Furusawa A., van Loock P. Quantum Teleportation and Entanglement: A Hybrid Approach to Optical Quantum Information Processing

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270 6 Quantum Teleportation of Gates

Formally, we may also interpret some of the schemes in Figure 6.16 as gate tele-

portations for arbitrary qumode gates U using nonlinear operations on linear re-

sources, where a Gaussian two-mode squeezed state is used as an EPR source and a

nonlinearly modiﬁed, two-mode Bell measurement is employed (see Figure 6.17).

For example, the scheme in Figure 6.17a can be formally described as

(U)

in,1

(β)

jψ

i˝

nD0

jn, ni

1,2

†

(β)Ujψ

i , (6.23)

using the deﬁnitions given earlier in this section and

(U)

(β)  (U

†

˝ 1)

Π(β)

(U ˝ 1), describing a projection onto the basis f(U

†

˝ 1)jΦ (β)ig for a ﬁxed U.

As such collective nonlinear measurements are not known to be experimentally

available, there seems to be no advantage of the schemes in Figure 6.17 besides

the conceptual insights they provide. The scheme in Figure 6.17a does not require

any correction operations other than displacements because it is equivalent to that

in Figure 6.16a where the gate U is trivially applied through teleportation by ﬁrst

performing U on jψ

i and then teleporting Ujψ

i using standard CV quantum

teleportation.

271

Cluster-Based Quantum Information Processing

In the preceding chapter, we introduced the concept of gate teleportation as an ex-

tension of state teleportation, and we presented various experiments that realized

gates through teleportation on qubits or qumodes. Similar to these gate telepor-

tations, the cluster-based schemes and experiments described in this chapter are

again m anifestations of the paradigm of measurement-based quantum computa-

tion where the unitary gates of the standard circuit model [5] are replaced by irre-

versible sequences of measurements [1] on an ofﬂine prepared resource state (see

Section 1.8 and Figure 1.11).

However, as mentioned before in the introduction to Chapter 6, cluster-based

quantum computing represents the ultimate form of measurement-based quan-

tum information processing: measurements are performed locally on the smallest

subsystems of the cluster resource state, that is, the individual qubits or qumodes

that form the cluster; the cluster itself embodies the maximal ofﬂine resource need-

ed because once it has been prepared, it remains ﬁxed during the computation with

measurements on it still offering sufﬁcient freedom to realize universal and hence

arbitrary quantum gates.

The locality of the measurements means that every entangling gate required for a

multi-qubit or multi-qumode computation is already embedded in the cluster state.

This is particularly useful when such entangling gates can only be probabilistically

implemented as in the single-photon-based approaches.

In this chapter, we will ﬁrst explain cluster computation on qubits (Section 7.1),

including the elementary teleportation steps needed (Section 7.1.1), their use for

Clifford and for universal quantum computing, and an experimental demon-

stration of a small cluster computation (Section 7.1.2). The analogous qumode

schemes are presented in Section 7.2, similarly divided into Section 7.2.1 on el-

ementary qumode teleportations, Section 7.2.2 on Gaussian computations, and

Section 7.2.3 describing various experiments in the CV Gaussian regime of cluster-

based quantum information processing. Gaussian regime here means that all

these experiments so far have demonstrated only Gaussian transformations using

cluster states.

An experiment that incorporates a non-Gaussian measurement into the cluster

computation and thereby leaves the regime of Gaussian gates as required for uni-

versal quantum computation is yet to be performed. Nonetheless, the experiment

Quantum Teleportation and Entanglement. Akira Furusawa, Peter van Loock

ISBN: 978-3-527-40930-3

272 7 Cluster-Based Quantum Information Processing

presented in Section 7.2.3.2 can be seen as an almost ultimate demonstration of

single-mode Gaussian cluster computation as it implements Fourier and squeez-

ing gates on independent, single-qumode optical states through a linear four-mode

cluster state which is universal for arbitrary single-qumode LUBO transformations.

7.1

Qubits

A multi-qubit cluster state is illustrated in Figure 1.11. Let us see how an elemen-

tary teleportation step between two adjacent qubits works.

