Continuous-variable gate teleportation and bosonic-code error correction

Blayney W. Walshe,1, ∗ Ben Q. Baragiola,1 Rafael N. Alexander,1, 2 and Nicolas C. Menicucci11Centre for Quantum Computation and Communication Technology,

School of Science, RMIT University, Melbourne, VIC 3000, Australia2Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA

We examine continuous-variable gate teleportation using entangled states made from pure prod-uct states sent through a beam splitter. We show that such states are Choi states for a (typically)non-unitary gate, and we derive the associated Kraus operator for teleportation, which can be usedto realize non-Gaussian, non-unitary quantum operations on an input state. With this result, weshow how gate teleportation is used to perform error correction on bosonic qubits encoded using theGottesman-Kitaev-Preskill code. This result is presented in the context of deterministically pro-duced macronode cluster states, generated by constant-depth linear optical networks, supplementedwith a probabilistic supply of GKP states. The upshot of our technique is that state injection forboth gate teleportation and error correction can be achieved without active squeezing operations—anexperimental bottleneck for quantum optical implementations.

I. INTRODUCTION

Recent strides in the experimental generation ofcontinuous-variable (CV) cluster states [1–3] prove thatthe use of CV measurement-based quantum computing(MBQC) is one of the most promising methods of achiev-ing fault-tolerant universal quantum computing. MBQCutilises a highly entangled resource—known as a clusterstate—as its substrate for quantum computing [4] andonly requires adaptive local measurements on the clusterstate to implement quantum gates.

Constructing cluster states using bosonic modes hasthe advantage of deterministic entanglement generationusing linear optics and is highly scalable [5]. Unfortu-nately, computation with CV cluster states is burdenedby intrinsic noise due to finite energy constraints [6]. De-spite this, CV MBQC on a cluster state generated fromsqueezed states meeting a threshold of 15-17 dB of mea-sured squeezing is fault-tolerant [7]—provided it is sup-plemented with a source of high-quality bosonic qubits.Furthermore, this squeezing threshold is not affected bydecoherence that manifests as anti-squeezing [8].

CV cluster states are multi-mode Gaussian states oflight specified by a complex-weighted graph [9]. Thestates originally considered in Refs. [10, 11] possessedsimple graphs (e.g., a 2D square lattice); however, knownmethods for their generation require either inline squeez-ing (active transformations on states other than the vac-uum) or a linear optics network that grows with the sys-tem size. Related states with a multi-layered graph struc-ture can be generated in a more experimentally feasibleway with circuits consisting of offline squeezing and lo-cal constant depth linear optics [2, 3, 5, 12–17]. Suchstates have been generated on a large scale, both in one[1, 18, 19] and two dimensions [2, 3]. MBQC on thesemulti-layered states respects the tensor product struc-ture of macronodes, with physical lattice sites made up

∗ blayneyw@gmail.com

of one mode from each layer [6, 20].The fundamental primitive of CV MBQC schemes with

multi-layered graphs is CV teleportation [21, 22]. Gaus-sian homodyne detection teleports quantum informa-tion from one node to another with a fidelity dependingon the quality of the shared entanglement between thenodes. Ideally, nodes share a maximally entangled EPRstate, which allows for perfect teleporation. Up to lo-cal phase delays, macronode continuous-variable clusterstates (CVCSs) are indeed just a collection of approx-imate EPR states stitched together at macronodes bynon-local measurements (more specifically, a sequence of50:50 beam splitters followed by homodyne detection onall modes). Gaussian CV MBQC schemes leverage thechoice of rotated homodyne bases as degrees of freedomto realize Gaussian operations on the teleported state.This was recently demonstrated in Ref. [23].

In this work, we provide a method to go beyond Gaus-sian operations by replacing each entangled pair with amore general state—two arbitrary pure states coupled ona beam splitter, see Fig. 1. If either (or both) of thesestates is non-Gaussian, teleportation can realize a non-Gaussian gate, extending multi-mode Gaussian resourcesto universal resources for quantum computation. Our ap-proach runs parallel to and extends the standard descrip-tion of gate teleportation [24–26], where a unitary gateto be teleported is applied to one half of a maximally en-tangled state. Using the entangled-state method here ad-ditionally allows teleportation of non-unitary operations,such as projections, which are useful for bosonic-code er-ror correction. This approach is compatible with a widevariety of non-Gaussian resources [27–30] and does notrequire active squeezing operations, making it convenientfor conventional quantum optics setups.

We apply our analysis to a particular example: gate-teleported error correction of the Gottesman-Kitaev-Preskill (GKP) code. GKP states allow encoding ofdigital quantum information in the continuous Hilbertspace of a CV mode [31]. Remarkably, this single non-Gaussian resource extends multi-mode Gaussian compu-tation to universal and fault-tolerant quantum computa-

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Figure 1. Schematic of a macronode wire, which is a chain ofmaximally entangled pairs (modes connected by beam split-ters) that meet at two-mode sites called macronodes. Non-local measurements of each macronode, achievable by an addi-tional beam splitter and local measurements at each macron-ode (not indicated), teleport an input state along the macron-ode wire (from right to left). In this work, we replace eachentangled pair with a Kraus state consisting of two arbitrarystates, |ψ〉 and |φ〉, coupled on a beam splitter. This realizesa teleported CV gate, which can be non-unitary, dependingon the Kraus state. A single instance of this CV gate tele-portation, which we call the teleportation gadget, is enclosedin the dashed box. The macronode-wire circuit diagram forgate teleportation is given in Fig. 2.

tion [32, 33].GKP states have recently been realized in trapped

ion systems [34] and superconducting qubit architectures[35], and multiple novel approaches have been proposedfor their generation in optics [36–42]. We show that tele-porting through entangled pairs made from a probabilis-tic source of GKP grid states implements GKP error cor-rection.

Thus, the entire scheme for universal and fault-tolerantquantum computing consists only of offline preparation ofGKP and squeezed vacuum states, constant-depth linearoptics, and homodyne detection. This is an improvementover prior proposals that require active transformationsto combine GKP states with CV cluster states [7, 16,43]. Finally, we note that our analysis extends to higherdimensional CV cluster states that have two- and three-dimensional structure.

II. MACRONODE-BASED CV CLUSTERSTATES

A continuous-variable cluster state (CVCS) is a large,Gaussian state across many modes that can serve as a re-source for CV teleportation [10] and measurement-basedquantum computing (MBQC) [7]. Canonical CVCSs areconstructed using eigq⊗q interactions to successively en-tangle momentum eigenstates, and Gaussian homodynemeasurements of the modes are used to teleport and en-act Gaussian operations on encoded information [10].An alternative construction employing constant-depthpassive elements (beam splitters and phase delays) andfinitely squeezed states, called a macronode CVCS, pro-vides a blueprint for experimental generation of a CVCS[6] that makes better use of the fundamental quantum re-source in such states: squeezing [44, 45]. Here, two-modesqueezed states are generated using beam splitters; these

entangled pairs can then be coupled to others using morebeam splitters to create macronode CVCSs with morecomplicated structures. In this construction, certain col-lections of physical modes are grouped into macronodes,across which logical encoded information is distributednonlocally. Processing the information within a macron-ode requires homodyne measurement of its constituentphysical modes.

The smallest nontrivial macronode CVCS, themacron-ode wire (see Figure 1), has two modes per macronode.Larger and more complex macronode CVCSs containmore modes per macronode [2, 3, 17] but can be viewedquantitatively as tools to route information along con-figurable macronode wires [20]. Macronode wires havebeen generated experimentally from both temporal andspectral modes [1, 18, 19] and recently adaptive homo-dyne measurements have been used to implement Gaus-sian operations on a temporal-mode macronode wire [23].

A. Structure of a macronode wire

A macronode-based CV cluster state [13] differs froma canonical one [10, 11] by the fact that multiple modesoccupy one logical site within the overarching graph rep-resenting the entanglement structure of the state. (Forfurther details, see Ref. [13].) These modes are logi-cally grouped together and referred to as a macronodeto emphasise that it comprises multiple “micronodes”,each of which is an individual mode. All of the ex-periments demonstrating large-scale continuous-variablecluster states have generated macronode-based clusterstates [1–3, 18, 19, 23].

We use the following notation to describe the individ-ual modes within a macronode-based CV cluster state.For each local mode a, we define the position and mo-mentum quadratures, q = 1√

2(a+a†) and p = −i√

2(a−a†),

respectively, satisfying [q, p] = i. This means thatthe vacuum variance in both quadratures of any modeis⟨q2⟩vac =

⟨p2⟩vac = 1/2, which can be interpreted as

equivalent to the convention that ~ = 1.Each quadrature has an associated set of eigenstates,|s〉q and |t〉p, satisfying q|s〉q = s|s〉q and p|t〉p = t|t〉p.Information in a one-dimensional macronode-based CVcluster state is encoded in the distributed symmetric(+) and antisymmetric (−) subspace of the two localmodes in a single macronode, arising from a change in thetensor-product structure due to the beam splitter trans-formation coupling the two modes prior to the final mea-surements [6]. The balanced beam splitter convention weuse here,

Bjk := e−iπ4 (qj⊗pk−pj⊗qk), (1)

generates the distributed quadrature operators at eachmacronode:

q± := 1√2(qa ± qb) (2)

3

p± := 1√2(pa ± pb). (3)

Specifically, B†ab(qa, qb, pa, pb)Bab = (q−, q+, p−, p+), re-spectively. Note that the beam splitter we use is not sym-metric; B†jk = Bkj 6= Bjk. In circuit diagrams (which weuse extensively below), we represent the beam splitter asan arrow pointing from mode j to mode k:

��

jBjk =

k

(4)

A macronode wire is a chain of entangled pairs that arelogically linked together at macronodes, a grouping oflocal modes. Non-local measurements on the macron-odes, a beam splitter interaction followed by local mea-surements at each local modes, implements a string ofsequential teleportations (from right to left), see Fig. 1.For more information, see Ref. [5, 6].

