Optical cluster state: Difference between revisions

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'''Optical [[cluster state]]sstates''' are a proposed tool to achieve quantum computational universality in [[linear optical quantum computing]] (LOQC).<ref>{{cite journal | last1=Kok, | first1=Pieter | last2=Munro | first2=W. J. | last3=Nemoto|author3-link= Kae Nemoto | first3=Kae | last4=Ralph | first4=T. C. | last5=Dowling | first5=Jonathan P. et| allast6=Milburn | first6=G., [https://arxivJ.org/abs/quant-ph/0512071 "| title=Linear optical quantum computing with photonic qubits"], ''Rev.| Mod.journal=Reviews Phys.'',of March,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.

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Optical [[cluster state]]s are a proposed tool to achieve quantum computational universality in [[linear optical quantum computing]] (LOQC).<ref>Kok, P. et al., [https://arxiv.org/abs/quant-ph/0512071 "Linear optical computing with photonic qubits"], ''Rev. Mod. Phys.'', March, 2007.</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==

On a [[silicon photonic]] chip, one of the most common platforms for implementing LOQC, there are two typical choices for encoding [[quantum information]], though many more options exist.<ref>Rudolph, [https://arxiv.org/abs/1607.08535 "Why I am optimistic about the silicon-photonic route to quantum computing"], ''APL Photonics'', 2017.</ref> Photons have useful degrees of freedom in the spatial modes of the possible photon paths or in the [[photon polarization|polarization]] of the photons themselves. The way in which a [[cluster state]] is generated varies with which encoding has been chosen for implementation.

Storing information in the spatial modes of the photon paths is often referred to as dual rail encoding. In a simple case, one might consider the situation where a photon has two possible paths, a horizontal path with [[creation and annihilation operators|creation operator]] <math>a^\dagger</math> and a vertical path with creation operator <math>b^\dagger</math>, where the logical zero and one states are then represented by

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::<math>b^\dagger |0_a,0_b\rangle=|0_a,1_b\rangle=|1\rangle_L</math>.

Single qubit operations are then performed by [[beam splitter]]s, which allow manipulation of the relative superposition weights of the modes, and phase shifters, which allow manipulation of the relative phases of the two modes. This type of encoding lends itself to the Nielsen protocol for generating cluster states. In encoding with [[photon polarization]], logical zero and one can be encoded via the horizontal and vertical states of a photon, e.g.

::<math>|H\rangle=|0\rangle_L</math>

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::<math>|V\rangle=|1\rangle_L</math>.

Given this encoding, single qubit operations can be performed using [[waveplate]]s. This encoding can be used with the Browne-Rudolph protocol.

===Nielsen protocol===

In 2004, Nielsen proposed a protocol to create cluster states,<ref>{{cite journal | last=Nielsen, M.,| first=Michael [https://arxivA.org/pdf/quant-ph/0402005 "| title=Optical quantumQuantum computationComputation usingUsing clusterCluster states"],States ''Rev.| Mod.journal=Physical Phys.'',Review July,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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===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 | last1=Kok, | first1=Pieter | last2=Munro | first2=W. J. | last3=Nemoto | first3=Kae|author3-link= Kae Nemoto | last4=Ralph | first4=T. C. | last5=Dowling | first5=Jonathan P. et| allast6=Milburn | first6=G., [https://arxivJ.org/abs/quant-ph/0512071 "| title=Linear optical quantum computing with photonic qubits"], ''Rev.| Mod.journal=Reviews Phys.'',of March,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

::<math>|H,a\rangle\rightarrow\frac{1}{\sqrt{2}}(|H,b\rangle+|V,a\rangle)</math>

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===Browne-Rudolph protocol===

An alternative approach to building cluster states that focuses entirely on photon polarization is the Browne-Rudolph protocol.<ref>{{cite journal | last1=Browne, D| first1=Daniel E., | last2=Rudolph, T.,| [https://arxiv.org/abs/quant-ph/0405157first2=Terry | "title=Resource-efficientEfficient linearLinear opticalOptical quantumQuantum computation"],Computation ''Phys.| Rev.journal=Physical Lett.'',Review June,Letters | 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====

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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</math>.

for <math>m=0,1</math> denoting the measurement outcome as either the <math>+1</math> eigenstate of Pauli-X for <math>m=0</math> or the <math>-1</math> eigenstate for <math>m=1</math>. A two qubit state <math>|\phi\rangle</math> connected by a pair of controlled-Z operations to the state <math>CZ|+\rangle^{\otimes 2}</math> yeildsyields a two-qubit operation on the teleported <math>|\phi\rangle</math> state after measuring the original qubits:

::<math>(\langle+|Z^{m_1}\otimes\langle+|Z^{m_2}\otimes I)CZ_{1,3}CZ_{2,4}(|\phi\rangle\otimes CZ|+\rangle^{\otimes 2})=\frac{1}{2}CZ(HZ^{m_1}\otimes HZ^{m_2})|\phi\rangle</math>.

for measurement outcomes <math>m_1</math> and <math>m_2</math>. This basic concept extends to arbitrarily many qubits, and thus computation is performed by the byproduct operators of teleportation down a chain. Adjusting the desired single-qubit gates is simply a matter of adjusting the measurement basis on each qubit, and non-Pauli measurements are necessary for universal quantum computation.

==Experimental Implementations==

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===Polarization encoding===

Polarization entangled photon pairs have also been produced on-chip.<ref>{{cite journal | last1=Matsuda, N.| etfirst1=Nobuyuki al.,| [https://www.nature.com/articles/srep00817last2=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"], ''Nature'',| November,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...2..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>.

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The polarization rotator is then designed with the specific dimensions such that horizontal polarization is switched to vertical polarization. Thus any pairs of photons generated before the rotator exit the waveguide with vertical polarization and any pairs generated on the other end of the wire exit the waveguide still having horizontal polarization. Mathematically, the process is, up to overall normalization,

::<math>|\alpha\rangle\rightarrow|\alpha'\rangle+|H_s,H_i\rangle\rightarrow|\alpha'\rangle+|V_s,V_i\rangle\rightarrow|\alpha''\rangle+|H_s,H_i\rangle+e^{i\phi}|V_s,V_i\rangle</math>.

Assuming that equal space on each side of the rotator makes spontaneous four-wave mixing equally likely one each side, the output state of the photons is maximally entangled:

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[[Category:Quantum information science]]

[[Category:Quantum optics]]