Journal of Physics B: Atomic, Molecular and Optical Physics

TOPICAL REVIEW • OPEN ACCESS

Continuous-variable quantum computing in the quantum opticalfrequency combTo cite this article: Olivier Pfister 2020 J. Phys. B: At. Mol. Opt. Phys. 53 012001

View the article online for updates and enhancements.

This content was downloaded from IP address 199.111.221.198 on 25/11/2019 at 19:23

Topical Review

Continuous-variable quantum computing inthe quantum optical frequency comb

Olivier Pfister

Department of Physics, University of Virginia, 382 McCormick Road, Charlottesville, VA 22903, UnitedStates of America

E-mail: olivier.pfister@gmail.com

Received 16 September 2018, revised 25 August 2019Accepted for publication 29 October 2019Published 25 November 2019

AbstractThis topical review introduces the theoretical and experimental advances in continuous-variable(CV)—i.e. qumode-based in lieu of qubit-based—large-scale, fault-tolerant quantum computingand quantum simulation. An introduction to the physics and mathematics of multipartiteentangled CV cluster states is given, and their connection to experimental concepts is delineated.Paths toward fault tolerance are also presented. It is the hope of the author that this review attractmore contributors to the field and promote its extension to the promising technology ofintegrated quantum photonics.

Keywords: continuous-variable quantum information, qumode, quantum optics, continuous-variable entanglement, measurement-based quantum computing, one-way quantum computing,cluster state

1. Introduction

1.1. Quantum computing: revolutionary promise and dauntingchallenge

Quantum computing promises exponential speedup for spe-cific tasks [1], most notably integer factoring [2] and quantumsimulation [3]. Originally proposed by Feynman, quantumsimulation features an exponential speedup over classicalcomputing for the calculation of N quantum systems by sol-ving their Schrödinger equation, which entails diagonalizing a2N×2N Hamiltonian, i.e. an exponential scaling of thecomplexity of the problem. However, if one were able toexquisitely control, while correcting or staying clear ofdecoherence, N ‘model’ qubits in the laboratory and if onewere able to ‘dial’ at will interactions between them toimplement the precise Hamiltonian to be studied, then theproblem would become polynomial in the number of qubits,

as these naturally evolve following quantum laws and can beread out using physical measurements (which are included inthis polynomial scaling).

A scalable qubit implementation is therefore the crux ofthe power of Feynman’s quantum simulator but it is essentialto be clear on what scalability means. It is sometimes claimedthat simply increasing the dimension of Hilbert space, e.g.passing from 2-state qubits to d-state qudits, can be useful forquantum simulation. If the number N of logic units is notincreased in the process, this argument cannot constitute agenuine scalability claim. Indeed, while it is true that therespective Hilbert dimensions verify >d 2N N if d>2, thisconstitutes polynomial, not exponential, scaling when N isheld constant. As explained above, scaling the quantumcomputer size with N is what confers a quantum simulator itsexponential power, not increasing the Hilbert space of a fixednumber of qudits. Hence scalability should always be taken inthe sense of an increase of the number of logic units N,regardless of their individual Hilbert-space dimension.

Quantum simulation could open the door to scientificdiscoveries by solving currently intractable problems such asground state of spin arrays, the Bose–Hubbard model (pavingthe way to the discovery of room-temperature

Journal of Physics B: Atomic, Molecular and Optical Physics

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 (16pp) https://doi.org/10.1088/1361-6455/ab526f

Original content from this work may be used under the termsof the Creative Commons Attribution 3.0 licence. Any

further distribution of this work must maintain attribution to the author(s) andthe title of the work, journal citation and DOI.

0953-4075/20/012001+16$33.00 © 2019 IOP Publishing Ltd Printed in the UK1

superconductors), simulating quantum field theory [4, 5], andquantum chemistry, from the calculation of energy levels[6, 7] to optimizing chemical reactions of fundamental soci-etal importance, such as nitrogen fixation [8] and possiblydiscovering new ones (carbon sequestration). In addition,applications to machine learning have also been envi-saged [9, 10].

The realization of practical quantum computing, utilizing(i), a large enough number of qubits that are, (ii), resilient todecoherence, has been known to be a daunting challenge fromthe inception of the field [11]. Significant progress, however,has been made on the decoherence front, where cutting edgeimplementations using trapped-ion and superconductingqubits have now reached the levels of fidelity required for theimplementation of quantum error correction [12–14]. On thescalability front, quantum control involving arbitrary unitarieshas been demonstrated in the 16-dimensional ground hyper-fine manifold of cesium [15], 2D atomic arrays of 49 atomswere demonstrated [16], and quantum simulation has beenachieved over 51 Rydberg atoms [17] and 53 ions [18]. Veryrecently, a quantum circuit was sampled over 53 super-conducting qubits [19].

While some trapped-ion and superconducting qubits havereached impressive levels of fidelity for quantum gates, theyare not yet large-scale quantum systems [20]. It has thereforebeen judiciously proposed to assess the potential of Noisy,Intermediate-Size (50–100 qubit), Quantum (NISQ) compu-ters to achieve quantum advantage, i.e. beyond-classicalperformance [21]. However, the NISQ concept does notcapture the whole field.

1.2. Quantum computing over continuous variables: a premiumon scalability

Another line of experimental research has placed scalability atthe forefront, relying on the remarkable ability of opticalparametric oscillators (OPOs) to produce very large numbersof entangled quantum fields [22, 23]. Experimental resultsconfirmed this with thousands and a million of entangledquantum modes—a.k.a. qumodes [23–26]— respectively inthe frequency [27–30] and temporal [31, 32] domains. In thisreview paper, we focus on the frequency domain imple-mentation in the quantum optical frequency comb (QOFC) ofa single OPO, whose scalability mirrors that of its classicalcounterpart, the OFC of a mode- and carrier-envelope-phase-locked laser, which emits thousands to millions of classicallycoherent fields [33, 34].

The groundbreaking scalability of the QOFC (as well asof ‘temporal combs’ of pulsed quantum fields) comes with therequirement of adopting continuous-variable (CV) quantuminformation (QI), encoded over dense qumodes rather thanover discrete qubits or qudits. Continuous-variable quantumcomputing (CVQC) was proposed 20 years ago [35], offersexponential speedup over classical computing as discrete QCdoes [36], and does not suffer from any fundamental impos-sibility regarding fault tolerance and quantum error correc-tion [37, 38].

Although quite recent compared to qubit platforms,CVQC has found a natural implementation in the mature fieldof quantum optics, from the original demonstration of Ein-stein–Podolsky–Rosen (EPR) entanglement [39, 40] tounconditional quantum teleportation [41], quantum densecoding [42], quantum secret sharing [43], quantum key dis-tribution [44], as well as the CNOT-equivalent CSUM [45]quantum gate in 2008, 13 years after the first qubit-entanglinggates in cavity QED [46] and trapped ions [47].

The goal of this review paper is to introduce CVQI to alarger audience already familiar with qubit-based QI, with aneye in particular on the upcoming development of integratedquantum photonics [48, 49] and its extension to the CVdomain [50–54], which one might expect to enable scalableQI in the same way as integrated electronics enabled scalableclassical information. Integration begets scalability in thesense that it will be easier to implement on the order of 104

photodetectors on chip than in bulk optics. Moreover, guidedoptics offers much higher nonlinear coupling strengths due tothe confinement of the field. The main challenge will be tolower the loss level in integrated optics in order to reachdecoherence levels compatible with quantum error correction.

2. Continuous-variable quantum computing

2.1. Continuous-variable QI: quadrature eigenstates

CVQC was proposed in 1999 by Lloyd and Braunstein [35]and the exponential speedup of CVQC, the equivalent of theGottesman–Knill theorem for qubit-based QC [55], was for-mulated by Bartlett et al [36]. Comprehensive reviews ofCVQI were written by Braunstein and van Loock [56] and byWeedbrook et al [26].

In quantum optical implementations of CVQI, thequantum variables are the ‘amplitude’ and ‘phase’ quadratureoperators of the quantum electromagnetic field,

= +Q a a1

21( ) ( )†

= -Pi

a a1

2, 2( ) ( )†

which are mathematical analogs of the position andmomentum observables of a quantum harmonic oscillator ofannihilation operator a. The term ‘quadrature’ refers to thefree evolution of the position and momentum of the harmonicoscillator. A correspondence between qubit- and qumode-based QI exists and is presented in table 2.1 [36]. The startingpoint is the use of the continuous amplitude eigenbasis

ñ Îq q{∣ } of Q as the computational basis. All discrete sumsbecome integrals and the Hadamard transform over qubitscorresponds to the quantum Fourier transform over qumodes.The Pauli group for qubits corresponds to the Weyl–Hei-senberg group of all displacements in phase space, generatedby quadrature translation operators. Controlled gates andentanglement follow straightforwardly, and have beendemonstrated in the laboratory as standalone operations

2

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

[45, 57]. Bell bipartite entangled states correspond to EPRentangled states [39] (table 1) which are joint eigenstates ofthe commuting two-mode operators -Q Q1 2 and +P P1 2

[58] (note that swapping the plus and minus sign in theseoperators and states is also possible). That means the mea-surement noise, or quantum standard deviation, for theseoperators is zero:

D - =Q Q 0 31 2( ) ( )

D + =P P 0. 41 2( ) ( )

In CVQI lingo, these ‘EPR’ operators are also called var-iance-based entanglement witnesses [59] or nullifiers. As wewill later see, nullifiers also denote the operatorial logarithmsof stabilizers in CV quantum graph theory.