B. Measurement-based CV computation with amacronode CVCS wire

In an ideal setting, measurement-based computingwith a macronode CVCS begins by preparing every localmode in either a position or a momentum eigenstate, |0〉qor |0〉p. Pairs of adjacent modes are first coupled witha beam splitter to generate infinitely squeezed two-modesqueezed states of the form:

��

|0〉paBab|0〉pa ⊗ |0〉qb =

|0〉qb

(5)

In physical settings, quadrature eigenstates are replacedby their finite-energy, squeezed approximations, whichwill be discussed in Sec. III A 1.

Homodyne measurements of a macronode’s two con-stituent local modes (followed by outcome-dependent dis-placements) teleport a state encoded in the symmetricmode of that macronode to the symmetric mode of thenext macronode. The macronode wire’s utility for quan-tum information processing arises from the fact that mea-suring in rotated bases over two successive macronodesadditionally implements any single-mode Gaussian uni-tary gate [6]. The essentials of this procedure are outlinedbelow.

We define a rotated momentum quadrature,

pθ := R†(θ)pR(θ) = q sin θ + p cos θ , (6)

where the phase-delay operator

R(θ) := eiθa†a (7)

generates an anti-clockwise rotation by θ in phase space.A measurement of pθ with outcome m, realized via ho-modyne detection, corresponds to a projection onto thequadrature eigenstate,

|m〉pθ := R†(θ)|m〉p , (8)

where pθ|m〉pθ = m|m〉pθ .1

Measuring both local modes in a single macronode inrotated momentum quadratures given by measurementangles θa and θb teleports the state to the next macronodeand implements the operation [1, 15, 46]

D(µ)V (θa, θb) , (9)

with measurement-basis-dependent Gaussian unitary

V (θa, θb) := R(θ+ − π2 )S(tan θ−)R(θ+), (10)

and displacement D(µ) with outcome-dependent ampli-tude

µ = −maeiθb +mbe

iθa

sin(θa − θb), (11)

which can be corrected with active Gaussian shifts.2 Theparameters

θ± := 12 (θa ± θb) (12)

are symmetric and antisymmetric combinations of themeasurement angles, and we define a (nonstandard)squeezing operator,

S(ζ) := R(Im ln ζ)e−i2 (Re ln ζ)(qp+pq). (13)

This is just an ordinary squeezing operator with squeez-ing parameter r = ln |ζ| generalized to allow negative val-ues of ζ, which can arise in Eq. (10). For ζ < 0, Eq. (13)describes squeezing followed by a π phase delay, whichis equivalent to a double Fourier transform or a parityoperation.3 Note that this operator produces squeezing(and conjugate antisqueezing) along the principal posi-tion and momentum axes; squeezing other quadraturesis achieved by following this operator by a phase delay,Eq. (7). Note that two teleportation steps through themacronode wire (Figure 1) are sufficient to perform twodifferent gates of the form of Eq. (9) and thus an arbi-trary Gaussian unitary [6].

C. Connection to maximally entangled EPR states

In the ideal macronode CVCS construction, the Krausstate is composed of local modes, shared betweenmacronodes, prepared in 0-momentum and 0-position

1 The phase-delayed eigenstates are defined via a Heisenberg-picture transformation, just as in Refs. [6, 10, 16].

2 Various works have slightly different definitions of these opera-tions. A derivation of Eq. (9) and further discussion can be foundin Appendix B.

3 This squeezing operator is designed such that S†(ζ)qS(ζ) = ζq

and S†(ζ)pS(ζ) = ζ−1p for all ζ ∈ R6=0.

4

eigenstates, Eq. (5), and then coupled on a beam splitter.Here, we show that the resulting state is a member of acomplete set of maximally entangled two-mode states.

Position and momentum eigenstates each form a res-olution of the identity over a single mode, thus it isstraightforward to construct a tensor-product represen-tation of the identity across two modes. Consider thefollowing tensor-product basis,

I1 ⊗ I2 =

∫∫ds dt |t〉p1p1

〈t| ⊗ |s〉q2q2〈s| . (14)

Any unitary transformation on this expression also givesa resolution of the identity and produces a new, typicallyentangled, basis. Specifically, we consider the unitarye−iq⊗p, which serves as the CV analog to a CNOT gate[47], to define a complete set of shifted maximally entan-gled EPR states,

|EPR(s, t)〉 :=e−iq1⊗p2 |t〉p1⊗ |s〉q2 (15a)

=1√2π

∫dr eirt |r〉q1 ⊗ |s+ r〉q2 , (15b)

satisfying 〈EPR(s, t)|EPR(s′, t′)〉 = δ(s−s′)δ(t−t′). Theset of shifted EPR states resolves the identity over twomodes,

I1 ⊗ I2 =

∫∫ds dt |EPR(s, t)〉〈EPR(s, t)| , (16)

comprising an entangled two-mode basis that comple-ments the tensor-product basis above, Eq. (14).

The states in Eq. (15) can also be written as a mo-mentum shift, Z(t) := eitq, and a position shift, X(s) :=e−isp, across the two modes as

|EPR(s, t)〉 = Z1(t)X2(s) |EPR〉 , (17)

where |EPR〉 := |EPR(0, 0)〉, is a canonical EPR state,with position-position representation,

|EPR〉 :=1√2π

∫dr |r〉q1 ⊗ |r〉q2 . (18)

This state, being perfectly correlated in position and per-fectly anti-correlated in momentum, is the continuous-variable analog of the Bell state |Φ+〉 = 1√

2

∑j=0,1 |j〉 ⊗

|j〉. The canonical EPR state can be represented as|EPR〉 = CX12(1)|0〉p⊗ |0〉q, where the entangling gate is aCV controlled-X gate of weight g,

CXjk(g) := e−igqj⊗pk , (19)

with control mode j and target mode k. This gives thecircuit identity:

• |0〉p|EPR〉 :=

|0〉q=

1√2πI

. (20)

The factor 1√2π

results from the normalization of the EPRstate in Eq. (18).4 Note that we use the convention thatquantum circuits proceed in time from right-to-left, suchthat a circuit-description of a state respects the sameordering as its partner equation.

1. Bouncing operators on an EPR state

Let O be a single-mode operator with position-spacerepresentation

O =

∫∫ds ds′O(s, s′)|s〉qq〈s

′| , (21)

where O(s, s′) = q〈s|O|s′〉q. Applying this operator toone mode of an EPR state gives

|ΨO〉 := (O ⊗ I) |EPR〉 (22a)

=1√2π

∫∫ds ds′O(s, s′)|s〉q1 ⊗ |s

′〉q2 . (22b)

The state |ΨO〉 is called the Choi state for O and is arepresentation of O through the Choi-Jamiołkowski iso-morphism [48? ]. It derives from the position-space ex-pansion of O simply by replacing |s〉qq〈s′| → |s〉q1 ⊗ |s

′〉q2(up to normalization). Teleporting through a Choi stateapplies the associated operator, which we call the tele-ported gate [24]. This forms the basis for gate teleporta-tion and the Kraus operator derived in Sec. III.

An important feature of maximally entangled states isthat a local operator acting on one mode can be moved tothe other (the operator is modified in the process), andthe resulting two-mode state is identical to the original.This operation, which we call bouncing, is described bythe circuit,

O=

OT

, (23)

where O is an operator on the top mode, and OT (thebounced operator) is the transpose of O taken in thebasis where the maximally entangled state is perfectlycorrelated—the position-position basis for the EPR statein Eq. (18). Thus, the Choi state above can be repre-sented in different, yet equivalent ways.

4 For qudits, the normalization of the maximally entangled statesused for teleportation, 1√

d, is typically absorbed into the “angle-

bracket” notation for these states in a circuit diagram [24] andalso ignored in the teleported gate. For CV systems consideredhere, we make the association d→ 2π (for δ-normed states) andexplicitly include this EPR normalization in teleported gates sothat the resulting Kraus operators are properly normalized.

5

The transposed position and momentum operators,qT = q and pT = −p, can be found by computing theirmatrix elements in the position basis,

qT(t,s) = t δ(t− s) = q(t,s)

pT(t,s) = i δ′(t− s) = −p(t,s),(24)

where x(t,s) := q〈t|x|s〉q.5 These relations allow a straight-forward bounce of a displacement operator,

D(α) := eαa−α∗a† = ei

√2(αI q−αRp) , (25)

with complex phase-space displacement α = αR + iαI ,through the EPR state in Eq. (18):

D1(α) |EPR〉 = D2(−α∗) |EPR〉 . (26)

More general single-mode operators generated by powersof q and p can also be bounced using Eqs. (24). Note thatEq. (26) can straightforwardly be used to bounce positionand momentum shifts by decomposing the displacementoperator,

D(α) = eiαRαI X(√

2αR)Z(√

2αI). (27)

2. Entangled states on a beam splitter

In a macronode wire, pairs of modes—one each fromneighboring macronodes—are combined on a beam split-ter. When these modes are prepared in position and mo-mentum eigenstates, respectively, the result is a shiftedEPR state, Eq. (15), whose entanglement lies at the heartof continuous-variable teleportation protocols.

