Linear optical quantum computing with photonic qubits
Pieter Kok
*
Department of Materials, Oxford University, Oxford OX1 3PH, United Kingdom
and Hewlett-Packard Laboratories, Filton Road Stoke Gifford, Bristol BS34 8QZ,
United Kingdom
W. J. Munro
Hewlett-Packard Laboratories, Filton Road Stoke Gifford, Bristol BS34 8QZ,
United Kingdom
Kae Nemoto
National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
T. C. Ralph
Centre for Quantum Computer Technology, University of Queensland, St. Lucia,
Queensland 4072, Australia
Jonathan P. Dowling
Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803, USA
and Institute for Quantum Studies, Department of Physics, Texas A&M University,
College Park, Texas 77843-4242, USA
G. J. Milburn
Centre for Quantum Computer Technology, University of Queensland, St. Lucia,
Queensland 4072, Australia
共Published 24 January 2007; corrected 30 May 2007兲
Linear optics with photon counting is a prominent candidate for practical quantum computing. The
protocol by Knill, Laflamme, and Milburn 关2001, Nature 共London兲 409,46兴 explicitly demonstrates
that efficient scalable quantum computing with single photons, linear optical elements, and projective
measurements is possible. Subsequently, several improvements on this protocol have started to bridge
the gap between theoretical scalability and practical implementation. The original theory and its
improvements are reviewed, and a few examples of experimental two-qubit gates are given. The use
of realistic components, the errors they induce in the computation, and how these errors can be
corrected is discussed.
DOI: 10.1103/RevModPhys.79.135 PACS number共s兲: 03.67.Lx, 42.50.Dv, 03.65.Ud, 42.79.Ta
CONTENTS
I. Quantum Computing with Light 136
A. Linear quantum optics 136
B. N-port interferometers and optical circuits 138
C. Qubits in linear optics 139
D. Early optical quantum computers and nonlinearities 140
II. A New Paradigm for Optical Quantum Computing 142
A. Elementary gates 142
B. Parity gates and entangled ancillae 144
C. Experimental demonstrations of gates 145
D. Characterization of linear optics gates 147
E. General probabilistic nonlinear gates 148
F. Scalable optical circuits and quantum teleportation 149
G. The Knill-Laflamme-Milburn protocol 150
H. Error correction of the probabilistic gates 151
III. Improvement on the KLM Protocol 152
A. Cluster states in optical quantum computing 153
B. The Yoran-Reznik protocol 154
C. The Nielsen protocol 155
D. The Browne-Rudolph protocol 156
E. Circuit-based optical quantum computing revisited 157
IV. Realistic Optical Components and Their Errors 158
A. Photon detectors 158
B. Photon sources 160
C. Circuit errors and quantum memories 165
V. General Error Correction 166
A. Correcting for photon loss 167
B. General error correction in LOQC 169
VI. Outlook: Beyond Linear Optics 170
Acknowledgments 170
References 171
*
Electronic address: pieter.kok@materials.ox.ac.uk
REVIEWS OF MODERN PHYSICS, VOLUME 79, JANUARY–MARCH 2007
0034-6861/2007/79共1兲/135共40兲 ©2007 The American Physical Society135
I. QUANTUM COMPUTING WITH LIGHT
Quantum computing has attracted much attention
over the last 10 to 15 years, partly because of its promise
of superfast factoring and its potential for the efficient
simulation of quantum dynamics. There are many differ-
ent architectures for quantum computers based on many
different physical systems. These include atom- and ion-
trap quantum computing, superconducting charge and
flux qubits, nuclear magnetic resonance, spin- and
charge-based quantum dots, nuclear spin quantum com-
puting, and optical quantum computing 关for a recent
overview, see Spiller et al. 共2006兲兴. All these systems
have their own advantages in quantum information pro-
cessing. However, even though there may now be a few
front-runners, such as ion-trap and superconducting
quantum computing, no physical implementation seems
to have a clear edge over others at this point. This is an
indication that the technology is still in its infancy.