7.1.1

Elementary Qubit Teleportations

Consider the qubit circuit in Figure 7.1. The output state of that circuit is

jψi (7.1)

for an arbitrary input state jψi. In other words, up to a Pauli correction operation

depending on the measurement result m for the measurement on the upper qubit

and up to a Hadamard gate, the resulting state is Z

jψi. For arbitrary angles θ,

arbitrary Z rotations are applied upon the input state.

Now the crucial point is that, instead of the unitary circuit shown in Figure 7.1a,

the unitary operator that describes the desired Z-rotation gate can be absorbed in-

to the measurement apparatus as shown in Figure 7.1b. This way any Z rotation

can be enacted by choosing a suitably rotated measurement basis. By concatenat-

†

(a)

(b)

Figure 7.1 Elementary “one-bit” teleportation

circuit for qubit cluster computation. (b) the

gate represents a horizontal edge connect-

ing two nodes of the cluster state. An input

state jψi is teleported into the left node, and

a subsequent, local single-qubit measurement

in the binary basis fZ

†

j˙ig leaves the second

node, up to a Hadamard gate H and a Pauli

correction X depending on result m D 0, 1,

in the unitarily evolved state Z

jψi,withthe

rotation angle θ for a Z-rotation exp(iθ Z /2)

controlled by the actual choice θ for the mea-

surement basis. (a) shows the equivalent

circuit in which the measurement is decom-

posed into unitary gates and a computational

basis measurement.

7.1 Qubits 273

XZHX

Figure 7.2 Two cascading elementary qubit teleportations.

ing these elementary teleportation steps, one can apply sequences of single-qubit

rotations such as for two steps (see Figure 7.2)

jψi . (7.2)

Now, it depends on the commuting properties of the rotation gate (if it is Clifford

or not, see Section 1.8) as to whether and how the Pauli corrections can be applied.

Assume the ﬁrst measurement result is m

D 1. In this case, we have

θ

X D XZ

(7.3)

such that the desired gate Z

ends up in front of the state vector jψi only provided

the actual measurement basis was chosen different from Z

, namely, according

to a rotated basis with Z

θ

. Whether or not this basis change is necessary in the

second teleportation step depends on the measurement outcome of the ﬁrst step.

Therefore, in general, feedforward is required, where the later measurement bases

must be adjusted according to the earlier measurement outcomes. However, there

is an exception. Whenever the rotations are Clifford rotations, they would simply

commute through the Pauli correction operators; in which case only the required

Pauli correction would change and no more the subsequent measurement bases

(see Chapter 1). In this case, no feedforward is needed, and all measurements can

be performed at any time and in any order, even in parallel – a notion known as

Clifford parallelism.

7.1.2

Experiment: Qubit Cluster Computation

One of the most famous experiments of qubit cluster computation would be the

one performed by Walther et al. [149] which uses the cluster states explained in Sec-

tion 3.1.3.3. Since the cluster states are created by post-selection, this experiment

also works with post-selection. Moreover, there is no feedforward in the experi-

ment. Only those events with no need for feedforward are post-selected as shown

later. In this sense, it is a partial demonstration of cluster computation because

feedfoward is one of the most important features of cluster computation. In this

section, we will ﬁrst explain this experiment in detail.

Walther et al. performed two types of experiments which are cascaded single-

qubit operations and two-qubit quantum logic gates [149]. For the cascaded opera-

tion experiment, they performed two or three cascaded operations with a four-mode

274 7 Cluster-Based Quantum Information Processing

linear cluster state. These states are given in Eq. (3.25) in polarization qubit repre-

sentation and in Eq. (3.26) in the computational basis.

For two cascaded operations, they ﬁrst made a polarization measurement of the

photon in mode 1 in the fjCi

, ji

g basis (˙45

-linearly polarization) to pre-

pare a three-mode linear cluster state of modes 2–4. Of course, one should make

ameasurementinthefj0i, j1ig basis (computational basis) to make the ﬁrst qubit

of the linear four-mode cluster state disentangled and the rest intact. However, in

this particular experiment, they used “lab basis” which needs an extra Hadamard

transformation (jCi • j$i D jHi, ji • jli D jV i)withquarterwaveplates.