Consider momentum and position eigenstates, |t〉p and|s〉q, coupled by a beam splitter. As shown in AppendixA, the resulting entangled state is

B12 |t〉p1⊗ |s〉q2 =

√2eist |EPR(

√2s,√

2t)〉 , (28)

up to a fixed phase (which is irrelevant). The norm scal-ing by

√2 preserves inner products and ensures that the

completeness relation, Eq. (16), holds,

I1⊗I2 =

∫∫dt ds

(B12|t〉p1

⊗|s〉q2)(p1〈t|⊗q2〈s|B

†12

). (29)

We keep track of this factor throughout so that the Krausoperators we derive in Sec. III can be used to faithfullycalculate measurement probabilities.

A final manipulation of the state in Eq. (28) provides auseful form for the beam splitter entangled state. Pullingout the shifts [Eq. (17)], bouncing the momentum shiftZ1(√

2t) to the second mode [Eq. (26)], and combining

5 The relations follow from the symmetry of δ(·) and the asymme-try of its derivative δ′(·).

the two single-mode shifts into a displacement operator[Eq. (27)] gives

B12 |t〉p1⊗ |s〉q2 =

√2D2(s+ it) |EPR〉 . (30)

This state is represented by the circuit diagram,

��

|t〉p

|s〉q=

1√πD(s+ it)

, (31)

indicating that the beam splitter entangled state is theChoi state, Eq. (22), for the displacement D(s + it) (upto a factor 1√

π).

Thus, the specific entangled state between adjacentmacronodes in Eq. (5), B12 |0〉p1

⊗ |0〉q2 =√

2 |EPR〉,is the Choi state [48] for the identity, which underpins itsutility for teleportation from one macronode to the nextin the standard macronode computing protocol.

III. GATE TELEPORTATION WITH THEMACRONODE WIRE

The macronode procedure for quantum computation,described in Sec. II B, has two fundamental components.The first is the multimode entangled state itself—themacronode wire—and the second is the collection of ho-modyne measurements. Both of these components areGaussian, which limits effective operations on an inputstate to Gaussian operations. Here, we show that modi-fying one part of this procedure, the state of the macron-ode wire itself, allows us to implement non-Gaussian op-erations in a gate-teleportation fashion. The general-ized macronode procedure is shown in Figure 1. Thekey difference from the standard procedure is the re-placement of the quadrature eigenstates that comprisea CVCS macronode wire [see Eq. (5)] with arbitary purestates. Modes from adjacent macronodes are still mea-sured in rotated homodyne bases, with the ultimate ef-fect that the Gaussian operation, Eq. (10), is supple-mented with an additional ancilla-state-dependent oper-ation. This additional operation can be leveraged formany purposes, including teleportation-based error cor-rection on a CV-encoded qubit (using the Gottesman-Kitaev-Preskill code), which we describe in Sec. IVA.

A. Kraus operator for gate teleportation

We focus on the teleportation gadget within themacronode wire, indicated in the dashed box in Fig. 2.This gadget describes the teleportation of a quantumstate arriving from the previous macronode through thegiven macronode using two homodyne measurements af-ter a beam splitter—an effective EPR measurement.This teleportation is accompanied by the Gaussian op-eration V (θa, θb) in Eq. (10), determined by the bases in

6

· · ·

��

(in)

pθa〈ma|

��

· · ·

pθb〈mb|

��

|ψ〉

· · ·

��

|φ〉

(out) · · ·Figure 2. Circuit diagram for gate teleportation using themacronode wire in Fig. 1. In our convention, circuits proceedfrom right to left. Inside the dashed box is the teleportationgadget, which consists of three modes. Arbitrary ancillae onthe second and third mode, |ψ〉 and |φ〉, are combined on abeam splitter to form the Kraus state. The first two modes,which comprise a single macronode, are combined on an ad-ditional beam splitter and measured via homodyne detectionin rotated bases. We describe these measurements as projec-tions onto rotated momentum eigenstates, Eq. (8). This pro-cedure teleports a state from the first mode to the third modewith a (potentially nonunitary) gate applied that depends onthe measurement bases, the measurement outcomes, and theKraus state.

which the homodyne measurements are performed. Aswe show here, a non-Gaussian and non-unitary opera-tion can also be applied by preparing local modes of ad-jacent macronodes in the arbitrary pure, product state|ψ〉 ⊗ |φ〉 and then mixing them on a beam splitter. Theentire operation is described by the Kraus operator,

K(ma,mb) := p1,θa〈ma| ⊗ p2,θb

〈mb|B12B23 |ψ〉2 ⊗ |φ〉3 ,(32)

where ma and mb are the outcomes of the homodynemeasurements, and subscripts in each beam splitter, Bjk,label the modes it couples. This Kraus operator is de-scribed by the circuit

pθa〈ma|

��

(in)

pθb〈mb|

��

|ψ〉

(out) |φ〉

(33)

Again, we use the convention that time proceeds right-to-left in circuit diagrams. We refer to the joint state of thelast two modes, B23 |ψ〉2⊗|φ〉3, as the Kraus state, whichis the Choi state for a (typically non-unitary) teleportedgate whose form we determine below.

After an input state on the top wire ρin traverses theteleportation gadget as part of a macronode wire, it isteleported to the third mode and transformed accordingto the conditional map,

ρout =K(ma,mb)ρinK

†(ma,mb)

Pr(ma,mb), (34)

where the joint probability (density) of obtaining the out-comes {ma,mb} is

Pr(ma,mb) = Tr[K†(ma,mb)K(ma,mb)ρin] . (35)

As the modes in the macronode wire are successivelymeasured, a sequence of such Kraus operators is ap-plied to the input state, each with a set of measurementoutcomes and an applied operation depending on whichquadratures are measured and the ancillae at the beamsplitter, which comprise the Kraus state. It suffices toconsider the single Kraus operator in Eq. (32), as it con-tains all the necessary ingredients.

The homodyne measurements on the first two modesrealize an entangled Bell-type measurement with theextension that, by measuring rotated quadratures, thebasis-dependent Gaussian operation V (θa, θb) in Eq. (10)is applied. This is described by the following circuit iden-tity

pθa〈ma|

��

D(µ) 1√πV (θa, θb)

pθb〈mb|

= (36)

where the effect of the outcomes is a displacement by µ,given in Eq. (11).

We now focus on the Kraus state on the second andthird modes B23 |ψ〉2 ⊗ |φ〉3. First, we write the state|ψ〉 ⊗ |φ〉 in the tensor-product basis, Eq. (14), using therespective momentum and position wavefunctions,

ψ(t) := p〈t|ψ〉 and φ(s) := q〈s|φ〉. (37)

Then, we apply the beam splitter using Eq. (30) to ar-rive at an expression for the Kraus state in terms of anoperation on an |EPR〉 state,

B23 |ψ〉2 ⊗ |φ〉3 = B23

∫∫dtds ψ(t)φ(s) |t〉p2

⊗ |s〉q3

(38)

=√

2A3(ψ, φ) |EPR〉 . (39)

The single-mode operator A(ψ, φ), referred to as the tele-ported gate, arises in its Cahill-Glauber form [49, 50],with displacements weighted by the respective momen-tum and position wavefunctions of the input states,

A(ψ, φ) :=

∫∫d2α ψ(αI)φ(αR)D(α), (40)

for complex number α = αR + iαI . The Kraus state canbe reexpressed with the circuit identity,

��

|ψ〉

|φ〉=

1√πA(ψ, φ)

, (41)

where the 1√π

arises due to the circuit identity for the|EPR〉 state, Eq. (5). In this form, it is clear that the op-erator A(ψ, φ) is indeed a teleported gate in the contextof the standard teleportation gadget [24].

Combining the circuits for the measurements, (36), andthe Kraus state, (41), we express the teleportation gadget

7

in Eq. (33) as

D(µ) 1√πV (θa, θb) (in)

(out) 1√πA(ψ, φ)

(42)

Pulling the circuit taut,

(out) 1√πA(ψ, φ) D(µ) 1√

πV (θa, θb) (in)(43)

allows us to read off the Kraus operator directly by virtueof the right-to-left circuit convention,

K(ma,mb) =1

πA(ψ, φ)D(µ)V (θa, θb). (44)

While the first two operations, D(µ) and V (θa, θb)

[Eq. (10)], are Gaussian, the teleported gate A(ψ, φ)[Eq. (40)] can realize non-Gaussian operations when theancilla states, |ψ〉 and |φ〉, are themselves non-Gaussian.In addition, the operation realized by the teleported gatecan be non-unitary. We apply this powerful tool belowto realize teleportation-based error correction of a boson-ically encoded qubit.

Finally, we note that standard teleportation is achievedwhen the Kraus state is the canonical EPR state, and themeasurement angles are chosen to be θa = π

2 and θb = 0

so that V (θa, θb) = I.