Optical quantum systems are prominent candidates
for quantum computing, since they provide a natural in-
tegration of quantum computation and quantum com-
munication. There are several proposals for building
quantum computers that manipulate the state of light,
ranging from cat-state logic to encoding a qubit in a
harmonic oscillator and optical continuous-variable
quantum computing. Cat states are states of the form
兩
␣
典±兩−
␣
典, where 兩
␣
典 is a weak coherent state. The logical
qubits are determined by the sign 共⫾兲 of the relative
phase 共Ralph et al., 2003兲. Gottesman, Kitaev, and
Preskil proposed quantum error correction codes for
harmonic oscillators that are used to encode qubit states
and showed that fault-tolerant quantum computing is
possible using quantum optics 共Gottesman et al., 2001兲.
Lloyd and Braunstein showed how the concept of quan-
tum computing can be extended to continuous variables
and how electromagnetic fields are a natural physical
representation of this formalism 共Lloyd and Braunstein,
1999兲. For a review article on optical quantum informa-
tion processing with continuous variables, see Braun-
stein and van Loock 共2005兲.
In this review, we focus on quantum computing with
linear quantum optics and single photons. It has the ad-
vantage that the smallest unit of quantum information,
the photon, is potentially free from decoherence: The
quantum information stored in a photon tends to stay
there. The downside is that photons do not naturally
interact with each other, and in order to apply two-qubit
quantum gates such interactions are essential. There-
fore, if we are to construct an optical quantum com-
puter, we have to introduce an effective interaction be-
tween photons in one way or another. In Sec. I.D, we
review the use of so-called large cross-Kerr nonlineari-
ties to induce a single-photon controlled-
NOT operation.
However, naturally occurring nonlinearities of this sort
are many orders of magnitude too small for our pur-
poses. An alternative way to induce an effective interac-
tion between photons is to make projective measure-
ments with photodetectors. The difficulty with this
technique is that such optical quantum gates are proba-
bilistic: More often than not, the gate fails and destroys
the information in the quantum computation. This can
be circumvented by using an exponential number of op-
tical modes, but scalability requires only a polynomial
number of modes 共see also Sec. I.D兲. In 2001, Knill,
Laflamme, and Milburn 共2001兲 共KLM兲 constructed a
protocol in which probabilistic two-photon gates are
teleported into a quantum circuit with high probability.
Subsequent error correction in the quantum circuit is
used to bring the error rate down to fault-tolerant levels.
We describe the KLM protocol in detail in Sec. II.
Initially, the KLM protocol was designed as a proof
that linear optics and projective measurements allow for
scalable quantum computing in principle. However, it
subsequently spurred new experiments in quantum op-
tics, demonstrating the operation of high-fidelity proba-
bilistic two-photon gates. On the theoretical front, sev-
eral improvements of the protocol were proposed,
leading to ever smaller overhead costs on the computa-
tion. A number of these improvements is based on
cluster-state quantum computing, or the one-way quan-
tum computer. Recently, a circuit-based model was
shown to have similar scaling properties as the best-
known cluster-state model. In Sec. III, we describe sev-
eral improvements to linear optical quantum informa-
tion processing in considerable detail, and in Sec. IV, we
describe the issues involved in the use of realistic com-
ponents such as photon detectors, photon sources, and
quantum memories. Given these realistic components,
we discuss loss tolerance and general error correction
for linear optical quantum computing 共LOQC兲 in Sec. V.
We will restrict our discussion to the theory of single-
photon implementations of quantum information pro-
cessors, and assume some familiarity with the basic con-
cepts of quantum computing. For an introduction to
quantum computation and quantum information, see,
e.g., Nielsen and Chuang 共2000兲. In Sec. VI we conclude
with an outlook on other promising optical quantum in-
formation processing techniques, such as photonic band-
gap structures, weak cross-Kerr nonlinearities, and hy-
brid matter-photon systems. We start our review with a
short introduction to linear optics, N-port optical inter-
ferometers, and circuits, and define the different ver-
sions of the optical qubit.
A. Linear quantum optics
The basic building blocks of linear optics are beam
splitters, half- and quarter-wave plates, phase shifters,
etc. In this section we describe these devices mathemati-
cally and establish the convention that is used through-
out the rest of the paper.