2.2. Realistic CVQI: squeezed states

The EPR states have infinite energy, as evidenced by theinfinite integrals and sum in table 1, and are thereforeunphysical. One can make, however, arbitrarily goodapproximations of EPR states in the laboratory. These states

called two-mode squeezed (TMS) states [40]

åñ = ñ ñ ñ=

¥

¥-r

rn n eTMS

tanh

cosh2 EPR 0,0 ,

5n

n

r

r

01 2 12∣ ∣ ∣ ⟶ ∣ ( )

( )

where r is called the squeezing parameter. Equation (5)indicates that an EPR state is an infinitely squeezed TMSstate. A TMS state can be created directly by an opticalparametric amplifier (OPA), e.g. a doubly resonant OPObelow threshold [40, 60]. The corresponding TMS Hamilto-nian is

k= -H i a a a a , 6TMS 1 2 1 2( ) ( )† †

where r=κτ if τ is the interaction time. Solving the Hei-senberg equations for this Hamiltonian yields

t t =Q Q Q Q e 7r1 2 1 2( ) ( ) ( ) ( )

t t = P P P P e 8r1 2 1 2( ) ( ) ( ) ( )

which shows clearly that ¥r means infinite energy in thetotal field. For initial vacuum states, one gets the finitelysqueezed standard deviations

Table 1. Correspondence between qubit- and qumode-based quantum computing [36].

Qubit-based Qumode-based

Computational basisñ ñ0 , 1{∣ ∣ } ñ Îq q{∣ }

dá ñ = Îk ℓ k ℓ, , 0, 1kℓ∣ { } dá ¢ ñ = - ¢ ¢ Î q q q q q q, ,∣ ( )y y yñ = ñ + ñ0 10 1∣ ∣ ∣ òy yñ = ñdq q q∣ ( )∣

Conjugate basis

Hadamard transformed Fourier transformedñ = ñ ñ0 11

2∣ (∣ ∣ ) òñ = ñ Î

pp e q dq p,ipq1

2∣ ∣

Single-qubit/qumode group generators

Pauli group Weyl–Heisenberg group of phase-space displacementsá ñX Z, x vá ñ º á ñx v

xx

vvÎ Î

-Î Î X Z e e, ,i P i Q{ ( )} { ( )} { } { }

ñ = Å ñ =X j j j1 , 0, 1∣ ∣ x xñ = + ñX q q( )∣ ∣ñ = ñ =pZ j e j j, 0, 1ij∣ ∣ v ñ = ñvZ q e qi q( )∣ ∣ñ = ñX∣ ∣ x ñ = ñx-X p e pi p( )∣ ∣ñ = ñZ∣ ∣ v vñ = + ñZ p p( )∣ ∣

Controlled, entangling gates

CSUM:ñ ñ = ñ Å ñC j k j k jX 1 2 1 2∣ ∣ ∣ ∣ añ ¢ñ = ñ ¢ñ = ñ ¢ + ña-C q q e q q q q qi Q P

X 1 2 1 2 1 21 2∣ ∣ ∣ ∣ ∣ ∣CPHASE:

ñ ñ = ñ ñpC j k e j ki jkZ 1 2 1 2∣ ∣ ∣ ∣ ñ ¢ñ = ñ ¢ñ = ñ ¢ña a ¢C q q e q q e q qi Q Q i qq

Z 1 2 1 2 1 21 2∣ ∣ ∣ ∣ ∣ ∣

Bipartite entanglement

Bell state (unnormalized) EPR state (unnormalizable)ñ = å ñ ñ=B j jj00 12 0

11 2∣ ∣ ∣ ò òñ = ñ ñ = ñ - ñq q dq p p dpEPR 0,0 12 1 2 1 2∣ ( ) ∣ ∣ ∣ ∣

= å ñ ñ=¥ n nn 0 1 2∣ ∣ (Schmidt decomp.)

Bellbasis EPR basisñ = ñ ÎB Z X B k ℓ, , 0, 1kℓ

k ℓ12 1 1 00 12∣ ∣ { } v x v x v xñ = ñ Î Z XEPR , EPR 0,0 , ,12 1 1 12∣ ( ) ( ) ( )∣ ( )

= å Å ñ ñp=

Åe j ℓ jjik j ℓ

01

1 2∣ ∣( ) ò x= + ñ ñv x+e q q dqi q1 2∣ ∣( )

3

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

D - = -Q Q e 9r1 2( ) ( )

D + = -P P e . 10r1 2( ) ( )

We will show in section 4 that such entanglement witnessescan be generalized to obtain specific signatures of multipartiteentanglement from specific multimode squeezing, in part-icular for cluster entangled states [61, 62], which are essentialto one-way quantum computing (section 3). We now addressthe universality and fault tolerance of CVQC, and the influ-ence of finite squeezing on the latter.

2.3. Analogy between qubit- and qumode-based quantumcomputing: Clifford and Gaussian gates, exponential speedup,and fault tolerance

One does not contain the universal quantum gate set for QC[63], which we need to translate any qubit-based quantumalgorithm into CV form. This requires, first, a discussion ofthe CV equivalent of Clifford gates. This discussion is rele-vant to both the exponential speedup and the fault toleranceof CVQC.

2.3.1. Exponential speedup: the Gottesman–Knill theorem andits CVQC version. The crucial feature of QC is itsexponential speedup over classical computing for specificproblems. Such a speedup is present in CVQC as well. Thesufficient condition for an exponential speedup is given by theGottesman–Knill theorem [64], which states that a qubit-based quantum algorithm can be run efficiently on a classicalcomputer if based only on Clifford gates. Clifford gates leavethe Pauli group globally invariant, by definition. This meansthat the transform ¢ of a Pauli operator by a Cliffordoperator ,

¢ = - , 111 ( )

is also a Pauli operator. Gottesman showed that Clifford-gatealgorithms over N qubits can be modeled classically in theHeisenberg picture by tracking the evolution of only Noperators [64], which is exponentially efficient since theHilbert space of quantum states is of dimension 2N. However,this approach fails when the quantum algorithm contains atleast one non-Clifford gate, which requires in principle thatone consider the full 2N dimensionality of the Hilbert spacefor the classical simulation of the quantum algorithm. In thatcase, Feynman’s argument for QC (that the needed resource isonly N qubits versus 2N classical bits) is in full force andyields an exponential speedup for QC. An example of a non-Clifford gate is the π/4 rotation around z that transformsPauli operator X into +X Y 2( ) , which is not a Paulioperator since the Pauli group is a multiplicative one. The π/4rotation is present in the quantum Fourier transform in Shor’salgorithm, for example [63]. The π/4 rotation gate can beimplemented by gate teleportation [55], if one has prepared aHadamard eigenstate or ‘magic state’ [65].

For CVQC, one must then seek the group of transforma-tions that leave the Weyl–Heisenberg group globally invariant(i.e. the normalizer of the Weyl–Heisenberg group inmathematical terms). As Bartlett et alestablished [36], the

group that normalizes displacements is that of all Gaussianoperations, i.e. the group of unitary evolution operatorscorresponding to Hamiltonians at most of quadratic order inthe quantum fields (i.e. in creation/annihilation operators orquadrature operators). Such evolution operators possessGaussian Wigner functions and, when operating on aquantum state of a Gaussian Wigner function, leave itsGaussian character unchanged, even though the Wignerfunction of the evolved quantum state may change throughdisplacement, rotation, squeezing or shearing.

Hence, non-Gaussian operations are crucial to ensure thatCVQC yields an exponential speedup. Such operationscorrespond to quantum evolution under a Hamiltonian that’sof cubic order (or higher) in the fields. The CV analog of a π/4 rotation therefore has to have the form of a cubic phase gate,e.g. gi Qexp 3( ). Cubic Hamiltonians were also proven to bethe minimum-order necessary resource to enable the genera-tion of Hamiltonians of arbitrary order in Lloyd andBraunstein’s first proposal of CVQC [35]. A resource theoryof non-Gaussian QI was recently formulated [66].

