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'''Optical cluster states''' are a proposed tool to achieve quantum computational universality in [[linear optical quantum computing]] (LOQC).<ref>{{cite journal | last=Kok | first=Pieter | last2=Munro | first2=W. J. | last3=Nemoto|author3-link= Kae Nemoto | first3=Kae | last4=Ralph | first4=T. C. | last5=Dowling | first5=Jonathan P. | last6=Milburn | first6=G. J. | title=Linear optical quantum computing with photonic qubits | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=79 | issue=1 | date=2007-01-24 | issn=0034-6861 | doi=10.1103/revmodphys.79.135 | pages=135–174|arxiv=quant-ph/0512071}}</ref> As direct [[quantum entanglement|entangling]] operations with [[photon]]s often require [[nonlinear optics|nonlinear]] effects, probabilistic generation of entangled resource states has been proposed as an alternative path to the direct approach.
==Creation of the cluster state==
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===Yoran-Reznik protocol===
Among the first proposals of utilizing resource states for optical quantum computing was the Yoran-Reznik protocol in 2003.<ref>{{cite journal | last=Kok | first=Pieter | last2=Munro | first2=W. J. | last3=Nemoto | first3=Kae|author3-link= Kae Nemoto | last4=Ralph | first4=T. C. | last5=Dowling | first5=Jonathan P. | last6=Milburn | first6=G. J. | title=Linear optical quantum computing with photonic qubits | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=79 | issue=1 | date=2007-01-24 | issn=0034-6861 | doi=10.1103/revmodphys.79.135 | pages=135–174|arxiv=quant-ph/0512071}}</ref> While the proposed resource in this protocol was not exactly a cluster state, it brought many of the same key concepts to the attention of those considering the possibilities of optical quantum computing and still required connecting multiple separate one-dimensional chains of entangled photons via controlled-Z operations. This protocol is somewhat unique in that it utilizes both the spatial mode degree of freedom along with the polarization degree of freedom to help entanglement between qubits.
Given a horizontal path, denoted by <math>a</math>, and a vertical path, denoted by <math>b</math>, a 50:50 beam splitter connecting the paths followed by a <math>\pi/2</math>-phase shifter on path <math>a</math>, we can perform the transformations
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