Linear Optical Quantum Computing or Linear
Optics Quantum Computation (LOQC) is a paradigm
of quantum computation, allowing (under certain
conditions, described below) universal quantum
computation. LOQC uses photons as information
carriers, mainly uses linear optical elements
(including beam splitters, phase shifters,
and mirrors) to process quantum information,
and uses photon detectors and quantum memories
to detect and store quantum information.
== Overview ==
Although there are many other implementations
for quantum information processing (QIP) and
quantum computation, optical quantum systems
are prominent candidates, since they link
quantum computation and quantum communication
in the same framework. In optical systems
for quantum information processing, the unit
of light in a given mode—or photon—is
used to represent a qubit. Superpositions
of quantum states can be easily represented,
encrypted, transmitted and detected using
photons. Besides, linear optical elements
of optical systems may be the simplest building
blocks to realize quantum operations and quantum
gates. Each linear optical element equivalently
applies a unitary transformation on a finite
number of qubits. The system of finite linear
optical elements constructs a network of linear
optics, which can realize any quantum circuit
diagram or quantum network based on the quantum
circuit model. Quantum computing with continuous
variables is also possible under the linear
optics scheme.The universality of 1- and 2-bit
gates to implement arbitrary quantum computation
has been proven. Up to
N
×
N
{\displaystyle N\times N}
unitary matrix operations (
U
(
N
)
{\displaystyle U(N)}
) can be realized by only using mirrors, beam
splitters and phase shifters (this is also
a starting point of boson sampling and of
computational complexity analysis for LOQC).
It points out that each
U
(
N
)
{\displaystyle U(N)}
operator with
N
{\displaystyle N}
inputs and
N
{\displaystyle N}
outputs can be constructed via
O
(
N
2
)
{\displaystyle {\mathcal {O}}(N^{2})}
linear optical elements. Based on the reason
of universality and complexity, LOQC usually
only uses mirrors, beam splitters, phase shifters
and their combinations such as Mach-Zehnder
interferometers with phase shifts to implement
arbitrary quantum operators. If using a non-deterministic
scheme, this fact also implies that LOQC could
be resource-inefficient in terms of the number
of optical elements and time steps needed
to implement a certain quantum gate or circuit,
which is a major drawback of LOQC.
Operations via linear optical elements (beam
splitters, mirrors and phase shifters, in
this case) preserve the photon statistics
of input light. For example, a coherent (classical)
light input produces a coherent light output;
a superposition of quantum states input yields
a quantum light state output. Due to this
reason, people usually use single photon source
case to analyze the effect of linear optical
elements and operators. Multi-photon cases
can be implied through some statistical transformations.
An intrinsic problem in using photons as information
carriers is that photons hardly interact with
each other. This potentially causes a scalability
problem for LOQC, since nonlinear operations
are hard to implement, which can increase
the complexity of operators and hence can
increase the resources required to realize
a given computational function. One way to
solve this problem is to bring nonlinear devices
into the quantum network. For instance, the
Kerr effect can be applied into LOQC to make
a single-photon controlled-NOT and other operations.
=== KLM protocol ===
It was believed that adding nonlinearity to
the linear optical network was sufficient
to realize efficient quantum computation.
However, to implement nonlinear optical effects
is a difficult task. In 2000, Knill, Laflamme
and Milburn proved that it is possible to
create universal quantum computers solely
with linear optical tools. Their work has
become known as the "KLM scheme" or "KLM protocol",
which uses linear optical elements, single
photon sources and photon detectors as resources
to construct a quantum computation scheme
involving only ancilla resources, quantum
teleportations and error corrections. It uses
another way of efficient quantum computation
with linear optical systems, and promotes
nonlinear operations solely with linear optical
elements.At its root, the KLM scheme induces
an effective interaction between photons by
making projective measurements with photodetectors,
which falls into the category of non-deterministic
quantum computation. It is based on a non-linear
sign shift between two qubits that uses two
ancilla photons and post-selection. It is
also based on the demonstrations that the
probability of success of the quantum gates
can be made close to one by using entangled
states prepared non-deterministically and
quantum teleportation with single-qubit operations
Otherwise, without a high enough success rate
of a single quantum gate unit, it may require
an exponential amount of computing resources.