1. Damped-ancillae Kraus operator

The Kraus operator, Eq. (44), implements opera-tions determined by the wavefunctions of the input an-cillae that comprise the Kraus state. In macronodeCVCS quantum computing, each ancilla is prepared in asqueezed vacuum state [10] that approximates a quadra-ture eigenstate to a degree that depends on the level ofsqueezing. In a standard description, these states are de-scribed by unitarily squeezing the vacuum state, via S(ζ)in Eq. (13). This gives the squeezed vacuum state, whichwe write in the notation of Ref. [10],

|0; ζ〉q :=1

(πζ2)1/4

∫ds e

−s2

2ζ2 |s〉q = S(ζ) |0〉 , (45)

where |0〉 (with no subscript) is the vacuum state, andS(ζ) is the squeezing operator in Eq. (13), with squeezingfactor ζ. The measured variance of this state is

⟨q2⟩

=

ζ2/2. ζ = 1 reproduces the vacuum state, and thelimit ζ → 0+ gives a normalized version of |0〉q. We alsodefine

|0; ζ〉p :=1

(πζ2)1/4

∫ds e

−s2

2ζ2 |s〉p = S(ζ−1) |0〉 , (46)

which analogously has⟨p2⟩

= ζ2/2. Note that this im-plies that

|0; ζ〉p = |0; ζ−1〉q . (47)

It will be helpful to list a few important relations foruse in what follows. First, let

ζ = e−r . (48)

Then, in the Fock basis {|n〉}∞n=0,

|0; ζ〉q =√

sech r

∞∑n=0

(− tanh r)n√

(2n)!

2nn!|2n〉 (49a)

=

√2ζ

1 + ζ2

∞∑n=0

(−1)n(

1− ζ2

1 + ζ2

)n √(2n)!

2nn!|2n〉 .

(49b)

Also, |0; ζ〉p is the same but without the (−1)n. Fi-nally, by recognizing that 〈n|0〉q = ψ∗n(0), where ψn isthe nth Fock-state wave function, we can write the delta-normalised q-quadrature eigenstate in the Fock basis:

|0〉q = π−1/4∞∑n=0

(−1)n√

(2n)!

2nn!|2n〉 , (50)

and similarly for |0〉p but again without the (−1)n. Theseseparate expressions are required because the prefactorvanishes in the normalised case, Eqs. (49), when ζ → 0+

(r →∞).We now introduce an equivalent, non-unitary formu-

lation of squeezed vacuum states that ultimately yieldsa convenient form for the associated macronode Krausoperator. Moreover, this method gives a straightforwardway to include finite-squeezing noise in any state. Par-ticularly useful cases are unphysical ideal states, suchas the GKP states introduced in Sec. (IVA), and, moregenerally, any state constructed from a countable super-position of quadrature eigenstates.

The squeezed vacuum state in Eq. (46) can equiva-lently be described in terms of the non-unitary dampingoperator defined by6

N(β) := e−βn (51)

acting on an infinitely squeezed quadrature eigenstate.To derive the precise relation, compare Eq. (49) withEq. (50). The only difference, aside from the normali-sation, is a factor of (tanh r)n in the sum. This factorcan be restored by writing (tanh r)n/2|0〉q. Referring toEq. (51), we see that setting tanh r = e−2β would pro-duce the full relation

|0; ζ〉q =1√Nζ

N(β)|0〉q , (52)

6 The damping operator can also be considered a phase delay,Eq. (7), with imaginary delay angle θ = iβ.

8

with

β =1

2ln coth r =

1

2ln

(1 + ζ2

1− ζ2

). (53)

(Recall that ζ = e−r, as above.) The state is normalizedby setting

Nζ =1√

π(1− e−4β)=

cosh r√π

=1 + ζ2

2ζ√π, (54)

which is required since N(β) is not unitary. Analogously,|0; ζ〉p = 1√

NζN(β)|0〉p, with the exact same relations be-

tween ζ, r, and β as shown above.The damping operator has been used in macronode

CVCS constructions previously, because it allows one toseparate finite-squeezing effects from the ideal quadra-ture eigenstates [7, 20, 51]. The damping operator canthen be manipulated separately from the state on whichit acts. This fact plays a critical role when the ancillaethat comprise the Kraus state in the teleportation circuitin Eq. (33) are described as damping operators acting ona state,

|ψ〉 → 1√Nψ

N(β) |ψ〉 (55)

with normalization explicitly given by,

Nψ := 〈ψ| N(2β) |ψ〉 . (56)

For identical damping on both ancillae, the joint dampingoperator N(β)⊗N(β) = e−β(n1+n2) is a function of totalphoton number, n1 + n2, which is conserved by a beamsplitter. The joint damping operator commutes triviallythrough the beam splitter in Eq. (57), and single-modedamping operators can then be bounced trivially to thefinal mode, since nT = n follows from Eq. (24). Theresulting circuit identity is

��

N(β) |ψ〉

N(β) |φ〉=

N(β) 1√πA(ψ, φ) N(β)

(57)Using this result in the macronode gadget, Eq. (33), givesthe final Kraus operator

K(ma,mb) =

1

π√NφNψ

N(β)A(ψ, φ)N(β)D(µ)V (θa, θb) , (58)

with outcome-dependent displacement amplitude µ,Eq. (11). We emphasize here that this Kraus operator isa specific case of the general one derived above, Eq. (44),useful for situations when the ancillae states (that com-prise the Kraus state) can be expressed as in Eq. (55).

IV. APPLICATION: GKP ERRORCORRECTION IN THE MACRONODE WIRE

In its standard implementation, macronode-basedquantum computing is used to perform Gaussian oper-ations on an input state as it is teleported down themacronode wire [6]. Momentum and position quadra-ture eigenstates at the ancillae realize the intended Gaus-sian operation: D(µ)V (θa, θb). Such states contain infi-nite energy and are not normalizable; in physical settings,finitely squeezed approximations to them are used. Puresqueezed states at the ancillae, described by Eq. (52),yield the Kraus operator

K(ma,mb) =1

πNζN(2β)D(µ)V (θa, θb). (59)

Finite-squeezing noise, originating in the ancillae andgenerated by N(2β), accompanies the Gaussian opera-tion at each macronode. From a wavefunction perspec-tive, this noise describes an equal blurring in position andmomentum [6, 7]. Given a sufficient number of telepor-tation steps, this noise overwhelms the process entirely,and any trace of the input state is wiped out. Further, astrict no-go theorem establishes that no procedure basedon Gaussian operations can be used to correct this noise[52].

The finite-squeezing noise that builds up throughmacronode teleportation (as well as other types of noise)can be dealt with using bosonic codes [53–58], whichencode digital quantum information in the continuousHilbert space of a mode. We focus on the Gottesman-Kitaev-Preskill (GKP) encoding of qubit into a mode[31], whose states and structure are given below. Im-portant is the direct compatibility of the GKP code withthe macronode wire (as well as with more sophisticatedmacronode protocols [20]), due to the fact that encodedClifford gates and Pauli measurements are respectivelyrealized by Gaussian unitaries and homodyne detection.Further, Gaussian operations suffice for universality andfault tolerance [32, 33]. We show here that using GKPstates as ancillae at various locations in a macronode wireperforms GKP error correction, which projects built-upCV errors into potential logical errors.

A. The GKP encoding

For use in defining the GKP encoding, we first definea Dirac comb with spacing between spikes (period) of Tas

XT (s) :=√T

∞∑n=−∞

δ(s− nT ) . (60)

The normalization of a Dirac comb is convenient becauseits Fourier transform takes the form,

F [XT ](t) =

√2π

T

∞∑n=−∞

δ(t− n 2π

T

)= X2π/T (t) , (61)

9

with the Fourier transform defined as

F [f ](t) :=1√2π

∫ ∞−∞

ds e−istf(s). (62)

Scaling the argument of a Dirac comb by a real number agives a relation between Dirac combs of different period,

XT (as) = 1√aXT/a(s) , (63)

which follows using δ(as) = 1aδ(s).

The ideal square-lattice GKP encoding for a qubit con-sists of two computational-basis states given by

|jGKP〉 :=

∫dsX2

√π(s− j

√π)|s〉q (64)

= (2√π)1/2

∞∑n=−∞

|(2n+ j)√π〉q , (65)

with j ∈ {0, 1}. These states span a two-dimensional sub-space, allowing a GKP qubit to be encoded as |ψGKP〉 =c0 |0GKP〉+ c1 |1GKP〉. The position-space wavefunctionsfor the states |jL〉 are

ψj,GKP(s) = X2√π(s− j

√π), (66)

using the definition above for XT (see Fig. 3). Thisallows us to write the momentum-space wave functionsfor the computational basis states in Eq. (64) as

ψj,GKP(t) = F [ψj,GKP](t) = eij√πtX√

π(t), (67)

which are Dirac combs with half the period of those inEq. (66) and whose teeth have alternating phase for |1L〉.Therefore, we have the momentum expansion

|jGKP〉 =

∫dt eij

√πtX√

π(t)|t〉p (68)

= (√π)1/2

∞∑n=−∞

eijπn|n√π〉p (69)

From these expressions, position and momentum wave-functions for the dual-basis logical states, |±GKP〉 :=1√2

(|0GKP〉 ± |1GKP〉

), can be constructed. Using the

wavefunctions above, 〈jGKP|kGKP〉 ∝ δj,k, where the con-stant of proportionality is infinite. This is an artefact ofusing states with infinite norm. Physical approximationsto these states will not have such pathologies.