Thus, the measurement on fjCi

, ji

g basis here corresponds to a measurement

on the computational basis. They only took the case of jCi

and discarded the case

of ji

.Thisactioneliminatestheneedforfeedforward.

To make two cascaded operations, polarization measurements on B

(α) basis are

made in modes 2 and 3 where B

(α) Dfjα

, jα



g and jα

D (e

iα/2

j0i

iα/2

j1i

Here, only the cases of jCαi

were post-selected in the exper-

iment to obtain the input jCi

and to eliminate the feedforwards. The B

(α)-

basis measurement corresponds to single-qubit rotation around z-axis (j0i–j1iaxis)

(α) D exp(iασ

/2) followed by a Hadamard operation H D (σ

C σ

where fσ

, σ

g are the Pauli matrices.

Thus, the post-selected state of mode 4

(qubit 4) jψ

out

after the measurements becomes

jψ

out

D HR

(α

)HR

(α

)jCi

D R

(α

)jCi , (7.4)

where R

(α) D HR

(α)H D exp(iασ

/2). This sequence of two elementary

teleportations is an example of the general concatenation shown in Figure 7.2.

Again, in the experimental scenario, feedforward was not needed because those

cases without feedforward were post-selected.

Walther et al. performed the experiment with α

D π/2, π/4, 0 and α

D π/2,

which was realized by tuning of quarter wave plates and polarization angles of

polarizers just before the detectors in modes 1–4 (Figure 3.10). The results are

shown in Figure 7.3. The ﬁdelity of operations with the ideal cases were 0.86˙0.03,

0.85 ˙ 0.04, and 0.83 ˙ 0.03 for α

D π/2, π, and /4, 0, respectively [149].

Two-qubit-gate experiments were performed as follows. As explained in Sec-

tion 3.1.3.3, a four-qubit square cluster state can be created using post-selection.

Walther et al. performed a two-qubit gate operation with the four-mode square clus-

ter state where the gate operation is shown in Figure 7.4.

In this experiment, the input state is jCi

˝jCi

which is automatically post-

selected i n the experiment. The two-qubit gate operation performed by polarization

measurements of photons (qubits) in modes 1 and 4 of the square cluster state is

1) In [149], the deﬁnitions of B

(α)andj˙αi

are B

(α) DfjCαi

, jαi

g and

j˙αi

D (j0i

˙e

iα

j1i

2) Identical to fX, Y, Zg according to the alternate notation used throughout this book. Similarly,

(α)  Z

, etc. because here we choose to use the same notation as in [149].

7.1 Qubits 275

(a) (b)

Figure 7.3 The state of qubit 4 in mode 4 a f-

ter the measurements of qubits in modes 1–

3 [149]. Here, “lab basis” is used for mode 4,

which makes additional Hadamard operation

on the state of Eq. (7.4). The states of qubit 4

with α

D π/2, π/4, 0 are labeled 1, 2, and

3 in this ﬁgure, respectively, and α

D π/2.

(a) depicts the ideal case, and (b) depicts the

experimental results.

Controlled-phase gate

Figure 7.4 Scheme of two-qubit gate operation with a four-mode square cluster state demon-

strated by Walther et al. [149]. Polarization measurements were performed in modes 1 and 4.

the following:

jψ

out

D CPhase(H ˝ H)[R

(α

) ˝ R

(α

)]CPhasejCi˝ jCi , (7.5)

where CPhase is the controlled-phase-gate operation (identical to C

,seeSec-

tion 1.8),

CPhasejxi˝jyiD(1)

jxi˝jyi ,(x, y D 0, 1) . (7.6)

Note that the CPhase acts as an entangling gate in the ﬁrst operation for separable

inputs, but it acts as a disentangling gate in the second operation as in Figure 7.4

and Eq. (7.5). Experimentally, the case of α

D π and α

D 0 should result in the

operation jCi

˝jCi

!jCi

˝ji

. The corresponding results are shown in Fig-

ure 7.5a. Note the “lab basis” was used and jCi

˝ji

corresponds to jV i

˝jHi

where jV iDjliand jHiDj$ibecause modes 2 and 3 are switched in the “lab

basis” of this experiment. In Figure 7.5a, only the diagonal element of VH ap-

pears, consistent with the theoretical predictions and showing that the experiment

is successful depending on postselection.