Meanwhile, the KLM scheme is based on the
fact that proper quantum coding can reduce
the resources for obtaining accurately encoded
qubits efficiently with respect to the accuracy
achieved, and can make LOQC fault-tolerant
for photon loss, detector inefficiency and
phase decoherence. As a result, LOQC can be
robustly implemented through the KLM scheme
with a low enough resource requirement to
suggest practical scalability, making it as
promising a technology for QIP as other known
implementations.
=== Boson sampling ===
The more limited boson sampling model was
suggested and analyzed by Aaronson and Arkhipov
in 2013. It is not believed to be universal,
but can still solve problems that are believed
to be beyond the ability of classical computers,
such as the boson sampling problem.
== Elements of LOQC ==
DiVincenzo's criteria for quantum computation
and QIP give that a universal system for QIP
should satisfy at least the following requirements:
a scalable physical system with well characterized
qubits,
the ability to initialize the state of the
qubits to a simple fiducial state, such as
|
000
⋯
⟩
{\displaystyle |000\cdots \rangle }
,
long relevant decoherence times, much longer
than the gate operation time,
a "universal" set of quantum gates (this requirement
cannot be satisfied by a non-universal system),
a qubit-specific measurement capability;if
the system is also aiming for quantum communication,
it should also satisfy at least the following
two requirements:
the ability to interconvert stationary and
flying qubits, and
the ability to faithfully transmit flying
qubits between specified location.As a result
of using photons and linear optical circuits,
in general LOQC systems can easily satisfy
conditions 3, 6 and 7. The following sections
mainly focus on the implementations of quantum
information preparation, readout, manipulation,
scalability and error corrections, in order
to discuss the advantages and disadvantages
of LOQC as a candidate for QIP
=== 
Qubits and modes ===
A qubit is one of the fundamental QIP units.
A qubit state which can be represented by
α
|
0
⟩
+
β
|
1
⟩
{\displaystyle \alpha |0\rangle +\beta |1\rangle
}
is a superposition state which, if measured
in the orthonormal basis
{
|
0
⟩
,
|
1
⟩
}
{\displaystyle \{|0\rangle ,|1\rangle \}}
, has probability
|
α
|
2
{\displaystyle |\alpha |^{2}}
of being in the
|
0
⟩
{\displaystyle |0\rangle }
state and probability
|
β
|
2
{\displaystyle |\beta |^{2}}
of being in the
|
1
⟩
{\displaystyle |1\rangle }
state, where
|
α
|
2
+
|
β
|
2
=
1
{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}
is the normalization condition. An optical
mode is a distinguishable optical communication
channel, which is usually labeled by subscripts
of a quantum state. There are many ways to
define distinguishable optical communication
channels. For example, a set of modes could
be different polarization of light which can
be picked out with linear optical elements,
various frequencies, or a combination of the
two cases above.
In the KLM protocol, each of the photons is
usually in one of two modes, and the modes
are different between the photons (the possibility
that a mode is occupied by more than one photon
is zero). This is not the case only during
implementations of controlled quantum gates
such as CNOT. When the state of the system
is as described, the photons can be distinguished,
since they are in different modes, and therefore
a qubit state can be represented using a single
photon in two modes, vertical (V) and horizontal
(H): for example,
|
0
⟩
≡
|
0
,
1
⟩
V
H
{\displaystyle |0\rangle \equiv |0,1\rangle
_{VH}}
and
|
1
⟩
≡
|
1
,
0
⟩
V
H
{\displaystyle |1\rangle \equiv |1,0\rangle
_{VH}}
. It is common to refer to the states defined
via occupation of modes as Fock states.
In boson sampling, photons are not distinguished,
and therefore cannot directly represent the
qubit state. Instead, we represent the qudit
state of the entire quantum system by using
the Fock states of
M
{\displaystyle M}
modes which are occupied by
N
{\displaystyle N}
indistinguishable single photons (this is
a
(
M
+
N
−
1
M
)
{\displaystyle {\tbinom {M+N-1}{M}}}
-level quantum system).
=== State preparation ===
To prepare a desired multi-photon quantum
state for LOQC, a single-photon state is first
required. Therefore, non-linear optical elements,
such as single-photon generators and some
optical modules, will be employed. For example,
optical parametric down-conversion can be
used to conditionally generate the
|
1
⟩
≡
|
1
,
0
⟩
V
H
{\displaystyle |1\rangle \equiv |1,0\rangle
_{VH}}
state in the vertical polarization channel
at time
t
{\displaystyle t}
(subscripts are ignored for this single qubit
case). By using a conditional single-photon
source, the output state is guaranteed, although
this may require several attempts (depending
on the success rate). A joint multi-qubit
state can be prepared in a similar way. In
general, an arbitrary quantum state can be
generated for QIP with a proper set of photon
sources.