The question of feasible experimental implementation ofnon-Gaussian gates is, of course, essential. Rather thanattempting to directly realize high-order nonlinearities, whichis fraught with difficulties, other promising avenues are toexploit the nonpositivity of the Wigner function of Fock statesof nonzero photon number by using projection techniquessuch as photon-number-resolved (PNR) detection [67],photon subtraction [68, 69], or photon addition [70]. Gottes-man et al proposed the use of PNR detection as the sole non-Gaussian resource needed to generate Hadamard eigenstatesand to implement cubic phase gate [65].

In this context, the case of quantum simulation alsodeserves a word. When considering quantum Hamiltoniansone wishes to simulate, needless to say efficiently, theGottesman–Knill theorem also informs us on how difficult theclassical simulation can be: if the Hamiltonian is Gaussian,there will only be a linear overhead to solving the Heisenbergevolution equations; else, the classical calculation is likely tobenefit from quantum simulation in a major way since theoverhead will be, in principle, exponential. Of course, theabove consideration does not account for finding relevantapproximations for solving hard problems, which has been amajor part of physics for centuries.

Another open question is whether CV encodings can beof particular interest for the quantum simulation of continuoussystems, such as quantum fields. An interesting example isthe CV translation [71] of a proposal to quantum simulatequantum field theory [4].

2.3.2. Fault tolerance. To the best of current knowledge,non-Gaussian operations are critical to fault tolerance ofCVQC. It is a somewhat counterintuitive feature of CVQCthat, while the Clifford–Gaussian correspondence holds whenconsidering QC’s exponential speedup, it does not hold forother concepts such as Bell inequality violation [72],entanglement distillation [73], and quantum error correction[74]: all these operations can be implemented over qubits

4

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

using Clifford resources, but cannot be implemented overqumodes using solely Gaussian resources.

However, the use of non-Gaussian resources, such asFock states, PNR measurements, or field-cubic Hamiltonians[35] remedies the situation and removes all impossibilities.

Fault tolerance and quantum error correction deserve abit more detail. The effect of finite squeezing on faulttolerance is an important question, as one might be tempted toargue that the intrinsically ‘fuzzy’ wavefunctions ψ(q) used inCVQI will unavoidably lead to non-correctible errors, as isthe case for classical analog computing. The issue is, while itis deemed straightforward to distinguish the orthogonal ñ0∣and ñ1∣ states of a qubit, it is impossible to distinguish theorthogonal ñq∣ from d+ ñq q∣ if d ááD yq Q , even in a squeezedstate D µ -yQ rexp( ), D yQ being the standard deviation ofQ in state yñ∣ .

As an example, one can ask how many CV teleportation[41] steps can be concatenated before the teleportation fidelityreaches the classical limit of 50%1. The answer is

=n rexp 2max ( ) if r is the squeezing parameter of the TMSstates used as teleportation channels [75]. This meansnmax=2 for 3 dB of squeezing and 10 for 10 dB ofsqueezing. Other studies of uncorrected CVQC made clearthat quantum error correction is needed [76, 77] (but this isequally true of qubit-based platforms).

While it is not known whether it might be possible todirectly correct CV errors, a path to fault-tolerant CVQC doesexist: Gottesman et al proposed the discrete encoding of aqubit in an oscillator to address small CV drifts [65], and thisencoding was applied to CVQC by Menicucci to prove theexistence of a fault tolerance threshold [38]. Therefore,infinite squeezing is not a requirement for fault-tolerantCVQC. As we will mention later, the amount of squeezingrequired for fault tolerance is actually not unreasonable. TheGKP error encoding relies on the creation of GKP resourcestates that are comb-like in quadrature quantum phase space.Experimental realization of GKP states is an arduous endevor.Proposals have been made for the generation of optical GKPstates [78–81] and an experimental realization over thephononic vibration field of trapped ion was recentlyperformed [82].

Finally, interesting results have been obtained recently onimplementing QC over qubits using a GKP encoding: in thatsituation, Baragiola et al have shown that Gaussian operationsare enough, along with the GKP encoding, to achieveuniversal, fault-tolerant QC with no cubic-phase gate needed[83]. Also intriguing are recent investigations of subuniversalquantum computing à la boson sampling [84], such asGaussian boson sampling [85] and CV instantaneousquantum computing [86], which have both been proven tobe hard classically.

3. Measurement-based, one-way quantumcomputing

3.1. Introduction. Cluster states

An equivalent (but still universal) alternative to the circuitmodel of universal QC [63] is that of measurement-based QC[55] and, in particular, one-way QC, based on cluster entan-gled states [61]. One-way QC starts from a cluster state or‘quantum computing substrate,’ an entangled qubit latticewhich contains all the entanglement that can ever be neededby a quantum algorithm. Quantum computing can then pro-ceed solely by single-qubit measurements which inform feed-forward unitaries on the lattice neighbors [87].

From this description, it is clear that the concept ofcluster state is a central one. Indeed, while the cluster stateclearly must be a multipartite entangled state, it cannot be justany multipartite state. It is well known that multipartiteentanglement differs fundamentally from bipartite entangle-ment in that there exists distinct families of LOCC-equivalententangled states, where LOCC stands for local operations andclassical communication. For example, the W states, e.g.

ñ + ñ + ñ001 010 100∣ ∣ ∣ , and the Greenberger–Horne–Zei-linger (GHZ) states [88], e.g. ñ + ñ000 111∣ ∣ , are not LOCCequivalent [89]. Cluster states are not LOCC equivalent toeither GHZ or W states for 4 qubits and more.

A cluster state is canonically defined as qubits initializedin the +ñ∣ state and interacting via CZ gates in a 2D pattern,typically a square lattice (although other lattices are alsopossible) [61]. It is convenient to represent cluster states asgraphs, see figure 1. We will call such graphs ‘canonical’graphs throughout the paper—as opposed to the CV graphsthat we will define later. The effect of measurements on acluster can be understood easily as a multipartite general-ization of teleportation: in regular teleportation, a bipartiteentangled state (LOCC-equivalent to a cluster state) is thequantum resource and Alice’s choice of measurement basisdecides the quantum gate applied to the teleported state [90].In order to realize the universal QC gate set, the cluster statemust be a 2D lattice, such as a square lattice, so as to allow fortwo-qubit gates [87, 90, 91].

A simple illustration of measurement-based quantumprocessing is given by the teleportation gate sequence offigure 2.

The cluster state concept is at the heart of QC scalabilityas featured in this paper: if one is able to generate the wholecluster state in a scalable manner, then all QC needs to pro-ceed is single-qubit measurements and feedforward. This isthe CVQC approach this paper discusses in the QOFC. Wewill also mention implementations in the time domain whichare very similar is spirit.

Figure 1. Graphical representation of a cluster state: vertices denotequbits in the +ñ∣ state, edges denote CZ gates.

1 This is an especially meaningful question since the teleportation gate canbe used as a primitive for QC [55].

5

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

3.2. The CV cluster state

One-way CVQC has been formulated using CV cluster states[37, 92]. Using table 1, we can easily see how to generate CVcluster states from the qubit definition [62]: they are createdby applying CZ (or CPHASE) gates along a square lattice ofqumodes in phase-quadrature eigenstates = ñp 0∣ . For illus-tration, we can compare the literal expression of a two-qubitcluster state,

+ñ +ñ = ñ ñ+ ñ ñ + ñ ñ - ñ ñC 0 0

0 1 1 0 1 1 , 12Z 1 2 1 2

1 2 1 2 1 2

∣ ∣ ∣ ∣∣ ∣ ∣ ∣ ∣ ∣ ( )

to that of a two-qumode cluster state

p

= ñ ¢ = ñ

= ¢ ñ ¢ñ

a

a ¢

e p p

dq dq e q q

0 01

2. 13

i Q Q

i qq

1 2

1 2

1 2

∬

∣ ∣

∣ ∣ ( )

Again, these cluster states are infinitely squeezed, thusunphysical; in the laboratory, we can only create finitelyphase-squeezed states, which are created by single-modesqueezers, e.g. degenerate OPAs, and then apply the CPHASE,a.k.a. quantum nondemolition (in the backaction evadingsense [93–95]), or spring coupling, gates [57].

3.3. Bottom up scalability

This canonical method of building an N-qumode cluster statewould therefore require N degenerate OPAs for the initialstates and a couple of OPAs per entangling gate. While thismethod scales linearly with such experimental resources, it isnonetheless not the most efficient way to generate a CVcluster state. A first improvement consists in noticing that theN-mode cluster state is a Gaussian resource whose generationprotocol can be re-cast as a Bloch–Messiah decomposition[96] consisting in N single-mode squeezers ‘sandwiched’between two N-mode interferometers. When the input statesare vacuum ones, the first interferometer is irrelevant and anysuch Gaussian state can be created from N single-modesqueezers followed by one N-mode interferometer, whichgreatly simplifies the protocol by replacing all nonlinearoptical CPHASE gates with linear optical interferometers[97, 98]. One can, however, find even more compact methodsfor generating cluster states.