Similar to the case of the square cluster state, a two-qubit gate operation can be

performed with a four-mode linear cluster state. In this case, we make measure-

276 7 Cluster-Based Quantum Information Processing

(a) (b)

Figure 7.5 Experimental density matrices

of two-qubit gate operation with four-mode

square and horseshoe cluster states demon-

strated by Walther et al. [149]. The “lab basis”

is used. (a) results for the four-mode square

cluster state, (b) for the four-mode horseshoe

cluster state. Upper charts are for the real

parts of density matrices and lower charts are

for imaginary parts. The outputs for (a) and

(b) are a separable and entangled states, re-

spectively, and it is consistent with the theory.

Figure 7.6 Graph representation of a four-qubit horseshoe cluster

state.

ments of qubi ts 2 an d 3 and the four-mode linear cluster state is regarded as a

horseshoe cluster state shown in Figure 7.6. Here, the expression of the operation

with this state is

jψ

out

D (H ˝ H)[R

(α

) ˝ R

(α

)]CPhasejCi˝ jCi , (7.7)

as shown in Figure 7.7.

The important thing here is that the output state jψi

is an entangled state

while the output state jψi

from the operation in Eq. (7.5) is not an entangled

state. Experimentally, they tried the case of α

D α

D 0(bothmeasurement

results are jCi D C45

linear polarization) which should make the operation

jCi

˝jCi

! (j$i

˝jCi

Cjli

˝ji

2 D (j0i

˝jCi

Cj1i

˝ji

The experimental results are shown in Figure 7.5b. Note again that the “l ab basis”

was employed, requiring extra Hadamard operations in modes 1 and 4. The exper-

7.1 Qubits 277

Controlled-phase gate

Figure 7.7 Scheme of two-qubit gate operation with a four-mode horseshoe cluster state

demonstrated by Walther et al. [149]. Polarization measurements were performed in modes 2

and 3.

Photon 1

Linear

cluster

‘1’

‘0’

PBS

±α

Delay

150 ns

EOM 1 adapts

measurement basis

EOMs 1 and 3

apply error correction

Delay

300 ns

‘1’

‘0’

HWP

QWP

Output

‘1’

Photon 2

Photon 3

Logic 2

EOM switching scheme

Outcome

photon 2

Outcome

photon 3

Error

correction

None

Logic 1

Photon 4

Figure 7.8 Experimental setup for cluster computation with feedforward [292].

imental results are again consistent with the theoretical predictions and conﬁrm

that the experiment is successful based on postselection.

Thus far, we have described an experiment for qubit cluster computation with-

out the use of feedforward because such an experimental situation is much easier.

However, since feedforward is a crucial ingredient of universal cluster computa-

tion, we shall now discuss an example for this kind of advanced cluster experiment

including feedforward. The corresponding experiment of cluster computation with

feedforward was performed by Prevedel et al. [292].

Figure 7.8 shows the experimental setup where the cluster state used in this

experiment is the same as the ones used in Walther’s experiment explained up

to here (post-selection). In this sense, this experiment corresponds to Walther’s

experiment with feedforward.

First, Prevedel et al. performed the operation described in Eq. (7.4) with a

four-mode l inear cluster state. This is very similar to the experiment explained

above. They ﬁrst made a polarization measurement of the photon in mode 1 on

278 7 Cluster-Based Quantum Information Processing

CPhase

ancilla

input

B (

)

HR (

)

ancilla

B (

)

=0,1

HR (

)

Figure 7.9 Elementary-teleportation picture

for the experiment of Prevedel et al. similar to

Figure 7.2 (in the ﬁgure here and in the main

text of this section we use the same notation

as in the article of Prevedel et al. differing

from the notation used mostly throughout

this book); s

D 0, 1 correspond to the mea-

surement results jα

, jα



, respectively.

fjCi

, ji

g basis (˙45

-linearly polarization) and took the case of jCi

to prepare

a three-mode linear cluster state of modes 2–4. Then, they made consecutive po-

larization measurements of the photons in modes 2 and 3 with α

and α

to make

consecutive operations.