The quantum-mechanical plane-wave expansion of
the electromagnetic vector potential is usually expressed
in terms of the annihilation operators a
ˆ
j
共k兲 and their
adjoints, the creation operators:
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Rev. Mod. Phys., Vol. 79, No. 1, January–March 2007
A
共x,t兲 =
冕
d
3
k
冑
共2
兲
3
2
k
兺
j=1,2
⑀
j
共k兲a
ˆ
j
共k兲e
ikx−i
k
t
+ H.c.,
where j denotes the polarization in the Coulomb gauge,
⑀
j
is the corresponding polarization vector, and
indi-
cates the components of the four-vector. For the mo-
ment we suppress the polarization degree of freedom
and consider general properties of the creation and an-
nihilation operators. They bear their names because
they act in a specific way on the Fock states 兩n典:
a
ˆ
兩n典 =
冑
n兩n −1典 and a
ˆ
†
兩n典 =
冑
n +1兩n +1典, 共1兲
where we suppressed the k dependence. It is straightfor-
ward to show that n
ˆ
共k兲⬅a
ˆ
†
共k兲a
ˆ
共k兲 is the number opera-
tor n
ˆ
兩n典=n兩n典 for a given mode with momentum k. The
canonical commutation relations between a
ˆ
and a
ˆ
†
are
given by
关a
ˆ
共k兲,a
ˆ
†
共k
⬘
兲兴 =
␦
共k − k
⬘
兲,
共2兲
关a
ˆ
共k兲,a
ˆ
共k
⬘
兲兴 = 关a
ˆ
†
共k兲,a
ˆ
†
共k
⬘
兲兴 =0.
In the rest of this review, we denote information about
the spatial mode k by a subscript, since we will be con-
cerned not with the geometrical details of the interfer-
ometers we describe, but only with how the spatial
modes are connected. Also to avoid notational clutter
we will use operator carets only for nonunitary and non-
Hermitian operators, except in cases where omission of
the caret would lead to confusion.
An important optical component is the single-mode
phase shift. It changes the phase of the electromagnetic
field in a given mode:
a
ˆ
out
†
= e
i
a
ˆ
in
†
a
ˆ
in
a
ˆ
in
†
e
−i
a
ˆ
in
†
a
ˆ
in
= e
i
a
ˆ
in
†
, 共3兲
with the interaction Hamiltonian H
=
a
ˆ
in
†
a
ˆ
in
共here and
throughout this review we use the convention that ប =1
and time dependence is absorbed in
兲. This Hamil-
tonian is proportional to the number operator, which
means that the photon number is conserved. Physically,
a phase shifter is a slab of transparent material with an
index of refraction that is different from that of free
space.
Another important component is the beam splitter
共see Fig. 1兲. Physically, it consists of a semireflective mir-
ror: when light falls on this mirror, part will be reflected
and part will be transmitted 共Leonhardt, 1997兲. The
theory of the lossless beam splitter is central to LOQC
and was developed by Zeilinger 共1981兲 and Fearn and
Loudon 共1987兲. Lossy beam splitters were studied by
Barnett et al. 共1989兲. The transmission and reflection
properties of general dielectric media were studied by
Dowling 共1998兲. Let the two incoming modes on either
side of the beam splitter be denoted by a
ˆ
in
and b
ˆ
in
and
the outgoing modes by a
ˆ
out
and b
ˆ
out
. When we param-
etrize the probability amplitudes of these possibilities as
cos
and sin
and the relative phase as
, then the
beam splitter yields an evolution in operator form
a
ˆ
out
†
= cos
a
ˆ
in
†
+ ie
−i
sin
b
ˆ
in
†
,
共4兲
b
ˆ
out
†
= ie
i
sin
a
ˆ
in
†
+ cos
b
ˆ
in
†
.
The reflection and transmission coefficients R and T of
the beam splitter are R=sin
2
and T=1−R=cos
2
. The
relative phase shift ie
±i
ensures that the transformation
is unitary. Typically, we choose either
=0 or
=
/2.
Mathematically, the two parameters
and
represent
the angles of a rotation about two orthogonal axes in the
Poincaré sphere. The physical beam splitter can be de-
scribed by any choice of
and
, provided the correct
phase shifts are applied to the outgoing modes.