The two-dimensional projector onto the GKP subspaceis an important operator defined as

ΠGKP := |0GKP〉〈0GKP|+ |1GKP〉〈1GKP| , (70)

and can likewise be constructed from the dual-basis code-words |±GKP〉. Since the GKP eigenstates are a setof measure zero in the infinite-dimensional CV space,Eq. (70) is more appropriately a projector density sat-isfying (ΠGKP)2 ∝ ΠGKP with an infinite constant of

(a) (b)

Figure 3. Position wave functions for logical-0 (solid red) andlogical-1 (dashed blue) GKP states. (a) Ideal GKP states. (b)Approximate GKP states with squeezing sGKP = 18.6 dB.

proportionality. Nevertheless, in keeping with commonusage, we still refer to ΠGKP as a projector henceforth.7

A key feature of the square-lattice GKP code is thatthe Fourier transform operator F = R(π2 ) (a phase delay,Eq. (7), by π

2 ) executes a logical Hadamard gate withinthe codespace. This is a result of the choice of spikeperiod T = 2

√π for the position wavefunctions of the

computational-basis states, Eq. (66). By choosing a dif-ferent period, T =

√2π, we define another useful state,

|∅〉 :=

∫dsX√

2π(s)|s〉q =

∫dtX√

2π(t)|t〉p . (71)

The identical periodic structure in both position and mo-mentum means that |∅〉 is invariant under a Fouriertransform, F |∅〉 = |∅〉. This state was proposed inDuivenvoorden et al. [59] as a sensor to simultaneouslydetect small shifts in position and momentum. We focuson its quantum information properties and refer to |∅〉as a qunaught state (in analogy to and with similar pro-nunciation as a "qubit") due to the fact that it definesa one-dimensional subspace and carries no quantum in-formation.8 Nevertheless, qunaught states serve as theresource for error correcting the square-lattice GKP codein the teleportation gadget, Eq. (33), as we show below.

1. GKP Bell pairs

The GKP-encoded version of the qubit Bell state is

|Φ+GKP〉 := 1√

2

(|0GKP〉 ⊗ |0GKP〉+ |1GKP〉 ⊗ |1GKP〉

).

(72)This state can be constructed on a beam splitter usingtwo qunaught states, Eq. (71). First, we recognize that

7 In fact, similar behavior is seen for projectors onto position eigen-states: (|s〉qq〈s|)2 = δ(0)|s〉qq〈s|. These, too, are technically pro-jector densities.

8 One can, of course, define a rectangular GKP code for any spikespacing T in the computational-basis codewords. In this case,executing a Hadamard gate typically requires both a Fouriertransform and squeezing. Our focus here is the square-latticecode only.

10

a pure state can be expressed as

|ψ〉 =

∫ds ψ(s)|s〉q = ψ(q)

∫ds |s〉q =

√2πψ(q) |0〉p ,

(73)where ψ(s) is the position wavefunction and

∫ds|s〉q =√

2π|0〉p. Using this relation and the beam splitter trans-formation on the position operators, q± = 1√

2(q1± q2) in

Eq. (2), allows us to express the state as

B12 |∅〉 ⊗ |∅〉= 2πX√

2π(q+)X√2π(q−)|0〉p ⊗ |0〉p

=2π√

2

1∑j=0

X2√π(q1 + j

√π)X2

√π(q2 + j

√π)|0〉p ⊗ |0〉p

= |Φ+GKP〉 . (74)

The generation of this state is shown graphically in Fig. 4.The beam splitter rotates the two-mode state in the(q1, q2)-plane by π/4, and the result is described by a sumof two separate lattices, each with Dirac-comb-period2√π. The direct connection to a GKP Bell pair is found

using Eq. (74).

2. Approximate GKP states

Ideal GKP states, including the computational ba-sis states in Eq. (64), are unphysical, unnormalizablestates. Physical approximations to these states replaceeach spike in the position wavefunction [Eq. (66)] with asharp Gaussian, and then damp spikes far from the ori-gin with a broad Gaussian envelope.9 This can be mod-eled mathematically by applying the damping operator,Eq. (51), to the ideal states, which has the added benefitof treating the momentum wavefunction in an egalitarianway. The resulting approximate GKP states are

|jGKP〉 =1√Nj,GKP

N(β) |jGKP〉 . (75)

The damping spoils the strict orthogonality of the code-words, 〈0GKP|1GKP〉 6= 0, which is a generic featureof various other physical bosonic codes, such as catcodes [62]. Applying the damping operator to the GKPcodespace projector, Eq. (70), gives the (unnormalized)quasi-projector onto the subspace spanned by the ap-proximate codewords,

ΠGKP :=N(β)(|0GKP〉〈0GKP|+ |1GKP〉〈1GKP|

)N(β)

=N0,GKP |0GKP〉〈0GKP|+N1,GKP |1GKP〉〈1GKP| .(76)

9 There are many other ways to approximate ideal GKP states,see Ref. [60, 61] for more details.

(a) (b)

Figure 4. Generation of a GKP Bell pair by mixing twoqunaught states, Eq. (71), on a beam splitter. Shown in bothfigures is the two-mode position-position wavefunction, whereeach dot is a two-dimensional δ-function. (a) Product stateof two qunaughts, |∅〉 ⊗ |∅〉. (b) Encoded GKP Bell state,Eq. (74), generated after the qunaughts are entangled by thebeam splitter, which simply rotates the position-space wave-function shown in (a) by π/4 in the (q1, q2)-plane. This GKPBell state is a sum of two product states, |0GKP〉⊗|0GKP〉 and|1GKP〉 ⊗ |1GKP〉, indicated in red (solid) and blue (hollow),respectively.

In the limit of β → 0, this becomes the ideal GKP sub-space projector.10

For small damping β � 1, the normalized position-space wavefunction for these states is [60, 61]

ψj,GKP(s) ≈√

2

π1/4e−

κ2s2

2

∞∑n=−∞

e−(s−(2n+j)

√π)2

2∆2 , (77)

where the parameters that define the quality of a GKPstate, the spike variance ∆2 and the envelope varianceκ−2 [31], are related to the damping factor by

∆2 = κ2 = β. (78)

Figure 3 shows |0GKP〉 (solid) and |1GKP〉 (dashed)for β = 13.8 × 10−3 corresponding to GKP squeez-ing of sGKP = 18.6 dB by the relation: sGKP =−10 log10(∆2).11

B. Teleportation-based GKP error correction

The goal of GKP error correction is to eliminate CV-level noise that has corrupted an encoded state througha procedure that projects the noisy state back into theGKP codespace. Sources of noise include displacements[31], thermal noise including pure loss [63], dephasing[64], and inherent finite-squeezing noise that is particu-larly relevant in CVCS settings [7]. In Steane-style GKP

10 For β 6= 0, the operator ΠGKP is not a projector because(ΠGKP)2 6∝ ΠGKP. This is due to the non-orthogonality of theapproximate GKP codewords and is a separate issue from theinfinite constant noted in the discussion below Eq. (70).

11 Note that sGKP < 0 is squeezed and sGKP > 0 is anti-squeezed(with respect to vacuum).

11

error correction, the noisy encoded state is coupled toa GKP ancilla, which is measured in the q basis (usinghomodyne detection). The process is repeated with an-other GKP ancilla for the p basis. The two measurementoutcomes (referred to together as the syndrome) are fedinto a decoder that provides a recovery map consistingof shifts in position and momentum that, when activelyperformed, restore the GKP subspace and attempt tocorrect logical errors on the encoded qubit [31, 65].

In CVCS settings, Gaussian finite-squeezing noise ac-cumulates as an encoded state teleports from node tonode [6]. It was shown that fault tolerance in canon-ical CVCS quantum computing is possible by periodi-cally performing Steane-style GKP error correction [7].However, canonical CVCSs and associated GKP errorcorrection require two-mode entangling operations of theform eigq⊗q, which are hard to realize because they re-quire active squeezing. Experimentally accessible CVCSsare those used in macronode-CVCS QC, where the cou-pling requires only passive components—beam splittersand phase delays.

We show here that an alternate method, known asKnill-style CV error correction, proceeds by teleportingthe input state through two encoded ancillae—typicallyan encoded Bell pair [32]—performing the recovery mapautomatically. The syndrome is used purely to determinethe likelihood and type of logical-level error [58]. In fact,pre-measuring the |0GKP〉 ancillae used for Steane-styleerror correction in the canonical CVCS construction ofRef. [7] (and performing shifts) accordingly embeds themdirectly inline as a 2-qubit GKP cluster state. Teleport-ing through this inline resource state may then be inter-preted as Knill-style error correction. This enables twodifferent interpretations of the same physical setup fromRef. [7],

measure GKP modes

, (79)

either Steane-style (left) or Knill-style (right) error cor-rection. Here, empty circles are modes in squeezed mo-mentum states, non-empty circles are modes in GKPstates, and lines between modes are eiq⊗q gates.

Importantly, we show here that Knill-styleteleportation-based error correction is directly com-patible with macronode-based protocols, which useexperimentally accessible CVCSs. Further, we showthat the GKP states for error correction need not beplaced adjacent to one another in the cluster state, aflexibility that is crucial for experimental settings withprobablistically generated GKP states.