Figure 7.9 illustrates the operations according to the elementary-teleportation

picture. The output state of the photon in mode 4 can be written as

jψ

out

D σ

(α

)σ

(α

)jψi , (7.8)

corresponding to two cascaded elementary teleportations (see Figure 7.2). Here,

D 0, 1 correspond to the measurement results jα

, jα



, respectively.

In this experiment, they wanted to have the output of HR

(α

)HR

(α

)jψiD

(α

)jψi irrespective of the measurement results. Therefore, they used

feedforward.

Inthecaseofs

D s

D 0, the output state of mode 4 is R

(α

)jψi and

thus feedforwad is not needed. In the case of s

D 1ands

D 0, the output state of

mode 4 becomes

jψ

out

D HR

(α

)σ

(α

)jψi

D HR

(α

)HHσ

(α

)jψi

D R

(α

)σ

(α

)jψi

D σ

(α

)σ

(α

)jψi

D σ

(α

(α

)jψi . (7.9)

From the equation, one needs to switch the measurement basis of mode 3 from

(α

)toB

(α

)andmakeσ

operation for the output state jψ

out

to get

(α

)jψi.Inthecaseofs

D 0ands

D 1, the output state of mode 4

becomes

jψ

out

D σ

(α

)HR

(α

)jψi

D σ

(α

)jψi . (7.10)

Therefore, one needs to make a σ

operation for the output state jψ

out

to get

(α

)jψi.Inthecaseofs

D s

D 1, one needs to switch the measure-

ment basis of mode 3 from B

(α

)toB

(α

)andmakeaσ

and σ

operation.

7.1 Qubits 279

Outcome measured and ideal

0.2

0.4

0.6

0.8

1.0

With feed-forward Without feed-forward

Measurement Readout

Linear

(3)

cluster

234

With feed-forward Without feed-forward

Fidelity with target state

Outcome measured and ideal

(a)

(b) (c)

Figure 7.10 Experimental results of one-way

quantum computation with three-mode linear

cluster with and without feedforward [292].

“C”, “”denoteS

D 0, 1, respectively.

The linear qubit cluster state and the cir-

cuit it implements; (b) α

D α

Dπ/2,

D π/4 and α

D π/12.

Thus, the output state including feedforward can be described as

jψ

out

D σ

((1)

(α

)jψi , (7.11)

where the switching table of feedforward is shown in Figure 7.8 as an inset.

The experiment was realized as shown in Figure 7.8. The most important thing

here is that there are delay lines to compensate the time delay of photodetectors

and logic circuits. By using the delay lines, the consecutive measurements and

feedforwards were performed.

Experimental results of one-way quantum computation with three-mode linear

cluster with feedforward are shown in Figure 7.10. Here, the input state is jCi

The results are consistent with theory, conﬁrming success of the experiments de-

pendence on post-selection.

Prevedel et al. also tried to perform a two-qubit-gate operation with feedforward

with the same experimental setup shown in Figure 7.8. They used a four-mode lin-

ear cluster as a horseshoe cluster state. Similar to the single-qubit-gate operations

of Eq. (7.11), the performed two-qubit-gate operation can be described as

jψ

out

D (σ

˝σ

)(H ˝H)[R

(α

)˝R

(α

)]CPhasejψ

i˝jψ

i . (7.12)

Figure 7.11 shows the experimental results. In this ﬁgure, the results for S

D 1 and the input jCi

˝jCi

are shown. In the case with feedforward, the

output should be (j$i

˝jCi

Cjli

˝ji

2, which is of course the same as

the one of Walther’s experiment mentioned before. On the other hand, in the case

without feedforward, the output should become (jCi

˝jli

Cji

˝j$i

2in

thecaseofS

D S

D 1. The experimental results presented in Figure 7.11 agree

well with the theoretical prediction. However, we note again that a crucial ingredi-