Optical cluster state: Difference between revisions

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===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, NeilsenNielsen'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.

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====Type-I fusion====

In type-I fusion, photons with either vertical or horizontal polarization are injected into modes <math>a</math> and <math>b</math>, connected by a polarizing beam splitter. Each of the photons sent into this system is part of a Bell pair that this method will try to entangle. Upon passing through the polarizing beam splitter, the two photons will go opposite ways if they have the same polarization or the same way if they have the sameopposite polarization, e.g.

::<math>|H_a,H_b\rangle\rightarrow|H_a,H_b\rangle</math>

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Once a cluster state has been successfully generated, computation can be done with the resource state directly by applying measurements to the qubits on the lattice. This is the model of [[one-way quantum computer|measurement-based quantum computation]] (MQC), and it is equivalent to the [[quantum circuit|circuit model]].

Logical operations in MQC come about from the byproduct operators that occur during [[quantum teleportation]]. For example, given a single qubit state <math>|\psi\rangle</math>, one can connect this qubit to a plus state (<math>|+\rangle=|0\rangle+|1\rangle</math>) via a two-qubit controlled-Z operation. Then, upon measuring the first qubit (the original <math>|\psi\rangle</math>) in the Pauli-X basis, the original state of the first qubit is teleported to the second qubit with a measurement outcome dependent extra rotation, which one can see from the partial inner product of the measurement acting on the two-qubit state:

::<math>

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===Polarization encoding===

Polarization entangled photon pairs have also been produced on-chip.<ref>{{cite journal | last1=Matsuda | first1=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 | 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...2E2..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>.