1. GKP error correction on the macronode wire

To demonstrate how the Kraus operator in Eq. (58)can be used for GKP error correction on the macron-ode wire, we first consider the ideal case, β = 0, with

qunaught states for both |ψ〉 and |φ〉. Using the qunaughtwavefunctions in the expression for the teleported gate,Eq. (40), gives

A(∅,∅) =

∫∫d2αX√

2π(αR)X√2π(αI)D(α) . (80)

This expression can be brought into a convenient formby decomposing the displacement operator into positionand momentum shifts with Eq. (27), recognizing thatthe resulting phase is trivial for qunaught wavefunctions,and then rewriting the two resulting integrals using theFourier relations,

ψ(q) =1√2π

∫dsF [ψ](s)Z(s) (81a)

F [ψ](p) =1√2π

∫dsψ(s)X(s) . (81b)

After these manipulations, the operator in Eq. (80) isrevealed to be the square-lattice GKP projector,

A(∅,∅) = π√

2X√π(p)X√

π(q) (82a)

= π√

2X√π(q)X√

π(p) (82b)

=

√π

2ΠGKP . (82c)

The final line is shown by expanding either of the previ-ous lines using XT (q) =

√T∑n δ(q − nT ), where

δ(q − nT ) = |nT 〉qq〈nT | , (83)

and an analogous expansion for XT (p). Then, with T =√π, we see that

X√π(q)X√

π(p) =1

2√π

ΠGKP , (84)

that is, these operators combine to form the GKP pro-jector [32] (up to an overall factor). The circuit diagramfor Eq. (82) is

��

|∅〉

|∅〉=

1√2ΠGKP

. (85)

Inserting Eq. (82) into the general form for the Krausoperator, Eq. (44), with β = 0 gives the Kraus operatorfor macronode-based GKP error correction,12

KGKP(ma,mb) =1√2π

ΠGKPD(µ)V (θa, θb). (86)

This expression shows that after the measurement-basis-dependent Gaussian unitary V (θa, θb) [Eq. (10)] andoutcome-dependent displacement D(µ), the state is pro-jected into the GKP subspace. The whole gadget can beinterpreted as teleporting through a square-lattice GKPBell pair [32], Eq. (72), with additional damping whenthe ancillae are imperfect, as we show below.

12 The factor preceding the GKP projector is a consequence of ourchosen normalization for ideal GKP states, Eqs. (65) and (69).

12

2. Approximate qunaught ancillae in the Kraus state

In physical settings, the qunaught ancillae containfinite-squeezing noise, which we model as damped idealqunaught states,

|∅〉 :=1√N∅

N(β) |∅〉 , (87)

with normalization factor,

N∅ := 〈∅| N(2β) |∅〉 , (88)

using the damping operator N(β) in Eq. (51). We thenuse the circuit identities above to write the beam splitterentangled state as joint damping on a GKP Bell pair,

��

|∅〉 N−1/2∅ N(β)

��

|∅〉

|∅〉=

N−1/2∅ N(β) |∅〉(89)

The macronode Kraus operator for this situation is ob-tained by making the replacement ,

ΠGKP →1

N∅N(β)ΠGKPN(β) =

1

N∅ΠGKP , (90)

in the ideal Kraus operator KGKP(ma,mb), Eq. (86), andΠGKP is given in Eq. (76).

The GKP error-correction Kraus operator transformsan arbitrary input pure state |Ψ〉 traversing the tele-portation gadget via the Kraus map in Eq. (34). Fordamped qunaught ancillae—that is, using the Kraus op-erator when Eq. (90) is used with Eq. (86)—results inthe GKP-encoded state,

|ΨGKP〉 = c0 |0GKP〉+ c1 |1GKP〉 (91)

whose qubit coefficients are given by

cj =1√

Pr(ma,mb)

Nj,GKP√2πN∅

〈jGKP| V (θa, θb)D(µ) |Ψ〉 ,

(92)

and the joint probability of obtaining the outcomes,Pr(ma,mb), is given by Eq. (35). The input state isactively projected into the GKP subspace, and the syn-drome information (the two homodyne measurement out-comes) is used by a GKP decoder to determine the like-lihood of a logical error. Effectively, this is a projectionof excess CV noise into GKP-logical noise. The benefitof a teleportation-based approach to error correction isthat the output state is already in the GKP subspace,meaning that corrections are purely logical operations(i.e., shifts by integer multiples of

√π). This differs from

Steane-style error correction, in which the corrections in-volve small displacements to realign the resulting statewith the GKP grid [31, 65]. For the case of no damping,β → 0, the ratio of normalization factors in Eq. (92) dis-appears, and the projection is onto the ideal, orthogonalGKP states |jGKP〉.

3. Partial GKP error correction

From Eqs. (82), we see that using qunaught statesfor both ancillae yields a separable operation: a pro-jection in

√π-periodic position followed by a projection

in√π-periodic momentum that together form the GKP

codespace projector. Interpreted logically, each of theseprojections corresponds to measurement and recovery ofone of the two GKP stabilizers. Here, we show that eachof these projections can take place separately, such thathalf of GKP error correction can occur at one macron-ode and the other half at another macronode—i.e. theperiodic q projection and then, later, the periodic p pro-jection. This is particularly useful for practical imple-mentations of GKP error correction using probabilisticsources of GKP states.

We consider two new cases, indicated graphically inFig. 5: (A) only mode a is prepared in a qunaught state,and (B) only mode b is prepared in a qunaught state. Thenon-qunaught mode is either a momentum-squeezed (A)or a position-squeezed (B) state following the standardmacronode procedure. The case (AB) where both an-cillae modes are qunaught states is considered above inSec. IVB1. The respective wave functions for the an-cillae, ψ(αI) and φ(αR), to be used in Eq. (40) for eachcase are given in the table in Fig. 5.

Case (A) gives

AGKP,A(ψ, φ) = 214√πX√

π(p) . (93)

Using Eq. (41), we get the following circuit identity:

��

|0〉p

|∅〉=

214X√

π(p). (94)

Using Eq. (44), this yields a Kraus operator that per-forms partial GKP error correction in the p quadrature,

KGKP,A(ma,mb) =2

14

√πX√

π(p)D(µ)V (θa, θb) . (95)

Case (B) is almost the same (with q ↔ p). Pluggingthe wave functions for case (B) into Eq. (40) gives

AGKP,B(ψ, φ) = 214√πX√

π(q) , (96)

whose circuit is

��

|∅〉

|0〉q=

214X√

π(q). (97)

This yields a Kraus operator that performs partial GKPerror correction in the q quadrature,

KGKP,B(ma,mb) =2

14

√πX√

π(q)D(µ)V (θa, θb) . (98)

13

local mode

beamsplitter

macronode

qunaught state

(A) (B) (AB)

a

b

Case (A) Case (B) Case (AB)

φ(αR) X√2π(αR) δ(αR) X√

2π(αR)

ψ(αI) δ(αI) X√2π(αI) X√

2π(αI)

Figure 5. Potential qunaught sites for GKP error correction.A macronode wire is depicted with the possible locations forqunaught states at either mode a (A) or b (B) within a tele-portation gadget. When qunaught states are placed at bothmodes, they comprise a GKP Bell pair shared between adja-cent macronodes (AB). The table shows the momentum andposition wave functions to be used in the Kraus operator,Eq. (44), for each case.

Each Kraus operator acts as a “filter” on the inputstate in either

√π-periodic momentum or position, re-

spectively. For measurement bases chosen such thatV (θa, θb) = I and displacements D(µ) at each macronodeaccounted for by active corrective shifts, the two partialGKP error corrections can occur at separated macron-odes. For ideal quadrature eigenstates and qunaughtstates, i.e., neither contains finite-squeezing noise, theresult is ideal GKP error correction.

We model physical states at the ancillae with dampedqunaught states, Eq. (87), and squeezed states describedby Eq. (52) as approximate quadrature eigenstates. Forboth types of state, we assume that the damping is equalso the damping operators can be extracted and the re-maining qunaught and quadrature eigenstates treated asideal.

The circuit identity for case (A) is

��

|0; ζ〉p N−1/2ζ N(β)

��

|0〉p

|∅〉=

N−1/2∅ N(β) |∅〉(99)

and the Kraus operator performs partial GKP error cor-rection in the p quadrature,

X√π(p)→ 1√

N∅NζN(β)X√

π(p)N(β) . (100)

Similarly, for case (B), the circuit identity is

��

|∅〉 N−1/2∅ N(β)

��

|∅〉

|0; ζ〉q=

N−1/2ζ N(β) |0〉q(101)

and the associated Kraus operator performs partial GKP

error correction in the q quadrature,

X√π(q)→ 1√

N∅NζN(β)X√

π(q)N(β) . (102)

The benefit of performing full GKP error correction util-ising a GKP Bell pair is simply that there is no buildupof finite-squeezing noise (and potential external noise)between the two partial GKP error corrections.

V. APPLICATION: GKP ERRORCORRECTION IN MACRONODE LATTICES

Above we analyzed the teleportation-gadget Kraus op-erator when one or more modes are prepared in theGKP qunaught state |∅〉. By combining this operatorwith the usual measurement-based evolution with Gaus-sian resource states, we can describe a hybrid resourcestate made up of both Gaussian and non-Gaussian parts.These results can be generalized to other CV clusterstates made from two-mode CV cluster states and 50:50beam splitters.

All multidimensional macronode CV cluster states pro-posed in Refs. [2, 3, 5, 14, 15, 17] can be constructedby arranging two-mode cluster states in a higher dimen-sional geometry and then producing a fully connected re-source by applying a local constant-depth circuit of 50:50beam splitters. In all of these cases, the resulting latticescontain embedded macronode wires that are used to im-plement single-mode gates. Said another way, each suchcluster state is equivalent to a collection of macronodewires coupled via additional 50:50 beam splitters. Bychoosing homodyne measurements appropriately, someof these beam splitters can be “cancelled out,” producinglocal regions in the lattice that are literally equivalentto a macronode wire. Details can be found in the ref-erences above. The upshot is that teleportation-basedGKP error correction, described in Sec. IVB, has appli-cations in all of the higher dimensional macronode-basedarchitectures known to date. This combines the single-mode Gaussian gates and GKP error correction of a sin-gle macronode wire with the ability to implement mul-timode Gaussian unitary gates, resulting in a universalset of resources [32] via passive linear optics and offlinepreparation of squeezed vacuum and GKP |∅〉 states.