3.4. Top-down scalability: CV cluster state generation inthe QOFC

In the above protocols, each single-mode squeezer is adegenerate OPA that is essentially an OPO cavity with amultitude of resonant modes, all of them but one unused!Instead of scaling to N qumodes using N OPO’s, it is thenpossible (and experimentally more tractable) to use the wholeQOFC of a single OPO. The first idea to use N-modesqueezing to generate N-mode entanglement involved onlyGHZ states [22]. It was then extended to proposing cluster-state generation [99] and, finally, proposing the generation ofa square cluster lattice in a single OPO [23]. Experimentalrealizations followed in the QOFC [27, 28, 100] as well as inthe pulsed ‘temporal comb’ regime [31, 32]. We now detailthe mathematical formalism used to describe CV clusterstates, as it also informs the methods for their generation,before describing experimental implementations of large-scale CV cluster states in the QOFC.

4. Graph states

4.1. Canonical graph states, stabilizers, and nullifiers

4.1.1. Qubits. As we mentioned earlier, qumode clusterstates can be represented as canonical graphs whose verticesare phase-squeezed states and edges are CPHASE gates. Theseare directly deduced from the qubit formalism using thecorrespondence of table 2.1 [36]. An important feature of anygraph state yñ∣ (over qubits, qudits, or qumodes) is that it is astabilizer state, i.e. is uniquely defined by a group ofoperators S that leave yñ∣ invariant:

y y" Î ñ = ñS S, . 14∣ ∣ ( )

For qubit cluster states, the multiplicative stabilizer group isgenerated by all possible products of its generators, which areconstructed by taking the Pauli X operator on any vertex andthe Pauli Z operator on all its graph neighbors:

=Î

X Z , 15jk

k

j calSspansj

⨂ ( ){ }\

where j denotes the neighborhood (i.e. the set of all edge-connected qubits) of qubit j. Again for illustration purposes,

Figure 2. Quantum teleportation using a cluster-state. Step 1: the state to be teleported is fused to the simplest possible cluster state. Step 2: ameasurement is made in the ñ∣ basis, projecting the neighbor qubit. Step 3: the (random) measurement result m=±1 is used to feedforward onto the neighbor qubit, thereby deterministically placing it into the state to be teleported.

6

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

applying this to the simple cluster state of equation (12) yields

+ñ +ñ = ñ ñ + ñ - ñ+ ñ ñ - ñ - ñ

X Z C 1 0 1 1

0 0 0 1 161 2 Z 1 2 1 2 1 2

1 2 1 2

( ) ∣ ∣ ∣ ∣ ∣ ( ∣ )∣ ∣ ∣ ( ∣ ) ( )

= +ñ +ñC . 17Z 1 2∣ ∣ ( )

These generators of the stabilizer group thus provide aHeisenberg picture of a cluster state (akin to that used inGottesman’s treatment of Clifford quantum algorithms [64])and constitute an efficient prescription for what observables tomeasure in the laboratory in order to certify that a cluster statewas made. As mentioned above, we call these observablesvariance-based entanglement witnesses: if all generators ofequation (15) are measured with no quantum noise, usingmany copies of yñ∣ , then yñ∣ is an eigenstate of the generators,i.e. a cluster state whose graph can be reconstructed from theedge structure of the neighborhoods. We now translate this toCV [97, 101].

4.1.2. Qumodes. As per table 2.1 [36], we have thecorrespondence

x = x-X X e 18i P( ) ( )

v = vZ Z e . 19i Q( ) ( )

The stabilizer group generators, which were both unitary andHermitian for qubits, are now only unitary for qumodes andcan be expressed as

åx v xvx

= - -Î Î

X Z i P Qexp . 20j jk

k k j jk

k

jk

j j

⎪⎪

⎪⎪

⎧⎨⎩

⎡⎣⎢⎢

⎤⎦⎥⎥

⎫⎬⎭

( ) ⨂ ( ) ( )

Its action on example state of equation (13) gives

x v a

p

= ñ ¢ = ñ

= ¢ ñ ¢ñx v a- ¢

X Z p p

e e dq dq e q q

C 0 01

221i P i Q i qq

1 2 Z 1 2

1 21 2 ∬

[ ( ) ( )] ( )∣ ∣

∣ ∣ ( )

px= ¢ + ñ ¢ña v¢+ ¢dq dq e q q

1

222i qq i q

1 2∬ ∣ ∣ ( )

p= ¢ ñ ¢ña v ax¢ ¢ -dq dq e e q q

1

223i qq iq

1 2∬ ∣ ∣ ( )( )

which yields stabilization for ϖ/ξ=α. (Note that qumodestabilizer states, unlike qubit ones, can have weighted-edgegraphs.)

As before, we will be interested in the variance-basedentanglement witnesses. These will be given by the Hermitianoperators that were exponentiated to give the stabilizers of astates yñ∣ as per equation (20). For stabilizers to have aneigenvalue of 1, these Hermitian operators must have a zeroeigenvalue and will be called nullifiers for this reason.Nullifiers are quadrature operators and can be directlymeasured using standard quantum optics tools such asbalanced homodyne detection and RF networks (splitters/combiners and phase shifters). Defining the vectors

= ¼Q Q Q, , NT

1( )

and = ¼P P P, , NT

1( )

for N qumodes, wecan then write the vector equation

y y- ñ = ñP QV 0 . 24( )∣ ∣ ( )

where V is the mathematical adjacency matrix of the clustergraph, whose entries Vij are nonzero if and only if there exitsan edge between vertices i and j.

4.2. (amiltonian) graph states and their connection tocanonical graph states

A different type of graph, the (amiltonian) graph, can alsobe defined. It is highly relevant experimentally and is relatableto V. The idea of the graph stems from the proposal togenerate multipartite entanglement using multimode squeez-ing of Hamiltonian [22, 102]

åk= -<

H i G a a a a . 25i j

ij i j i j( ) ( )† †

It is easy to show that this Hamiltonian yields the system ofequations of motion

k=dQ

dtQG , 26( )

where G is the matrix of entries Gij and is the adjacencymatrix of the graph. Diagonalizing G provides thesqueezing parameters (eigenvalues) and the squeezed multi-mode observables (eigenvectors). We will always assume theinitial state is the vacuum. See figure 3. The question of therelation of the graph to the canonical graph or, equiva-lently, of G to V, is not an easy one but it has been resolvedfor infinite [99] and finite [101] squeezing. An interestingtheorem is [103]

= =-G G G V. 271 ( )

4.3. An example: the GHZ graph

The first proposal of compact generation of multipartiteentanglement in the QOFC [22] was for a GHZ state. TheHamiltonian in this case has all possible TMS terms, whichleads to the following G matrix

k=G

0 1 11

11 1 0

. 28

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟( )

This G matrix corresponds to the complete graph, an exampleof which is shown in figure 4, left. Note that G is not self-

Figure 3. The different quantum graph types. Note that the startingstates are vacuum states in the -graph case, as =N a aj j j

† .

7

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

inverse in this case. Solving the Heisenberg equations [22]yields the nullifiers

å=+- -

=

P e P 29N r

j

N

j1

1

( )( )

= - " ¹-Q e Q Q j k, . 30jkr

j k( ) ( )

Note the remarkable squeezing boost by the mode number inequation (29) [22]. In the limit of infinite squeezing, thesenullifiers are those of the following GHZ state [104]:

òñ = ñ ñ ñdq q q qGHZ ... . 31N1 2∣ ∣ ∣ ∣ ( )

To get the canonical graph state from this, one can justFourier transform all qumodes but one, say qumode 1, i.e. doQ Pj j and -P Qj j for j>1, which only requires a π/2optical phase shift of qumodes j>1 (or a −π/2 shift ofqumode 1 alone) in the laboratory [105] and yields the nul-lifiers

å¢ = -+- -

>

P e P Q 32N r

j

N

j1

11

⎛⎝⎜⎜

⎞⎠⎟⎟ ( )( )

= - " ¹-Q e P Q j k, . 33jr

j1 1( ) ( )

These nullifiers can easily be seen to correspond to thecanonical graph of a GHZ state figure 4, right [106]. Thedifference in connectivity, or valence, of a QC cluster and aGHZ graph is significant: the cluster graph possesses a localstructure, i.e. one can define a set of nearest-neighbors, orneighborhood, for each qubit. In contrast, the GHZ state isnonlocal, its only neighborhood is the whole graph, and GHZstates have actually been shown to be ‘too entangled’ for one-way QC [107–109].

