finite levels of squeezing is described by nullifier terms of the resource cluster state.
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Notes
- 1.
Although we can use observables \(\hat{x}\) at the first and the second measurements which correspond to \(\kappa _1\) and \(\kappa _2\), we will omit these cases because universality will be achieved without using these observables.
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Ukai, R. (2015). Cluster States and One-Way Quantum Computation. In: Multi-Step Multi-Input One-Way Quantum Information Processing with Spatial and Temporal Modes of Light. Springer Theses. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55019-8_5
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DOI: https://doi.org/10.1007/978-4-431-55019-8_5
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