VI. CONCLUSION

We have presented the Kraus operator for CV gateteleportation based on teleporting through a class of en-tangled states that are prepared on a beam splitter. Theteleported gate depends on the ancilla states at the beamsplitter, which themselves can be non-Gaussian, enablingimplementation of teleported non-Gaussian operations.Our analysis focuses on the macronode wire, but it is di-rectly mappable onto higher-dimensional macronode CV

14

cluster states because all known macronode cluster statesare equivalent to a collection of macronode wires coupledvia additional 50:50 beam splitters.

We use the derived formalism to propose ateleportation-based scheme for GKP error-correction re-quiring only constant-depth linear optics to implement,removing the requirement of previous work on activesqueezing operations. Further, our GKP error-correctionscheme is compatible with nondeterministic state prepa-ration [? ]. This simplifies the implementation of prac-tical GKP error correction—the final ingredient neededby CV cluster state architectures for universal and fault-tolerant computation—bringing it in line with the exper-imentally available resources.

ACKNOWLEDGMENTS

This work was supported by the Australian Re-search Council Centre of Excellence for Quantum Com-putation and Communication Technology (Project No.CE170100012). R.N.A. is supported by National ScienceFoundation Award No. PHY-1630114.

Appendix A: Entangling quadrature eigenstates on abeam splitter

We start with the LDU and UDL decompositions of arotation matrix R(θ) valid for θ 6= nπ/2 for odd n,

R(θ) =

(cos θ − sin θsin θ cos θ

)=

(1 0

tan θ 1

)(cos θ 0

0 sec θ

)(1 − tan θ0 1

)(A1a)

=

(1 − tan θ0 1

)(sec θ 0

0 cos θ

)(1 0

tan θ 1

). (A1b)

Each decomposes R(θ) into a product of symplectic ma-trices describing a shear, squeezing, and another shear.

Decomposing the symplectic representation of theHeisenberg action of the beam splitter Bjk that acts onthe vector of quadratures (q1, q2, p1, p2)T [20], namely(

R(π4 ) 00 R(π4 )

)

=

1 0 0 01 1 0 00 0 1 −10 0 0 1

1√2

0 0 0

0√

2 0 0

0 0√

2 00 0 0 1√

2

1 −1 0 0

0 1 0 00 0 1 00 0 1 1

,

(A2)

yields, up to an overall phase, a unitary decompositionof the beam splitter:

B12 = e−iq1⊗p2 [S†1(√

2)⊗ S2(√

2)]eip1⊗q2 . (A3)

We can write this as a circuit identity (again, up to anoverall phase),

��=

• S†(√

2)

S(√

2)

, (A4)

where the two-mode gates are each a CV controlled-Xgate, Eq. (19), of weight g = ±1, with the open circleindicating weight −1.13

Consider momentum and position eigenstates, |t〉p and|s〉q, coupled by a beam splitter. Using the above decom-position, the resulting entangled state is

B12 |t〉p1⊗ |s〉q2 = eiste−iq1⊗p2 S†1(

√2) |t〉p1

⊗ S2(√

2) |s〉q2=√

2eiste−iq1⊗p2 |√

2t〉p1⊗ |√

2s〉q2=√

2eist |EPR(√

2s,√

2t)〉 , (A5)

again up to a fixed phase. The norm scaling by√

2 arisesfrom the action of the squeezing operator on a quadratureeigenstate:

S(ζ)|s〉q = |ζ|1/2 |ζs〉q , (A6a)

S(ζ)|t〉p = |ζ|−1/2 |ζ−1t〉p , (A6b)

where S(ζ) is defined in Eq. (13). The prefactor ensuresthat inner products are preserved and that projectorsonto squeezed quadrature eigenstates resolve the single-mode identity when integrated over s (or t).

Appendix B: Derivation of the gates for rotatedhomodyne measurements

The results of the previous Appendix can be general-ized to the case where eigenstates of rotated quadratures,such as those in Eq. (8), are coupled on a beam split-ter. The resulting state is equivalent to a set of single-mode Gaussian operations and a displacement acting onan EPR state. Important for our purposes is the factthat the Hermitian conjugate of this state gives a descrip-tion of the two-mode entangled homodyne measurementin the macronode teleportation gadget, Eq. (33). Theoperations on the EPR state are indeed those that al-low universal Gaussian quantum computation with stan-dard (Gaussian) macronode cluster states: the operatorV (θa, θb) in Eq. (10), as well as a displacement D(µ).Here, we give a derivation of these operations with spe-cific attention to how they arise in this measurement. Inthis context, note that they arise solely from the choice

13 Again, notice that the right-to-left circuit convention makes suchconversions straightforward: Eqs. (A2), (A3), and (A4) are allordered the same way.

15

of homodyne measurement angles and outcomes and aredistinct from the gate teleported by the Kraus state (al-though we use the same formalism in its derivation).

Rotated momentum and position quadratures,

pθ := R†(θ)pR(θ) = sin θq + cos θp , (B1)

qθ := R†(θ)qR(θ) = cos θq − sin θp , (B2)

have respective eigenstates pθ|t〉pθ = t|t〉pθ and qθ|s〉qθ =

s|s〉qθ . We express these states in a convenient way usingthe decompositions of the rotation operator in Eqs. (A1),

|t〉pθ := R†(θ)Z(t)|0〉p (B3)

= D†(− is√

2e−iθ

)R†(θ)|0〉p (B4)

=√| sec θ|D†

(− it√

2e−iθ

)P (− tan θ)|0〉p , (B5)

|s〉qθ := R†(θ)X(s)|0〉q (B6)

= D†(− s√

2e−iθ

)R†(θ)|0〉q (B7)

=√| sec θ|D†

(− s√

2e−iθ

)Pp(tan θ)|0〉q. (B8)

In the final line of each expression are unitary shear op-erators, defined respectively as

P (σ) := eiσ2 q

2

(B9)

Pp(σ) := e−iσ2 p

2

. (B10)

Note that each shear does nothing to a particular 0-eigenstate, P (σ)|0〉q = |0〉q and Pp(σ)|0〉p = |0〉p, whichallowed for the simplifcations in Eqs. (B5) and (B8). Thefactor in front of each expression arises from the action ofsqueezing operators on quadrature eigenstates, Eq. (A6).

We now focus on the rotated measurements in themacronode teleportation gadget, Eq. (33), that are writ-ten as a projection onto a two-mode entangled state,

pθa〈ma|

��pθb〈mb|

. (B11)

As the ultimate goal is to write this expression in termsof an EPR state of the form in Eq. (30) , we include inthe rotation angle on the top mode an additional π/2, sothat it describes a rotation from the q axis of that mode,

θ′a := θa − π/2 . (B12)

For convenience, we take the Hermitian conjugate of theexpression (which reverses the direction of the beam split-ter), to get

R†(θ′a) |ma〉qOO

R†(θb) |mb〉p. (B13)

We split each phase delay operator into symmetric andanti-symmetric parts, R(θ) = R(θ+)R(θ−), with

θ′± :=θ′a ± θb

2, (B14)

since the common delays (rotations) commute with anybeam splitter,

R†(θ′a)OO

R†(θb)

=R†(θ′+) R†(θ′−)

R†(θ′+)

OO

R†(−θ′−)

(B15)

Finally, we extract the displacements from each statein Eq. (B13), commute them past the beam splitter, anduse Eqs. (B5) and (B8) to express the remaining rotatedquadrature eigenstates as sheared eigenstates,

R†(θ′+) D†(−β+)√| sec θ′−|Pp(tan θ′−) |0〉q

R†(θ′+) D†(β−)

OO

√| sec θ′−|P (tan θ′−) |0〉p

(B16)where the displacement amplitudes are

β± :=1

2(mae

−iθ′− ± imbeiθ′−) . (B17)

We now recognize that the right-hand side of the circuitis a Kraus state of the form in Eq. (41) with the ancillaegiven by

|ψ〉 =√| sec θ′−|P (tan θ′−)|0〉p , (B18)

|φ〉 =√| sec θ′−|Pp(tan θ′−)|0〉q . (B19)

We can now directly evaluate the teleported gate thatthey enact using Eq. (40). This requires obtaining wave-functions for |ψ〉 and |φ〉. The following are easy to eval-uate directly using the forms above:

ψ(s) := q〈s|ψ〉=1√

2π| cos θ′−|ei2 (tan θ

′−)s

2

, (B20)

φ(t) := p〈t|φ〉=1√

2π| cos θ′−|e−

i2 (tan θ

′−)t

2

. (B21)

Taking the appropriate Fourier transforms, Eq. (62), weobtain the other two, which we need for Eq. (40):

ψ(t) = F [ψ](t) =1√

2π| cos θ′−|ei2 (cot θ

′−)t

2

, (B22)

φ(s) = F−1[φ](s) =

1√2π| cos θ′−|

e−i2 (cot θ

′−)s

2

. (B23)

Substituting these wavefunctions into the teleportedgate, Eq. (40) (t = αI , s = αR), gives the Weyl repre-sentation of some operator O,

O =1

2π| cos θ′−|

∫d2α e−

i2 (cot θ

′−)(α

2R−α

2I)D(α) (B24)

with characteristic function [66],

Tr[OD†(α)] =1

2| cos θ′−|e−

i2 (cot θ

′−)(α

2R−α

2I) . (B25)

16

We now show that the rotated squeezing operator O =R†(π4 )S(ζ)R(π4 ), yields the same characteristic function,and we find the squeezing ζ as a function of θ′−. Evalua-tion of this characteristic function,

Tr[R†(π4 )S(ζ)R(π4 )D†(α)] = Tr[S(ζ)D†(αeiπ4 )] , (B26)

with the trace taken in the position basis gives∫dt q〈t|S(ζ)D†

(αei

π4

)|t〉q =

√|ζ|

|1− ζ|e−

i2

1+ζ1−ζ (α

2R−α

2I) .