In general the Hamiltonian H
BS
of the beam-splitter
evolution in Eq. 共4兲 is given by
H
BS
=
e
i
a
ˆ
in
†
b
ˆ
in
+
e
−i
a
ˆ
in
b
ˆ
in
†
. 共5兲
Since the operator H
BS
commutes with the total number
operator 关H
BS
,n
ˆ
兴=0, the photon number is conserved in
the lossless beam splitter, as one would expect.
The same mathematical description applies to the
evolution due to a polarization rotation, physically
implemented by quarter- and half-wave plates. Instead
of having two different spatial modes a
in
and b
in
, the two
incoming modes have different polarizations. We write
a
ˆ
in
→ a
ˆ
x
and b
ˆ
in
→ a
ˆ
y
for some orthogonal set of coordi-
nates x and y 共i.e., 具x兩y典=0兲. The parameters
and
are
now angles of rotation:
a
ˆ
x
⬘
†
= cos
a
ˆ
x
†
+ ie
−i
sin
a
ˆ
y
†
,
共6兲
a
ˆ
y
⬘
†
= ie
i
sin
a
ˆ
x
†
+ cos
a
ˆ
y
†
.
This evolution has the same Hamiltonian as the beam
splitter, and it formalizes the equivalence between the
so-called polarization and dual-rail logic. These transfor-
mations are sufficient to implement any photonic single-
qubit operation 共Simon and Mukunda, 1990兲.
The last linear optical element that we highlight here
is the polarizing beam splitter 共PBS兲. In circuit diagrams,
it is usually drawn as a box around a regular beam split-
ter 关see Fig. 2共a兲兴. If the PBS is cut to separate horizontal
and vertical polarization, the transformation of the in-
coming modes 共a
in
and b
in
兲 yields the following outgoing
modes 共a
out
and b
out
兲:
FIG. 1. 共Color online兲 The beam splitter with transmission am-
plitude cos
.
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a
ˆ
in,H
→ a
ˆ
out,H
and a
ˆ
in,V
→ b
ˆ
out,V
,
共7兲
b
ˆ
in,H
→ b
ˆ
out,H
and b
ˆ
in,V
→ a
ˆ
out,V
.
Using quarter-wave plates and polarizers, we can also
construct a PBS for different polarization directions
共e.g., L and R兲, in which case we make the substitution
H↔ L, V ↔ R. Diagrammatically a PBS with a different
polarization typically has a circle drawn inside the box
关Fig. 2共b兲兴.
At this point, we should devote a few words to the
term “linear optics.” Typically this denotes the set of
optical elements whose interaction Hamiltonian is bilin-
ear in the creation and annihilation operators:
H =
兺
jk
A
jk
a
ˆ
j
†
a
ˆ
k
. 共8兲
An operator of this form commutes with the total num-
ber operator and has the property that a simple mode
transformation of creation operators into a linear com-
bination of other creation operators affects only the ma-
trix A, but does not introduce terms that are quadratic
共or higher兲 in the creation or annihilation operators.
However, from a field-theoretic point of view, the most
general linear transformation of creation and annihila-
tion operators is defined by the Bogoliubov transforma-
tion
a
ˆ
j
→
兺
k
u
jk
a
ˆ
k
+ v
jk
a
ˆ
k
†
. 共9兲
Clearly, when such a transformation is substituted into
Eq. 共8兲 this will give rise to terms such as a
ˆ
j
a
ˆ
k
and a
ˆ
j
†
a
ˆ
k
†
,
i.e., squeezing. The number of photons is not conserved
in such a process. For the purpose of this review, we
exclude squeezing as a resource other than as a method
for generating single photons.
With the linear optical elements introduced in this
section we can build large optical networks. In particu-
lar, we can make computational circuits by using known
states as the input and measuring the output states. Next
we will study these optical circuits in more detail.
B. N-port interferometers and optical circuits
An optical circuit can be thought of as a black box
with incoming and outgoing modes of the electromag-
netic field. The black box transforms a state of the in-
coming modes into a different state of outgoing modes.
The modes might be mixed by beam splitters, or they
might pick up a relative phase shift or polarization rota-
tion. These operations all belong to a class of optical
components that preserve the photon number, as de-
scribed in the previous section. In addition, the box may
include measurement devices, the outcomes of which
may modify optical components on the remaining
modes. This is called feedforward detection, and it is an
important technique that can increase the efficiency of a
device 共Clausen et al., 2003; Lapaire et al., 2003兲.