Note that this graph is actually equivalent, under localunitaries, to a complete canonical graph [106], which makesthe graph and the canonical graphs identical even though

this is not mandated by the theorem of equation (27) in thiscase because G is not self-inverse.

4.4. Finite squeezing

A natural question then arises of the general relationship—ifany—between matrices G and V. This question has beenanswered in several chronological steps. First, it was shownthat multimode squeezing always produces a cluster state,which yielded a general (though not bijective) relationbetween G and V [99]. This relation then yielded G=V inthe notable particular case G=G−1 [103].

Finally, Menicucci et al generalized, in a foundationalpaper, the CV cluster state formalism to finite squeezing byusing the symplectic formalism and complex adjacencymatrices = + iUZ V , where V is the canonicalgraph adjacency matrix and U contains finite squeezingeffects. This yields

y y- ñ = ñP QZ 0 , 34G G( )∣ ∣ ( )

where y ñG∣ is a finitely squeezed cluster state. Moreover, itwas shown that

= kt-ieZ , 35G2 ( )

where τ is the interaction time of the Hamiltonian ofequation (25). Taking a self-inverse graph =G2 yields

kt kt= -iZ Gcosh 2 sinh 2 , 36[ ( ) ( ) ] ( )

which can be proven, for a bicolorable graph, to be equivalentto the Z graph

kt kt¢ = + = ¢ + ¢i iZ G V Utanh 2 sech 2 , 37( ) ( ) ( )

thereby confirming the equivalence of ¢V and G for self-inverse matrices—a sufficient but not necessary condition asthe GHZ example showed.

Figure 4. Left, the graph of a GHZ state equation (28) for N=18. Right, the corresponding canonical graph obtained from the nullifiers ofequations (32) and (33).

8

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

4.5. Fault tolerance, high-dimensional lattices, and topologicalqumodes

Raussendorf formulated fault tolerance for qubit-based one-way quantum computing and showed that using topologicalerror encoding over cluster states yields a remarkable faulttolerance threshold at the 1.4% error probability level fordepolarizing errors, using three-dimensional lattices [110].

As it turns out, the generation of CV cluster state latticesof higher dimension, or valence, is relatively straightforwardover the QOFC: n-hypercubic-lattice CV cluster states can begenerated by the interference of n identical OPOs, using afractal construction method [30]. However, it is not clear howthe expected needed non-Gaussian nature of the quantumerror correcting resource will be factored in, in this case, sincen-hypercubic-lattice CV cluster states are still Gaussianresources.

Topological properties of qumode states have also beenexplored theoretically with a proposal to measure entangle-ment entropy of topological structures such as the toriccode [111].

As we mentioned earlier, a fault tolerant CV cluster-statearchitecture was also proposed by Menicucci using the GKPencoding [38]. This work determined, for the first time, thesqueezing required to build the GKP resource states for givenfault tolerant thresholds. The fact that threshold values forsqueezing exist at all was actually the main discovery of [38]:the existence proof of a CVQC fault tolerance threshold. Thecorresponding squeezing values for different error ratethresholds (corresponding to different encodings [112]) aregiven in table 2. It is worthwhile at this stage to point out thatthe current record level of optical squeezing is 15 dB (for asingle mode) [113].

This result has inspired more theoretical work to nowoptimize this threshold to lower values. This result wasrecently improved by showing that excess technical noise inexcess of the reciprocal of the squeezing level—which is asignature of impurity of the squeezed state—does not affectthe QC outcome [112]. Other recent work has shown thatfault-tolerant CVQC could be reachable on the order of 10 dBsqueezing, using different architectures [114, 115]. Anotheravenue deserving of theoretical work is the possibility of non-Gaussian error correcting resources other than GKP states,such as Fock states, which could benefit from the coming ofage of PNR detection [67]. Note that non-Gaussian binomial,a.k.a. ‘kitten-state,’ error encoding has also been done in thecontext of superconducting qubits [116].

The takeaway here is that there are no fundamental limitsto fault-tolerant one-way CVQC, even if a great deal oftheoretical and experimental work remains to be done.

5. Experimental realizations of CV clusterentanglement

The first experimental generation of photonic CV clusterstates used ‘bottom up’ quantum-circuit like approaches [97],based on the Bloch–Messiah decomposition [96] whichyielded four-mode [98, 117] and eight-mode [118] clusterstates, using several OPOs and linear optical transformations.In this approach, the number of OPOs is proportional to thenumber of entangled modes.

An alternative, ‘top down’ approach was proposed, firstin the frequency domain [22, 99], then in the time domain[101, 119]. Such approaches only require one or two OPOs togenerate TMS states, over the OPO’s QOFC, or in pulsedmode trains. Their scalability is therefore very promising.

5.1. Toroidal square lattice proposal

The initial proposal for generating a square-lattice clusterstate in a single OPO is depicted in figure 5 [23, 120]. Thiswork overcame a no-go theorem for creating linear-chain andsquare-lattice cluster states in the QOFC [120]. The solution,as always with impossibility proofs, was to think outside ofthe box and expand the context of said proof by addinganother degree of freedom, polarization, to the frequencylabel of qumodes. Conceptually, this allowed the replacementof the regular G matrix over QOFC qumodes with a moregeneral matrix whose entries are 2×2 polarization blocks.Such a general matrix is not subject to the aforementioned no-go theorem and can be used to build universal CV clusterstates.

The proposal to implement the needed polarization-blockG called for a doubly resonant OPO containing a speciallyengineered periodically poled KTiOPO4 (KTP) crystal, pha-sematching the 3 different pump/signal/signal polarizationsets ZZZ, ZYY, and YZY/YYZ, all with equal couplingstrengths. This crystal was successfully designed anddemonstrated experimentally [121].

A slightly inconvenient aspect of this proposal was thefairly complicated 15-mode pump field with orthogonal ±45°polarization components and nontrivial frequency spacings(figure 6) that would require sophisticated phase modulationtechniques to produce, e.g. single-sideband modulators [122]at multiple frequencies. Although possible, the impracticalityof this method encouraged the exploration of other avenuesfor top-down generation of cluster states, starting with smallersized cluster graphs. This architecture remains, however,remarkably compact and might still be implementable in thefuture.

Table 2. Upper-bound [38] squeezing thresholds corresponding tofault tolerance thresholds.

Desired error rate threshold (encoding-dependent) 10−2 10−4 10−6

MAXIMUM required squeezing thresh-old (dB)

15.6 18.7 20.5

9

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

5.2. Many squares

In 2011, an experiment successfully implemented a 2008proposal [103] for creating multiple 2×2 cluster states,figure 7. Although the cluster state size was small, there was

still a novel element of scalability to this work, in the numberof copies of the state: 15 copies of the 4-qumode square stateswere generated simultaneously in the QOFC and verified[27]. This was the first demonstration of CV cluster stategeneration over a large scale [123]. The OPO comprised two

Figure 5. The square lattice cluster state proposed in [23]. Left, the resulting canonical CV cluster graph, identical to the graph (self-inverse G). Right inset, architecture detail: each white vertex is a set of 4 the black vertices which represent individual TEM00 cavityqumodes, labeled by 2 frequencies and 2 orthogonal polarizations. The blue and yellow edges denote the ZYY, ZZZ, and YZY/YYZnonlinear interactions. Reprinted figure with permission from [23], Copyright 2008 by the American Physical Society.

Figure 6. Pump spectrum (scaled by 1/2) of the OPO generating the toroidal cluster state of figure 5. Parameters s, t are integer multiples ofthe free spectral range of the OPO cavity, and±(blue,red) denote ±45° polarizations.

Figure 7. Bottom, the quantum OFC of a single OPO (horizontal axis is frequency, line pairs denote orthogonally polarized modes that arefrequency degenerate). Top, canonical CV graph states generated in [27]. Reprinted figure with permission from [27], Copyright 2011 by theAmerican Physical Society.

10

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

KTP crystals: one PPKTP crystal phasematched the ZZZ andYZY/YZ interactions simultaneously and coupled 2 fre-quencies and 2 polarizations with a single pump frequency,placing them into ring cluster states; the other crystal wasidentical to the PPKTP one but unpoled and rotated 90° withrespect to it, this to ensure the crucial requirement of polar-ization degeneracy of cavity modes at the same frequency.

