A squeezed vacuum created by SPDC is a beam of even

Number of photons; when a single photon or two pho-

Tons are subtracted from it, the resulting state is a

‘Schr¨odinger kitten’—a superposition of coherent states

of opposite phase | ± α i ≡ | α i ± |− α i, where α ∼ 1

(Ref. 67). These ‘kittens’ are almost orthogonal to each

other and can be used as logical qubits: |0iL=| − α i;

|1iL=|+ α i. Since | ± α iare CV states, this can be re-

garded as hybrid qubit-CV QIP. These Schr¨odinger kit-

Tens have been created in the lab68–70.

Squeezing bandwidth is the most important factor for

handling these ‘kittens’. This is because the avalanche

Photodiodes typically used for the photon subtraction

5

a

Ancilla

ψ

ψ

X

Entanglement

Measurement and

Feedforward

CNOT (qubit) QND (CV)

b

ψ

ˆ

U ψ

X

ˆ

Uc

ˆ

U

X

ˆ

Uc

State preparation

()

ˆ0Sr

3

Ikx

Dx e x

∫

Squeezed vacuum

Cubic phase state

Universal squeezer

Cubic phase gate

xx

p

c

ψ

ˆ

U ψ

X

0p=

ψ

ˆ

U ψ

X

0p=

Generalized measurement

†

ˆˆ ˆ

U pU

FIG. 3: a, Generalized quantum teleportation. b, O ff -line

Quantum information processing with the generalized quan-

Tum teleportation. c, Generalized quantum teleportation is

Applied to one-way quantum computation with cluster states.

A measurement is generalized as ˆ

U†ˆpˆ

U.

Have a much wider bandwidth than that of the squeezer,

and thus bandwidth of the ‘kittens’ is the full bandwidth

of the squeezer. To handle the ‘kittens’, the bandwidth of

QIP should be broader than that of the ‘kittens’. More

Generally the bandwidth determines the speed of QIP.

Since a cavity is usually used for the enhancement of

Nonlinearity to achieve high level of squeezing, the band-

width is limited by the cavity bandwidth: ∼ 100 MHz at

Most. For broadband quantum teleportation and QIP,

We should therefore not use a cavity. An alternative is to

Use a waveguide to enhance the nonlinearity, which has

Been done with waveguided periodically poled LiNbO3

(PPLN)71. The bandwidth of squeezing and entangle-

Ment in this case is only limited by the bandwidth of the

phase matching condition in principle: ∼ 10THz.

Finally, single photons can be created via single photon

Subtraction from weekly squeezed vacuum, or so-called

‘photon pairs’ (two photons minus one photon gives one

Photon). Again, to handle single photon polarisation

Qubits, the bandwidth of QIP should be broader than

That of the single photons. In this way one can handle

Polarised-photon qubits in a CV context, if the band-

Width is broad enough. This is hybridisation of qubits

and CVs, the fi rst step of which was recently demon-

strated via CV teleportation of ‘Schr¨odinger kittens’ cre-

Ated with photon subtraction72.

Generalized quantum teleportation

The concept of quantum teleportation has been extended

To generalized quantum teleportation76,77 (for both qubit

and CV regimes), which can be applied to o ff -line QIP.

a

0p=

†

22

ˆˆ ˆ

U pU

ψ

0p=

†

11

ˆˆ ˆ

U pU

3 2 1

ˆˆˆ

U U U ψ

X2

0p=

X1

X3

†

33

ˆˆ ˆ

U pU

1 2 3 4

ψ

3 2 1

ˆˆˆ

U U U ψ

2 3 42 3 4

da p a x a p a = = =

∫

()

101

2+ + + − −

() ()

11

0 1, 0 1

22

+ = + −=−

Cluster state

Multipartite entanglement

b

Mode cluster states

Linear 4-mode

T-shape















→−

→−−

→−−

→−

0

ˆˆ

0

ˆˆˆ

0

ˆˆˆ

0

ˆˆ

34

423

312

21

xp

Xxp

Xxp

xp

Diamond-shape

12

2 1 2 3

32

42

ˆˆ 0

ˆˆˆˆ 0

ˆˆ 0

ˆˆ 0

px

P x x x

px

px

−→



 − − − →



 −→



 −→



1 2 3 41 2 3 4

1 2 3

4

1 2 3

4

1

3 4

2

1

3 4

2

1 3 4

234

3 1 2

4 1 2

ˆˆˆ 0

ˆˆˆ 0

ˆˆˆ 0

ˆˆˆ 0

P x x

P x x

P x x

P x x

− − →



 − − →



 − − →



 − − →



FIG. 4: a, One-way quantum computation with cluster states.

This example is on a three-mode linear cluster state. b, Ex-

Perimentally created CV cluster states73. These are simulta-

neous eigenstates of the operators listed near the fi gures. For

qubits, a cluster state is an eigenstates of so-called ‘stabiliz-

ers’: ˆ σ (i)

xNi0 ∈ N(i)ˆ σ (i0)

z, where ˆ σ xis Pauli-X operator, N(i) is

the set of nearest neighbor qubits of qubit i, and ˆ σ zis Pauli-Z

Operator74. For CVs, they are again the eigenstates of stabi-

lizers: ˆ

Xi(si)Qi0 ∈ N(i)ˆ

Pi0(si), where ˆ

X(s) and ˆ

P(s) are the

X- and p-translation operators, respectively, and N(i) is the

Set of nearest neighbor modes of mode i75. For example, the

three-qubit linear cluster state is (|+0+i+|−1−i)/√2, where

|±i = (|0i ± |1i)/√2, and the three-mode linear CV cluster

state is R∞

−∞ da|p=ai|x=ai|p=ai.

The essence of the scheme (Fig. 3a) is o ff -line: pre-

existing entanglement between the input | ψ iand an an-

Cilla followed by measurement and feedforward. An ex-

ample is shown in Fig. 3b in which the ancilla is a speci fi c

state ˆ

U|ci, where the unitary operation ˆ

Uis the one we

Want to apply on the input. The crucial advantage of

the scheme is that the di ffi culty of operation is con fi ned

in state preparation of the ancilla, since the fi delity of

Teleportation itself is rather high: one does not have to

Make the unitary operation for an arbitrary input, in-

stead it is enough to make it on a particular state |ci,

Which is much easier than the case for an arbitrary input.