Optical circuits can also be thought of as a general
unitary transformation on N modes, followed by the de-
tection of a subset of these modes 共followed by unitary
transformation on the remaining modes, detection, and
so on兲. The interferometric part of this circuit is also
called an N-port interferometer. N-ports yield a unitary
transformation U of the spatial field modes a
k
, with j,k
苸 兵1, ... ,N其:
b
ˆ
k
→
兺
j=1
N
U
jk
a
ˆ
j
and b
ˆ
k
†
→
兺
j=1
N
U
jk
*
a
ˆ
j
†
, 共10兲
where the incoming modes of the N-port are denoted by
a
j
and the outgoing modes by b
j
. The explicit form of U
is given by the repeated application of transformations
like those given by Eqs. 共3兲, 共4兲, and 共6兲.
The two-mode operators L
ˆ
+
=a
ˆ
†
b
ˆ
, L
ˆ
−
=a
ˆ
b
ˆ
†
, and L
ˆ
0
=共a
ˆ
†
a
ˆ
−b
ˆ
†
b
ˆ
兲/2 form an su共2兲 Lie algebra:
关L
ˆ
0
,L
ˆ
±
兴 =±L
ˆ
±
and 关L
ˆ
+
,L
ˆ
−
兴 =2L
ˆ
0
. 共11兲
This means that any two-mode interferometer exhibits
U共2兲 symmetry.
1
In general, an N-port interferometer
can be described by a transformation from the group
U共N兲. Reck et al. 共1994兲 demonstrated that the converse
is also true, i.e., that any unitary transformation of N
optical modes can be implemented efficiently with an
N-port interferometer. They showed how a general
U共N兲 element can be broken down into SU共2兲 elements,
for which we have a complete physical representation in
terms of beam splitters and phase shifters 共see Fig. 3兲.
The primitive element is a matrix T
pq
defined on the
modes p and q, which corresponds to a beam splitter
and phase shifts. Implicit in this notation is the identity
operator on the rest of the optical modes, such that
T
pq
⬅T
pq
丢 1
rest
. We then have
U共N兲 ⫻ T
N,N−1
⫻ ¯ ⫻ T
N,1
=U共N −1兲 丣 e
i
, 共12兲
where
is a single-mode phase. Concatenating this pro-
cedure leads to a full decomposition of U共N兲 into T el-
ements, which in turn are part of SU共2兲. The maximum
1
Two remarks: Lie algebras are typically denoted in lower
case, while the group itself is denoted in upper case. Second,
single-mode phase shifts break the special symmetry 共 det U
=1兲, which is why an interferometer is described by U共N兲,
rather than SU共N兲.
FIG. 2. 共Color online兲 The polarizing beam splitter in different
polarization bases. 共a兲 The horizontal-vertical basis. 共b兲 The
diagonal basis.
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number of beam-splitter elements T that are needed is
N共N−1兲/2. This procedure is thus manifestly scalable.
Subsequently, Jex et al. 共1995兲 and Törmä et al. 共1995,
1996兲 showed how to construct multimode Hamiltonians
that generate these unitary mode transformations, and a
three-path Mach-Zehnder interferometer was demon-
strated experimentally by Weihs et al. 共1996兲. A good
introduction to linear optical networks has been given
by Leonhardt 共2003兲, and a determination of effective
Hamiltonians has been given by Leonhardt and Neu-
maier 共2004兲. For a treatment of optical networks in
terms of their permanents, see Scheel 共2004兲. Optical
circuits in a 共general兲 relativistic setting have been de-
scribed by Kok and Braunstein 共2006兲.
C. Qubits in linear optics
Formally, a qubit is a quantum system that is de-
scribed by the fundamental representation of the SU共2兲
symmetry group. We saw above that two optical modes
form a natural implementation of this symmetry. In gen-
eral, two modes with fixed total photon number n fur-
nish natural irreducible representations of this group
with the dimension of the representation given by n +1
共Biedenharn and Louck, 1981兲. It is at this point not
specified whether we should use spatial or polarization
modes. In linear optical quantum computing, the qubit
of choice is usually taken to be a single photon that has
the choice of two different modes 兩0典
L
=兩1典 丢 兩0典⬅兩1,0典
and 兩1典
L
=兩0典 丢 兩1典⬅兩0,1典. This is called a dual-rail qubit.