5.3. Dual-rail quantum wire

Scalability of the size of the cluster state was finally achieved—remarkably, while keeping the scalability feature of numberof copies—by adapting in the frequency domain an initialproposal of Menicucci et al for sequential CVQC using time-defined qumodes [119, 124]. The crux of the idea is to startwith TMS states, which we will loosely call EPR pairs fromnow on, as the primary building blocks and to ‘knit up’ acluster state chain, or ‘quantum wire,’ by entangling qumodesfrom different pairs. This is described in figure 8 for theoriginally proposed temporal approach and in figure 9 for thespectral approach. In the time-domain, spatially separatedEPR pairs are created by interfering two single-modesqueezed states in quadrature to create EPR pairs. Then onequmode of one EPR pair goes through a delay line beforeinterfering with one qumode from the next EPR pair at abalanced beam splitter, resulting in a dual-rail quantum wirestructure. This was realized experimentally by Akira Fur-usawa’s group at the University of Tokyo, reaching initialwire lengths of 104 qumodes [31] and later one millionqumodes [32], accessible sequentially, 2 at a time, seefigure 8. Note this sequential aspect is compatible with QCand has been dubbed the ‘Wallace and Gromit approach,’ asexplained in [119].

In the frequency domain version of the original scheme,figure 9(a), all entangled qumodes are generated simulta-neously. Rather than straddling two distinct spatial paths andmany temporal bins, as in figure 8, the EPR pairs straddle twoorthogonal polarization states and many frequencies: they arecreated in two sets, at two orthogonal linear polarizations, andthe pairs at one polarization are shifted with respect to the pairat another polarization by frequency shifting the independentpump fields that create each pair set, figure 9(a). Note that,unlike the delay-line shift in the temporal approach, this fre-quency shift is a lossless operation. All EPR pairs are emittedin the same cavity mode and are subjected to balanced beamsplitting by undergoing a 45° polarization rotation in a half-wave plate, thereby generating a (slightly different, see edge

colors, which denote weight signs) dual-rail quantum wire,figure 9(b), before impinging on a polarizing beam splitter.

Figure 10 depicts the whole experiment. Three ultrastableCW Nd:YAG lasers (1 kHz emission linewidth) were used toprovide tunable pump fields, as well as local oscillator (LO)fields for squeezing detection. They also served as frequencyreferences for the two optical cavities in the setup, whoseresonance frequencies were locked to them by the Pound–Drever–Hall method [125]. All 3 lasers were also phaselocked to one another in order to ensure their required preciserelative frequency relationships. The filter cavity for the LOfield had exactly the same free spectral range (FSR) as theOPO cavity in order to select LO fields corresponding to only2 OPO QOFC frequencies, at which the balanced homodynedetection setups, one for each polarization, provided a two-tone quantum noise signal. An RF network then reconstructedthe nullifier noise, which was found to be squeezed by 3.2(2) dB across the measurement range of 60 modes. Thismeasurement range was determined—and limited—by themaximum bandwidth of 15 GHz of the electro-optic mod-ulator used to create the LO fields, which corresponded to 30QOFC modes (spaced by 0.95 GHz) at each polarization. Thismeasurement range did not span the whole generation range,which we believed to be at least 3.2 THz from a measurement

Figure 8. Temporal quantum wire generation, from [31]. See text. Reprinted by permission from Springer Nature Customer Service CentreGmbH: Springer Nature, Nature Photonics [31] 2013.

Figure 9. Spectral quantum wire generation, from [28]. (a) the initial graph in the QOFC. The arrows mark the pumps’ half frequencies.(b) The reordered frequencies make the chain structure appear. Thegrayed ovals represent balanced beamsplitter interactions. At thebottom is the measured 60-qumode CV cluster state. Reprinted figurewith permission from [28], Copyright 2014 by the American PhysicalSociety.

11

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

of the phasematching bandwidth of the OPO PPKTP crystal,as shown in figure 11 [29]. This measurement used the sum-frequency generation (SFG) of two stable diode lasers, tun-able from 1050 to 1080 nm, and tuned symmetrically inopposite directions from 1064 nm so as to give a constant532 nm SFG wavelength, corresponding to that of the OPOpump. In fact, while this measurement showed that the non-linear interaction inside the OPO, i.e. the squeezing, hasconstant strength over that 3.2 THz range, corresponding to6700 OPO qumodes, it still did not capture the whole pha-sematching range as one of the diode lasers ran out of tuningrange. The theoretical expectation is closer to 4–5 THz [29],i.e. on the order of 104 modes. It is worth mentioning that thisPPKTP ZZZ quasiphasematching bandwidth should increaseeven more at longer wavelength, to the order of 10 THz forthe 775 nm/1550 nm interaction.

Another limitation will come into play before one runsout of phasematching bandwidth: because of the dispersion ofthe OPO crystals, the QOFC’s FSR will become chirped andqumodes far way from the pump’s half-frequency will shiftout of OPO resonance. However, this can be remedied byusing a slightly spectrally broadened pump field [29].

5.4. Dual-rail quantum wires in the QOFC

As was mentioned above, the scalability feature of thisscheme does not solely apply to the state size, i.e. the numberof qumodes per state, it also applies to the number of copiesof the state. To see this in the temporal scheme of figure 8, allone has to do is to consider a temporal delay that is an integermultiple of the mode spacing. Equivalently, in the spectralscheme of figure 9, one needs detune the pump half-fre-quencies by an integer multiple of the OPO FSR. This is

depicted in figure 12, which presents three equivalent ver-sions of the same graph. The experimental demonstrationof this method yielded two independent quantum wires of 30(measured) qumodes each generated by a single OPO [28].

An advantage of the spectral implementation is that thelarge delays required for scaling to large number of wirescorrespond to large pump detuning, which can be imple-mented losslessly, in contrast to the temporal implementationif one uses a fiber-based delay line. However, and to be fair,the spectral implementation is ultimately limited by the pha-sematching bandwidth, i.e. on the order of 104 modes in ourcase, whereas the temporal implementation is only limited bythe characteristic stability time of the experiment, whichsuffers no fundamental limit, being a purely technical issue.

5.5. Square and hypercubic cluster states in the QOFC

The temporal CVQC scheme uses two commensurate delaysto ‘knit up’ the square lattice cluster state required for uni-versal QC. Experimental implementations of this proposal inthe temporal domains were announced very recently andconstitute exciting progress, even though scalability waslimited by the losses in the temporal delays [126, 127].

In the spectral domain, the generation of an N×Nsquare lattice was proposed by interfering two QOFCs, onehosting a single wire (half pump detuning of 1 FSR) with onehosting N independent wires (half pump detuning of N FSRs)[30]. While this was initially the transposition of the originaltemporal idea [124], it was discovered in the process that onecan expand it to access yet another type of scalability: that ofthe valence of the cluster graph, from 1D to 2D to hypercubic4D and above, simply by using 1 QOFC per dimension andgeneralized interferometers in a fractal procedure [30]. Again,

Figure 10. Experimental setup for spectral quantum wire generation. Four servo loops are required to phaselock all CW Nd:YAG laserstogether and stabilize the OPO and filter cavities.

12

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

this approach will not suffer from losses in the QOFC shiftsbut will be capped by the spectral bandwidth of theinteraction.

5.6. Hybrid temporal and spectral approach

In an effort to obtain the best out of both worlds, a hybridapproach was proposed in which spectral quantum wiresundergo temporal delays and beam splitting to yield squarelattice cluster states defined both in the frequency and timedomains [128]. The protocol is depicted in figure 13.

It combines the quantum wire generation of [28] with thetemporal entanglement of [124]. This creates a square latticeof the ‘temporal’ balanced beamsplitter applies to every otherfrequency mode. Sorting ‘even’ from ‘odd’ frequencies in thequantum domain can actually be achieved fairly easily using a

properly unbalanced Mach–Zehnder interferometer [129].The musical score approximation holds in this case, i.e. thetemporal evolution only takes place over time scales muchlonger than the reciprocal of the linewidth of a qumode. Adetailed study of the CVQC protocol in this case showed thatsuch states are, indeed, QC-universal [128].

5.7. Other implementations of entanglement in the QOFC

Other degrees of freedom such as transverse spatial modeswere used to generate cluster states by Ping Koy Lam’s groupat the Australian National University [130]. In the single OPOQOFC, the group of Nicolas Treps at Sorbonne Universitédemonstrated an elegant approach to the generation of QOFCentanglement, by using synchronous pumping, i.e. a mode-locked OFC pump field whose repetition rate is equal to that

Figure 11. Top, Theoretical phasematching of the SFG process w w w wµ -I I IZ p Z Z p( ) ( ) ( ), versus signal frequency w and crystaltemperature. The crossing ridges are due to the more narrowly quasiphasematched SHG interactions. Bottom, Experimental phasematchingdata. The laser wavelength was scanned from 1058 to 1070 nm, the temperature of the crystal was scanned from 15 °C to 40 °C (11 differenttemperatures). About 30 data points of different wavelengths were measured at each temperature. The 3D plot was obtained by interpolation(Mathematica) of the data points. The measured SFG bandwidth is 3.178(2) THz, at quasi-constant efficiency, around 23 °C.