(B27)Since the functional form of this expression is the same asthat in Eq. (B25), we can read off the relations betweenparameters,

cot θ′− =1 + ζ

1− ζ−→ ζ = − tan

(θ′− − π

4

). (B28)

Thus, we have shown that, for the Kraus state on theright-hand side of the circuit in Eq. (B16), the teleportedgate is a π

4 -rotated squeezing operation.Combining this result with the remaining rotations and

displacements, Eq. (B16) becomes

R†(θ′+) D†(−β+) R†(π4 ) 1√πS(ζ) R(π4 )

R†(θ′+) D†(β−)

,

(B29)Note that the teleported gate is on the top wire here dueto the beam splitter orientation in the state, Eq. (B13).Bouncing all the operators to the top wire using rulesdescribed in Eqs. (24) and (26), and then combining thetwo displacements reduces this to

R†(θ′+ + π4 ) 1√

πS(ζ) R†(θ′+ − π

4 ) D†(µ),

(B30)where the final displacement amplitude µ can be de-rived by bouncing all operators to the top modeand commuting both displacements to the end, giv-ing µ = −eiθ

′+ [β+ cosh (− ln ζ)− iβ∗+ sinh (− ln ζ) + β∗−].

The phase eiθ′+ at the front arises from bouncing and

then commuting the final R†(θ′+) through all displace-ments, the β∗− results from bouncing the D†(β−) to thetop wire, and the remaining terms result from commut-ing D†(−β+) through the rotated squeezing operator. Inthe end, µ evaluates to

µ = −maeiθb +mbe

iθa

sin(θa − θb). (B31)

Writing Eq. (B30) in terms of (combinations of) theoriginal measurement angles using Eqs. (B12) and (B14)gives the equivalent circuit,

1√πV †(θa, θb) D†(µ)

, (B32)

where we define the operator [Eq. (10)]

V (θa, θb) := R(θ+ − π2 )S(tan θ−)R(θ+) , (B33)

with θ± = 12 (θa ± θb) [Eq. (12)]. Note that we have

ignored an overall phase in this derivation, which resultsfrom combining the two displacements into µ.

Taking the Hermitian conjugate of the circuit inEq. (B32) gives a description of the two-mode entangledmeasurement in Eq. (B11),

pθa〈ma|

��

D(µ) 1√πV (θa, θb)

pθb〈mb|

= . (B34)

This circuit contains the operators used in the main textand those in Refs. [1, 23, 46? ]. Note that when theKraus state in the full teleportation gadget is nontrivial,then the outcome-dependent displacement D(µ) shouldbe commuted past the teleported gate A(ψ, φ) in orderto determine the necessary correction at the end of thegadget.

We present several variations of the circuit in Eq. (B34)by commuting the outcome-dependent displacement todifferent places among the other operators. For the casewhere the displacement acts first, we find

1√πV (θa, θb) D(µ′)

(B35)

with amplitude

µ′ =−mae

−iθb +mbe−iθa

sin(θa − θb). (B36)

We now discuss a different final form in order to com-plete the connection to the literature [6, 16]. Pullingout a Fourier transform F † = R(−π2 ) in Eq. (B33),V (θa, θb) = F †V ′′(θa, θb), we now consider the case wherethe displacement lies between the two other gates,

F † D(µ′′) 1√πV ′′(θa, θb)

(B37)

and has amplitude

µ′′ =−imae

iθb − imbeiθa

sin(θa − θb). (B38)

This form is suitable for comparison with Refs. [6, 16],as V ′′(θa, θb) and D(µ′′) agree with those works up toone difference: their circuit does not contain the Fouriertransform F †.14 This is because teleportation gadget in

14 Note that Ref. [6] has an error in the denominator of the dis-placement operator: it appears in that reference as sin θ−, whilethe correct quantity is sin(θa − θb).

17

those works uses a two-mode canonical CV cluster stateas its Kraus state, |CVCS〉 := eiq⊗q|0〉p ⊗ |0〉p, which isrelated to the canonical EPR state by R(π4 ) on each mode(or, equivalently, F on either mode). Since q = −F pF †,this state can be represented with the following circuitidentity,

• |0〉p • |0〉p

• |0〉p=

F F † |0〉p. (B39)

Acting with the Fourier transform gives F †|0〉p = |0〉q,and using Eq. (20), the resulting circuit is

|CVCS〉 :=• |0〉p

• |0〉p=

1√2πF

, (B40)

recovering the well-known fact that teleportation withcanonical CV cluster states is accompanied by a Fouriertransform [12]. Thus, when using this Kraus statein conjunction with the measurement as described byEq. (B37), the two Fourier transform operators cancelin the final Kraus operator, F F † = I.

Appendix C: Glossary of circuit identities

Here, we provide a succinct summary of the circuit identities in the main text. Note that all circuits are to be readright (input) to left (output).

• Canonical maximally entangled EPR state, Eq. (20):

• |0〉p|EPR〉 :=

|0〉q=

1√2πI

(C1)

• Canonical two-mode CV cluster state, Eq. (B40):

|CVCS〉 :=

• |0〉p

• |0〉p=

1√2πF

(C2)

• Square-lattice GKP Bell state, Eq. (85):

|Φ+GKP〉 =

��

|∅〉

|∅〉=

1√2

ΠGKP(C3)

with qunaught state |∅〉, Eq. (71), and GKP projector ΠGKP, Eq. (70).

• Kraus state for a partial GKP projector (p quadrature), Eq. (94):

��

|0〉p

|∅〉=

214X√

π(p)(C4)

• Kraus state for a partial GKP projector (q quadrature), Eq. (97):

��

|∅〉

|0〉q=

214X√

π(q)(C5)

18

• Decomposition of the beam splitter, Bjk in Eq. (1), in terms of Gaussian gates, Eq. (A4), using notationintroduced in Eq. (4):

��=

• S†(√

2)

S(√

2)

(C6)

• Bouncing operations from one mode to another through an EPR state, Eq. (23):

O=

OT

(C7)

with the transpose taken in the position basis.

• State produced by mixing a t-momentum and s-position eigenstate on a beam splitter, Eq. (31):

��

|t〉p

|s〉q=

1√πD(s+ it)

(C8)

• Entangled two-mode measurement of rotated quadratures with outcomes ma and mb, Eq. (36):

pθa〈ma|

��

D(µ) 1√πV (θa, θb)

pθb〈mb|

= (C9)

with Gaussian unitary V (θa, θb), Eq. (10), and outcome-dependent displacement with amplitude µ, Eq. (11).

• Beam splitter-entangled Kraus state for identical local damping on the ancillae, Eq. (57):

��

N(β) |ψ〉

N(β) |φ〉=

N(β) 1√πA(ψ, φ) N(β)

(C10)

for damping operator N(β) in Eq. (51). This state can be normalized with factor (NφNψ)−1/2.

• Gate-teleportation gadget with damped ancillae comprising the Kraus state, Eq. (58):

pθa〈ma|

��

(in)

pθb〈mb|

��

N(β) |ψ〉

(out) N(β) |φ〉

= (out) N(β) 1√πA(ψ, φ) N(β) D(µ) 1√

πV (θa, θb) (in) (C11)

• Case (AB)—state produced by mixing two damped qunaught states, Eq. (87), on a beam splitter, Eqs. (85) and(89):

��

|∅〉 N−1/2∅ N(β)

��

|∅〉

|∅〉=

N−1/2∅ N(β) |∅〉=

N−1/2∅ N(β)1√2ΠGKP N−1/2∅ N(β)

(C12)

The final form uses Eq. (C3) and (C7) above.

19

• Case (A)—state produced by mixing a momentum-squeezed state, Eq. (52), and a damped qunaught state ona beam splitter, Eqs. (95) and (99):

��

|0; ζ〉p N−1/2ζ N(β)

��

|0〉p

|∅〉=

N−1/2∅ N(β) |∅〉=

N−1/2∅ N(β) 214X√

π(p) N−1/2ζ N(β)

(C13)

The final form uses Eq. (C4) and (C7) above.

• Case (B)—state produced by mixing a damped qunaught state and a position-squeezed state, Eq. (52), on abeam splitter, Eqs. (98) and (101):

��

|∅〉 N−1/2∅ N(β)

��

|∅〉

|0; ζ〉q=

N−1/2ζ N(β) |0〉q=

N−1/2ζ N(β) 214X√

π(q) N−1/2∅ N(β)

(C14)

The final form uses Eqs. (C5) and (C7) above.

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