When the two modes represent the internal polarization
degree of freedom of the photon 共兩0典
L
=兩H典 and 兩1典
L
=兩V典兲, we speak of a polarization qubit. In this review we
will reserve the term “dual rail” for a qubit with two
spatial modes. As we showed earlier, these two repre-
sentations are mathematically equivalent and we can
physically switch between them using polarization beam
splitters. In addition, some practical applications 共typi-
cally involving a dephasing channel such as a fiber兲 may
call for so-called time-bin qubits, in which the two com-
putational qubit values are “early” and “late” arrival
times in a detector. However, this degree of freedom
does not exhibit a natural internal SU共2兲 symmetry: Ar-
bitrary single-qubit operations are very difficult to
implement. In this review we will be concerned mainly
with polarization and dual-rail qubits.
In order to build a quantum computer, we need both
single-qubit and two-qubit operations. Single-qubit op-
erations are generated by the Pauli operators
x
,
y
, and
z
, in the sense that the operator exp共i
j
兲 is a rotation
about the j axis in the Bloch sphere with angle
.Aswe
have seen, these operations can be implemented with
phase shifters, beam splitters, and polarization rotations
on polarization and dual-rail qubits. In this review, we
will use the convention that
x
,
y
, and
z
denote physi-
cal processes, while we use X, Y, and Z for the corre-
sponding logical operations on the qubit. These two rep-
resentations become inequivalent when we deal with
logical qubits that are encoded in multiple physical qu-
bits.
Whereas single-qubit operations are straightforward
in the polarization and dual-rail representations, the
two-qubit gates are more problematic. Consider, for ex-
ample, the transformation from a state in the computa-
tional basis to a maximally entangled Bell state:
兩H,H典
ab
→
1
冑
2
共兩H,V典
cd
+ 兩V,H典
cd
兲. 共13兲
This is the type of transformation that requires a two-
qubit gate. In terms of the creation operators 共and ignor-
ing normalization兲, the linear optical circuit that is sup-
posed to create Bell states out of computational basis
states is described by a Bogoliubov transformation of
both creation operators
a
ˆ
H
†
b
ˆ
H
†
→
冉
兺
k=H,V
␣
k
c
ˆ
k
†
+

k
d
ˆ
k
†
冊冉
兺
k=H,V
␥
k
c
ˆ
k
†
+
␦
k
d
ˆ
k
†
冊
⫽ c
ˆ
H
†
d
ˆ
V
†
+ c
ˆ
V
†
d
ˆ
H
†
. 共14兲
It is immediately clear that the right-hand sides in both
lines cannot be made the same for any choice of
␣
k
,

k
,
␥
k
, and
␦
k
: The top line is a separable expression in the
creation operators, while the bottom line is an entangled
expression in the creation operators. Therefore, linear
optics alone cannot create maximal polarization en-
tanglement from single polarized photons in a determin-
istic manner 共Kok and Braunstein, 2000a兲. Entangle-
ment that is generated by changing the definition of our
subsystems in terms of the global field modes is in-
equivalent to the entanglement that is generated by ap-
plying true two-qubit gates to single-photon polarization
or dual-rail qubits.
Note also that if we choose our representation of the
qubit differently, we can implement a two-qubit transfor-
mation. Consider the single-rail qubit encoding 兩0典
L
=兩0典 and 兩1典
L
=兩1典. That is, the qubit is given by the
vacuum and single-photon state. We can then implement
the following 共unnormalized兲 transformation determinis-
tically:
兩1,0典 → 兩1,0典 + 兩0,1典. 共15兲
This is a 50:50 beam-splitter transformation. However, in
this representation the single-qubit operations cannot be
FIG. 3. 共Color online兲 Decomposing an N-port unitary U共N兲
into SU共2兲 group elements, i.e., beam splitters and phase
shifters. Moreover, this is an efficient process: the maximum
number of beam splitters needed is N共N−1兲/2.
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Kok et al.: Linear optical quantum computing with …
Rev. Mod. Phys., Vol. 79, No. 1, January–March 2007