13

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

of the OPO. Using a much broader emission range over whichon the order of 10 individually addressable qumodes weredefined, they demonstrated qumode-resolved multipartiteentanglement [100] and discovered counterintuitive propertiesof the propagation along a graph of non-Gaussian featuresinduced by photon addition and subtraction [131].

6. Conclusion

It has been the goal of this review to survey the work done sofar on CVQI in the QOFC, as well as with temporally andspatially defined qumodes. What’s next? As bulk-optics-based approaches continue to explore fundamental concepts,

we also look forward for CVQI to translate to integratedplatforms and for quantum photonics on chip to become areality. Much like integrated electronics has been the future ofelectronic technology, we want to bet on quantum photonicsto take this to the next level of scalability and device inte-gration. Quantum technology does not yet exist at the level ofreal-life applications as the challenges, especially that ofdecoherence, are daunting but it is a worthwhile goal, one thatwe hope to share with a growing number of researchers.

ORCID iDs

Olivier Pfister https://orcid.org/0000-0003-3386-9661

Figure 12. Better living through -graph automorphism: scaling of the number of quantum wires by detuning pump fields. The number ofquantum wires generated is 3 in this case, each wire stemming from interactions of one given color (after one additional beam splitter). Thered arrows denote the half frequencies of the two pump fields, separated by 3 FSR.

Figure 13. Hybrid spectro-temporal square grid CV cluster state proposal, from [128]. See text. The box on the upper right represents thespectral quantum wire experiment of [28]. Reprinted figure with permission from [128], Copyright 2016 by the American Physical Society.Reprinted figure with permission from [28], Copyright 2014 by the American Physical Society.

14

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

References

[1] Montanaro A 2016 NPJ Quantum Inf. 2 15023[2] Shor P W 1994 Algorithms for quantum computation:

discrete logarithms and factoring Proc., 35th AnnualSymposium on Foundations of Computer Science (Santa Fe,NM) ed S Goldwasser (Los Alamitos, CA: IEEE ComputerSociety Press) pp 124–34

[3] Feynman R P 1982 Int. J. Theor. Phys. 21 467–88[4] Jordan S P, Lee K S M and Preskill J 2012 Science 336

1130–3[5] Marshall K, Pooser R, Siopsis G and Weedbrook C 2015

Phys. Rev. A 91 032321[6] Alán A-G, Dutoi A D, Love P J and Head-Gordon M 2005

Science 309 1704[7] Huh J, Guerreschi G G, Peropadre B, McClean J R and

Aspuru-Guzik A 2015 Nat. Photon. 9 615[8] Reiher M, Wiebe N, Svore K M, Wecker D and Troyer M

2017 Proc. Natl Acad. Sci. 114 7555–60[9] Rebentrost P, Mohseni M and Lloyd S 2014 Phys. Rev. Lett.

113 130503[10] Carleo G and Troyer M 2017 Science 355 602–6[11] DiVincenzo D 2001 The physical implementation of quantum

computation Scalable Quantum Computers. Paving the wayto Realization (New York: Wiley) p 1

[12] Barends R et al 2014 Nature 508 500[13] Ofek N et al 2016 Nature 536 441[14] Gaebler J P et al 2016 Phys. Rev. Lett. 117 060505[15] Anderson B E, Sosa-Martinez H, Riofrío C A,

Deutsch I H and Jessen P S 2015 Phys. Rev. Lett. 114240401

[16] Saffman M 2016 J. Phys. B: At. Mol. Opt. Phys. 49202001

[17] Bernien H et al 2017 Nature 551 579[18] Zhang J, Pagano G, Hess P W, Kyprianidis A, Becker P,

Kaplan H, Gorshkov A V, Gong Z X and Monroe C 2017Nature 551 601

[19] Arute F et al 2019 Nature 574 505–10[20] Friis N et al 2018 Phys. Rev. X 8 021012[21] Preskill J 2018 Quantum 2 79[22] Pfister O, Feng S, Jennings G, Pooser R and Xie D 2004

Phys. Rev. A 70 020302[23] Menicucci N C, Flammia S T and Pfister O 2008 Phys. Rev.

Lett. 101 130501[24] Gottesman D 2008 Private communication[25] Furusawa A and van Loock P 2011 Quantum Teleportation

and Entanglement: A Hybrid Approach to Optical QuantumInformation Processing (New York: Wiley)

[26] Weedbrook C, Pirandola S, García-Patrón R, Cerf N J,Ralph T C, Shapiro J H and Lloyd S 2012 Rev. Mod. Phys.84 621–69

[27] Pysher M, Miwa Y, Shahrokhshahi R, Bloomer R andPfister O 2011 Phys. Rev. Lett. 107 030505

[28] Chen M, Menicucci N C and Pfister O 2014 Phys. Rev. Lett.112 120505

[29] Wang P, Fan W and Pfister O 2014 arXiv:1403.6631[physics.optics]

[30] Wang P, Chen M, Menicucci N C and Pfister O 2014 Phys.Rev. A 90 032325

[31] Yokoyama S, Ukai R, Armstrong S C, Sornphiphatphong C,Kaji T, Suzuki S, Yoshikawa J, Yonezawa H,Menicucci N C and Furusawa A 2013 Nat. Photon. 7 982

[32] Yoshikawa J I, Yokoyama S, Kaji T, Sornphiphatphong C,Shiozawa Y, Makino K and Furusawa A 2016 APLPhotonics 1 060801

[33] Hall J L 2006 Rev. Mod. Phys. 78 1279[34] Hänsch T W 2006 Rev. Mod. Phys. 78 1297[35] Lloyd S and Braunstein S L 1999 Phys. Rev. Lett. 82 1784

[36] Bartlett S D, Sanders B C, Braunstein S L and Nemoto K2002 Phys. Rev. Lett. 88 097904

[37] Menicucci N C, van Loock P, Gu M, Weedbrook C,Ralph T C and Nielsen M A 2006 Phys. Rev. Lett. 97 110501

[38] Menicucci N C 2014 Phys. Rev. Lett. 112 120504[39] Einstein A, Podolsky B and Rosen N 1935 Phys. Rev. 47 777[40] Ou Z Y, Pereira S F, Kimble H J and Peng K C 1992 Phys.

Rev. Lett. 68 3663[41] Furusawa A, Sørensen J L, Braunstein S L, Fuchs C A,

Kimble H J and Polzik E S 1998 Science 282 706[42] Li X, Pan Q, Jing J, Zhang J, Xie C and Peng K 2002 Phys.

Rev. Lett. 88 047904[43] Lau H K and Weedbrook C 2013 Phys. Rev. A 88 042313[44] Grosshans F, Van Assche G, Wenger J, Brouri R,

Cerf N J and Grangier P 2003 Nature 421 238[45] Yoshikawa J, Miwa Y, Huck A, Andersen U L,

van Loock P and Furusawa A 2008 Phys. Rev. Lett. 101250501

[46] Turchette Q A, Hood C J, Lange W, Mabuchi H andKimble H J 1995 Phys. Rev. Lett. 75 4710–3

[47] Monroe C, Meekhof D M, King B E, Itano W M andWineland D J 1995 Phys. Rev. Lett. 75 4714–7

[48] Tanzilli S, Martin A, Kaiser F, De Micheli M, Alibart O andOstrowsky D 2012 Laser Photon. Rev. 6 115–43

[49] Kues M, Reimer C, Lukens J M, Munro W J, Weiner A M,Moss D J and Morandotti R 2019 Nat. Photon. 13 170–9

[50] Fürst J U, Strekalov D V, Elser D, Aiello A, Andersen U L,Marquardt C and Leuchs G 2011 Phys. Rev. Lett. 106113901

[51] Dutt A, Luke K, Manipatruni S, Gaeta A L,Nussenzveig P and Lipson M 2015 Phys. Rev. Appl. 3044005

[52] Lenzini F et al 2018 Sci. Adv. 4 eaat9331[53] Mondain F, Lunghi T, Zavatta A, Gouzien E, Doutre F,

Micheli M D, Tanzilli S and D’Auria V 2019 Photon. Res. 7A36–9

[54] Vaidya V D et al 2019 Broadband quadrature-squeezedvacuum and nonclassical photon number correlations from ananophotonic device arXiv:1904.07833

[55] Gottesman D and Chuang I L 1999 Nature 402 390[56] Braunstein S L and van Loock P 2005 Rev. Mod. Phys.