=== Implementations of elementary quantum
gates ===
To achieve universal quantum computing, LOQC
should be capable of realizing a complete
set of universal gates. This can be achieved
in the KLM protocol but not in the boson sampling
model.
Ignoring error correction and other issues,
the basic principle in implementations of
elementary quantum gates using only mirrors,
beam splitters and phase shifters is that
by using these linear optical elements, one
can construct any arbitrary 1-qubit unitary
operation; in other words, those linear optical
elements support a complete set of operators
on any single qubit.
The unitary matrix associated with a beam
splitter
B
θ
,
ϕ
{\displaystyle \mathbf {B} _{\theta ,\phi
}}
is:
U
(
B
θ
,
ϕ
)
=
[
cos
⁡
θ
−
e
i
ϕ
sin
⁡
θ
e
−
i
ϕ
sin
⁡
θ
cos
⁡
θ
]
{\displaystyle U(\mathbf {B} _{\theta ,\phi
})={\begin{bmatrix}\cos \theta &-e^{i\phi
}\sin \theta \\e^{-i\phi }\sin \theta &\cos
\theta \end{bmatrix}}}
,where
θ
{\displaystyle \theta }
and
ϕ
{\displaystyle \phi }
are determined by the reflection amplitude
r
{\displaystyle r}
and the transmission amplitude
t
{\displaystyle t}
(relationship will be given later for a simpler
case). For a symmetric beam splitter, which
has a phase shift
ϕ
=
π
2
{\displaystyle \phi ={\frac {\pi }{2}}}
under the unitary transformation condition
|
t
|
2
+
|
r
|
2
=
1
{\displaystyle |t|^{2}+|r|^{2}=1}
and
t
∗
r
+
t
r
∗
=
0
{\displaystyle t^{*}r+tr^{*}=0}
, one can show that
U
(
B
θ
,
ϕ
=
π
2
)
=
[
t
r
r
t
]
=
[
cos
⁡
θ
−
i
sin
⁡
θ
−
i
sin
⁡
θ
cos
⁡
θ
]
=
cos
⁡
θ
I
^
−
i
sin
⁡
θ
σ
^
x
=
e
−
i
θ
σ
^
x
{\displaystyle U(\mathbf {B} _{\theta ,\phi
={\frac {\pi }{2}}})={\begin{bmatrix}t&r\\r&t\end{bmatrix}}={\begin{bmatrix}\cos
\theta &-i\sin \theta \\-i\sin \theta &\cos
\theta \end{bmatrix}}=\cos \theta {\hat {I}}-i\sin
\theta {\hat {\sigma }}_{x}=e^{-i\theta {\hat
{\sigma }}_{x}}}
,which is a rotation of the single qubit state
about the
x
{\displaystyle x}
-axis by
2
θ
=
2
cos
−
1
⁡
(
|
t
|
)
{\displaystyle 2\theta =2\cos ^{-1}(|t|)}
in the Bloch sphere.
A mirror is a special case where the reflecting
rate is 1, so that the corresponding unitary
operator is a rotation matrix given by
R
(
θ
)
=
[
cos
⁡
θ
−
sin
⁡
θ
sin
⁡
θ
cos
⁡
θ
]
{\displaystyle R(\theta )={\begin{bmatrix}\cos
\theta &-\sin \theta \\\sin \theta &\cos \theta
\\\end{bmatrix}}}
.For most cases of mirrors used in QIP, the
incident angle
θ
=
45
∘
{\displaystyle \theta =45^{\circ }}
.
Similarly, a phase shifter operator
P
ϕ
{\displaystyle \mathbf {P} _{\phi }}
associates with a unitary operator described
by
U
(
P
ϕ
)
=
e
i
ϕ
{\displaystyle U(\mathbf {P} _{\phi })=e^{i\phi
}}
, or, if written in a 2-mode format
U
(
P
ϕ
)
=
[
e
i
ϕ
0
0
1
]
=
[
e
i
ϕ
/
2
0
0
e
−
i
ϕ
/
2
]
(global phase ignored)
=
e
i
ϕ
2
σ
^
z
{\displaystyle U(\mathbf {P} _{\phi })={\begin{bmatrix}e^{i\phi
}&0\\0&1\end{bmatrix}}={\begin{bmatrix}e^{i\phi
/2}&0\\0&e^{-i\phi /2}\end{bmatrix}}{\text{(global
phase ignored)}}=e^{i{\frac {\phi }{2}}{\hat
{\sigma }}_{z}}}
,which is equivalent to a rotation of
−
ϕ
{\displaystyle -\phi }
about the
z
{\displaystyle z}
-axis.
