|
'''Optical cluster states''' are a proposed tool to achieve quantum computational universality in [[linear optical quantum computing]] (LOQC).<ref>{{cite journal | lastlast1=Kok | firstfirst1=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| bibcode=2007RvMP...79..135K | s2cid=119335959 }}</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==
===Nielsen protocol===
In 2004, Nielsen proposed a protocol to create cluster states,<ref>{{cite journal | last=Nielsen | first=Michael A. | title=Optical Quantum Computation Using Cluster States | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=93 | issue=4 | date=2004-07-21 | issn=0031-9007 | doi=10.1103/physrevlett.93.040503 | page=040503| pmid=15323741 |arxiv=quant-ph/0402005| bibcode=2004PhRvL..93d0503N | s2cid=7720448 }}</ref> borrowing techniques from the [[KLM protocol|Knill-Laflamme-Milburn protocol]] (KLM protocol) to probabilistically create controlled-Z connections between qubits which, when performed on a pair of <math>|+\rangle=|0\rangle+|1\rangle</math> states (normalization being ignored), forms the basis for cluster states. While the KLM protocol requires error correction and a fairly large number of modes in order to get very high probability two-qubit gate, Neilsen's protocol only requires a success probability per gate of greater than one half. Given that the success probability for a connection using <math>n</math> ancilla photons is <math>n^2/(n+1)^2</math>, relaxation of the success probability from nearly one to anything over one half presents a major advantage in resources, as well as simply reducing the number of required elements in the photonic circuit.
To see how Nielsen brought about this improvement, consider the photons being generated for qubits as vertices on a two dimensional grid, and the controlled-Z operations being probabilistically added edges between nearest neighbors. Using results from [[percolation theory]], it can be shown that as long as the probability of adding edges is above a certain threshold, there will exist a complete grid as a sub-graph with near unit probability. Because of this, Nielsen's protocol doesn't rely on every individual connection being successful, just enough of them that the connections between photons allow a grid.
===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 | lastlast1=Kok | firstfirst1=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| bibcode=2007RvMP...79..135K | s2cid=119335959 }}</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
===Browne-Rudolph protocol===
An alternative approach to building cluster states that focuses entirely on photon polarization is the Browne-Rudolph protocol.<ref>{{cite journal | lastlast1=Browne | firstfirst1=Daniel E. | last2=Rudolph | first2=Terry | title=Resource-Efficient Linear Optical Quantum Computation | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=95 | issue=1 | date=2005-06-27 | issn=0031-9007 | doi=10.1103/physrevlett.95.010501 | page=010501| pmid=16090595 |arxiv=quant-ph/0405157| bibcode=2005PhRvL..95a0501B | s2cid=27224760 }}</ref> This method rests on performing parity checks on a pair of photons to stitch together already entangled sets of photons, meaning that this protocol requires entangled photon sources. Browne and Rudolph proposed two ways of doing this, called type-I and type-II fusion.
====Type-I fusion====
===Polarization encoding===
Polarization entangled photon pairs have also been produced on-chip.<ref>{{cite journal | lastlast1=Matsuda | firstfirst1=Nobuyuki | last2=Le Jeannic | first2=Hanna | last3=Fukuda | first3=Hiroshi | last4=Tsuchizawa | first4=Tai | last5=Munro | first5=William John | last6=Shimizu | first6=Kaoru | last7=Yamada | first7=Koji | last8=Tokura | first8=Yasuhiro | last9=Takesue | first9=Hiroki |display-authors=5| title=A monolithically integrated polarization entangled photon pair source on a silicon chip | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=2 | issue=1 | date=2012-11-12 | issn=2045-2322 | doi=10.1038/srep00817|pmc=3495342 | page=817| pmid=23150781 | arxiv=1211.2885 | bibcode=2012NatSR...2E.817M |doi-access=free}}</ref> The setup involves a silicon wire waveguide that is split in half by a [[polarization rotator]]. This process, like the entanglement generation described for the dual rail encoding, makes use of the nonlinear process of spontaneous four-wave mixing, which can occur in the silicon wire on either side of the polarization rotator. However, the geometry of these wires are designed such that horizontal polarization is preferred in the conversion of laser pump photons to signal and idler photons. Thus when the photon pair is generated, both photons should have the same polarization, i.e.
::<math>|\psi\rangle=|H_s,H_i\rangle</math>.
| 