77 513[57] Miwa Y, Yoshikawa J-I, van Loock P and Furusawa A 2009

Physical Review A 80 050303[58] Bohr N 1935 Phys. Rev. 48 696[59] Hyllus P and Eisert J 2006 New J. Phys. 8 51[60] Walls D F and Milburn G J 1994 Quantum Optics (Berlin:

Springer)[61] Briegel H J and Raussendorf R 2001 Phys. Rev. Lett. 86 910[62] Zhang J and Braunstein S L 2006 Phys. Rev. A 73 032318[63] Nielsen M A and Chuang I L 2000 Quantum Computation

and Quantum Information (Cambridge: CambridgeUniversity Press)

[64] Gottesman D 1999 The Heisenberg representation of quantumcomputers Proc. 22nd Int. Colloquium on GroupTheoretical Methods in Physics ed S P C et al (Cambridge,MA: International Press) p 32 arXiv:quant-ph/9807006v1

[65] Gottesman D, Kitaev A and Preskill J 2001 Phys. Rev. A 64012310

[66] Zhuang Q, Shor P W and Shapiro J H 2018 Phys. Rev. A 97052317

[67] Lita A E, Miller A J and Nam S W 2008 Opt. Express 163032–40

[68] Wenger J, Tualle-Brouri R and Grangier P 2004 Phys. Rev.Lett. 92 153601

[69] Ourjoumtsev A, Tualle-Brouri R, Laurat J and Grangier P2006 Science 312 83–6

[70] Zavatta A, Viciani S and Bellini M 2004 Science 306 660

15

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review

[71] Marshall K, Pooser R, Siopsis G and Weedbrook C 2015Phys. Rev. A 92 063825

[72] Bell J S 1987 EPR correlations and EPW distributionsSpeakable and Unspeakable in Quantum Mechanics(Cambridge: Cambridge University Press) ch 21 pp 196–200

[73] Eisert J, Scheel S and Plenio M B 2002 Phys. Rev. Lett. 89137903

[74] Niset J, Fiurášek J and Cerf N J 2009 Phys. Rev. Lett. 102120501

[75] Yonezawa H, Furusawa A and van Loock P 2007 Phys. Rev.A 76 032305

[76] Ohliger M, Kieling K and Eisert J 2010 Phys. Rev. A 82042336

[77] Ohliger M and Eisert J 2012 Phys. Rev. A 85 062318[78] Vasconcelos H M, Sanz L and Glancy S 2010 Opt. Lett. 35

3261–3[79] Motes K R, Baragiola B Q, Gilchrist A and Menicucci N C

2017 Phys. Rev. A 95 053819[80] Weigand D J and Terhal B M 2018 Phys. Rev. A 97 022341[81] Eaton M, Nehra R and Pfister O 2019 New Journal of Physics

(In press) (https://doi.org/10.1088/1367-2630/ab5330)[82] Flühmann C, Nguyen T L, Marinelli M, Negnevitsky V,

Mehta K and Home J P 2019 Nature 566 513–7[83] Baragiola B Q, Pantaleoni G, Alexander R N, Karanjai A and

Menicucci N C 2019 arXiv:1903.00012 [quant-ph][84] Aaronson S and Arkhipov A 2010 The Computational

Complexity of Linear Optics Electronic Colloquium onComputational Complexity Report No. 170 pp 1–91

[85] Hamilton C S, Kruse R, Sansoni L, Barkhofen S,Silberhorn C and Jex I 2017 Phys. Rev. Lett. 119 170501

[86] Douce T, Markham D, Kashefi E, Diamanti E, Coudreau T,Milman P, van Loock P and Ferrini G 2017 Phys. Rev. Lett.118 070503

[87] Raussendorf R and Briegel H J 2001 Phys. Rev. Lett. 86 5188[88] Greenberger D M, Horne M A and Zeilinger A 1989 Going

beyond Bell’s theorem Bell’s Theorem, Quantum Theoryand Conceptions of the Universe ed M Kafatos (Dordrecht:Kluwer) p 69

[89] Dür W, Vidal G and Cirac J I 2000 Phys. Rev. A 62 062314[90] Jozsa R 2005 arXiv:quant-ph/0508124[91] Briegel H J, Browne D E, Dur W, Raussendorf R and

Van den Nest M 2009 Nat. Phys. 5 19–26[92] Gu M, Weedbrook C, Menicucci N C, Ralph T C and

van Loock P 2009 Phys. Rev. A 79 062318[93] Braginsky V B and Khalili F Y 1992 Quantum Measurement

(Cambridge: Cambridge University Press)[94] Caves C M, Thorne K S, Drever R W P, Sandberg V D and

Zimmermann M 1980 Rev. Mod. Phys. 52 341–92[95] Yurke B 1985 J. Opt. Soc. Am. B 2 732[96] Braunstein S L 2005 Phys. Rev. A 71 055801[97] van Loock P, Weedbrook C and Gu M 2007 Phys. Rev. A 76

032321[98] Yukawa M, Ukai R, van Loock P and Furusawa A 2008 Phys.

Rev. A 78 012301[99] Menicucci N C, Flammia S T, Zaidi H and Pfister O 2007

Phys. Rev. A 76 010302(R)[100] Roslund J, Medeiros de Araújo R, Jiang S, Fabre C and

Treps N 2014 Nat. Photon. 8 109[101] Menicucci N C, Flammia S T and van Loock P 2011 Phys.

Rev. A 83 042335

[102] Bradley A S, Olsen M K, Pfister O and Pooser R C 2005Phys. Rev. A 72 053805

[103] Zaidi H, Menicucci N C, Flammia S T, Bloomer R,Pysher M and Pfister O 2008 Laser Phys. 18 659

[104] van Loock P and Braunstein S L 2003 Multipartiteentanglement for continuous variables Quantum Informationwith Continuous Variables (New York: Springer)

[105] Braunstein S L 1998 Nature 394 47[106] Hein M, Eisert J and Briegel H J 2004 Phys. Rev. A 69

062311[107] Bacon D 2009 Physics 2 38[108] Bremner M J, Mora C and Winter A 2009 Phys. Rev. Lett.

102 190502[109] Gross D, Flammia S T and Eisert J 2009 Phys. Rev. Lett. 102

190501[110] Raussendorf R, Harrington J and Goyal K 2006 Ann. Phys.

321 2242–70[111] Demarie T F, Linjordet T, Menicucci N C and Brennen G K

2014 New J. Phys. 16 085011[112] Walshe B W, Mensen L J, Baragiola B Q and Menicucci N C

2019 Phys. Rev. A 010301[113] Vahlbruch H, Mehmet M, Danzmann K and Schnabel R 2016

Phys. Rev. Lett. 117 110801[114] Fukui K, Tomita A, Okamoto A and Fujii K 2018 Phys. Rev.

X 8 021054[115] Fukui K 2019 arXiv:1906.09767 [quant-ph][116] Michael M H, Silveri M, Brierley R T, Albert V V,

Salmilehto J, Jiang L and Girvin S M 2016 Phys. Rev. X 6031006

[117] Su X, Tan A, Jia X, Zhang J, Xie C and Peng K 2007 Phys.Rev. Lett. 98 070502

[118] Su X, Zhao Y, Hao S, Jia X, Xie C and Peng K 2012 Opt.Lett. 37 5178–80

[119] Menicucci N C, Ma X and Ralph T C 2010 Phys. Rev. Lett.104 250503

[120] Flammia S T, Menicucci N C and Pfister O 2009 J. Phys. B:At. Mol. Opt. Phys. 42 114009

[121] Pysher M, Bahabad A, Peng P, Arie A and Pfister O 2010Opt. Lett. 35 565

[122] Izutsu M, Shikama S and Sueta T 1981 IEEE J. QuantumElectron. 17 2225

[123] Miller J L 2011 Phys. Today 64 21[124] Menicucci N C 2011 Phys. Rev. A 83 062314[125] Drever R W P, Hall J L, Kowalski F V, Hough J, Ford G M,

Munley A J and Ward H 1983 Appl. Phys. B 31 97–105[126] Asavanant W et al 2019 Science 366 373–6[127] Larsen M V, Guo X, Breum C R, Neergaard-Nielsen J S and

Andersen U L 2019 Science 366 369–72[128] Alexander R N, Wang P, Sridhar N, Chen M, Pfister O and

Menicucci N C 2016 Phys. Rev. A 94 032327[129] Huntington E H, Milford G N, Robilliard C, Ralph T C,

Glöckl O, Andersen U L, Lorenz S and Leuchs G 2005Phys. Rev. A 71 041802

[130] Armstrong S, Morizur J F, Janousek J, Hage B, Treps N,Lam P K and Bachor H A 2012 Nat. Commun. 31026

[131] Walschaers M, Sarkar S, Parigi V and Treps N 2018 Phys.Rev. Lett. 121 220501

[132] Fukui Kosuke, Tomita Akihisa, Okamoto Atsushi andFujii Keisuke 2018 Physical Review X 8 021054

16

J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 012001 Topical Review