Since any two
S
U
(
2
)
{\displaystyle SU(2)}
rotations along orthogonal rotating axes can
generate arbitrary rotations in the Bloch
sphere, one can use a set of symmetric beam
splitters and mirrors to realize an arbitrary
S
U
(
2
)
{\displaystyle SU(2)}
operators for QIP. The figures below are examples
of implementing a Hadamard gate and a Pauli-X-gate
(NOT gate) by using beam splitters (illustrated
as rectangles connecting two sets of crossing
lines with parameters
θ
{\displaystyle \theta }
and
ϕ
{\displaystyle \phi }
) and mirrors (illustrated as rectangles connecting
two sets of crossing lines with parameter
R
(
θ
)
{\displaystyle R(\theta )}
).
In the above figures, a qubit is encoded using
two mode channels (horizontal lines):
|
0
⟩
{\displaystyle \left\vert 0\right\rangle }
represents a photon in the top mode, and
|
1
⟩
{\displaystyle \left\vert 1\right\rangle }
represents a photon in the bottom mode.
== Integrated photonic circuits for LOQC ==
In reality, assembling a whole bunch (possibly
on the order of
10
4
{\displaystyle 10^{4}}
) of beam splitters and phase shifters in
an optical experimental table is challenging
and unrealistic. To make LOQC functional,
useful and compact, one solution is to miniaturize
all linear optical elements, photon sources
and photon detectors, and to integrate them
onto a chip. If using a semiconductor platform,
single photon sources and photon detectors
can be easily integrated. To separate modes,
there have been integrated arrayed waveguide
grating (AWG) which are commonly used as optical
(de)multiplexers in wavelength division multiplexed
(WDM). In principle, beam splitters and other
linear optical elements can also be miniaturized
or replaced by equivalent nanophotonics elements.
Some progress in these endeavors can be found
in the literature, for example, Refs. In 2013,
the first integrated photonic circuit for
quantum information processing has been demonstrated
using photonic crystal waveguide to realize
the interaction between guided field and atoms.
== Implementations comparison ==
=== Comparison of the KLM protocol and the
boson sampling model ===
The advantage of the KLM protocol over the
boson sampling model is that while the KLM
protocol is a universal model, boson sampling
is not believed to be universal. On the other
hand, it seems that the scalability issues
in boson sampling are more manageable than
those in the KLM protocol.
In boson sampling only a single measurement
is allowed, a measurement of all the modes
at the end of the computation. The only scalability
problem in this model arises from the requirement
that all the photons arrive at the photon
detectors within a short-enough time interval
and with close-enough frequencies.In the KLM
protocol, there are non-deterministic quantum
gates, which are essential for the model to
be universal. These rely on gate teleportation,
where multiple probabilistic gates are prepared
offline and additional measurements are performed
mid-circuit. Those two factors are the cause
for additional scalability problems in the
KLM protocol.
In the KLM protocol the desired initial state
is one in which each of the photons is in
one of two modes, and the possibility that
a mode is occupied by more than one photon
is zero. In boson sampling, however, the desired
initial state is specific, requiring that
the first
N
{\displaystyle N}
modes are each occupied by a single photon
(
N
{\displaystyle N}
is the number of photons and
M
≥
N
{\displaystyle M\geq N}
is the number of modes) and all the other
states are empty.
=== Earlier models ===
Another, earlier model which relies on the
representation of several qubits by a single
photon is based on the work of C. Adami and
N. J. Cerf. By using both the location and
the polarization of photons, a single photon
in this model can represent several qubits;
however, as a result, CNOT-gate can only be
implemented between the two qubits represented
by the same photon.
The figures below are examples of making an
equivalent Hadamard-gate and CNOT-gate using
beam splitters (illustrated as rectangles
connecting two sets of crossing lines with
parameters
θ
{\displaystyle \theta }
and
ϕ
{\displaystyle \phi }
) and phase shifters (illustrated as rectangles
on a line with parameter
ϕ
{\displaystyle \phi }
).
In the optical realization of the CNOT gate,
the polarization and location are the control
and target qubit, respectively
