The KLM scheme or KLM protocol is an implementation of linear optical quantum computing (LOQC) developed in 2000 by Emanuel Knill, Raymond Laflamme and Gerard J. Milburn. This protocol allows for the creation of universal quantum computers using solely linear optical tools.[1] The KLM protocol 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.
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.[2] 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.[3] [4] Without a high enough success rate of a single quantum gate unit, it may require an exponential amount of computing resources. 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. 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 quantum information processing as other known implementations.
To avoid losing generality, the discussion below does not limit itself to a particular instance of mode representation. A state written as
|0,1\rangleVH
V
H
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\rangle\equiv|0,1\rangleVH
|1\rangle\equiv|1,0\rangleVH
Such notations are useful in quantum computing, quantum communication and quantum cryptography. For example, it is very easy to consider a loss of a single photon using these notations, simply by adding the vacuum state
|0,0\rangleVH
s=0
|1,0\rangle
| a, | |
| VH |
|0,1\rangle
| a | |
| VH |
|1,0\rangle
| b, | |
| VH |
|0,1\rangle
| b | |
| VH |
(|1,0\rangle
| a | |
| VH |
|0,1\rangle
| b | |
| VH |
-|0,1\rangle
| a | |
| VH |
|1,0\rangle
| b)/\sqrt2. | |
| VH |
In the KLM protocol, a quantum state can be read out or measured using photon detectors along selected modes. If a photodetector detects a photon signal in a given mode, it means the corresponding mode state is a 1-photon state before measuring. As discussed in KLM's proposal, photon loss and detection efficiency dramatically influence the reliability of the measurement results. The corresponding failure issue and error correction methods will be described later.
A left-pointed triangle will be used in circuit diagrams to represent the state readout operator in this article.
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\theta,\phi
U(B\theta,\phi) =\begin{bmatrix} \cos\theta&-ei\phi\sin\theta\\ e-i\phi\sin\theta&\cos\theta\end{bmatrix}
where
\theta
\phi
r
t
| \phi= | \pi |
| 2 |
|t|2+|r|2=1
t*r+tr*=0
| U(B | ||||
|
) =\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
| -i\theta\hat{\sigma | |
| \hat{\sigma} | |
| x} |
which is a rotation of the single qubit state about the
x
2\theta=2\cos-1(|t|)
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(\theta)=\begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}
\theta=45\circ
Similarly, a phase shifter operator
P\phi
| i\phi | |
| U(P | |
| \phi)=e |
U(P\phi)=\begin{bmatrix} ei\phi&0\\ 0&1\end{bmatrix}=\begin{bmatrix}ei\phi/2&0\\ 0&e-i\phi/2\end{bmatrix}
| |||||
| (globalphaseignored)=e | |||||
| z} |
which is equivalent to a rotation of
-\phi
z
Since any two 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
SU(2)
\theta
\phi
R(\theta)
In the above figures, a qubit is encoded using two mode channels (horizontal lines):
\left\vert0\right\rangle
\left\vert1\right\rangle
In the KLM scheme, qubit manipulations are realized via a series of non-deterministic operations with increasing probability of success. The first improvement to this implementation that will be discussed is the nondeterministic conditional sign flip gate.
An important element of the KLM scheme is the conditional sign flip or nonlinear sign flip gate (NS-gate) as shown in the figure below on the right. It gives a nonlinear phase shift on one mode conditioned on two ancilla modes.
In the picture on the right, the labels on the left of the bottom box indicate the modes. The output is accepted only if there is one photon in mode 2 and zero photons in mode 3 detected, where the ancilla modes 2 and 3 are prepared as the
|1,0\rangle2,3
x
x=-1
| \circ | |
| \theta | |
| 1=22.5 |
| \circ | |
| \phi | |
| 1=0 |
| \circ | |
| \theta | |
| 2=65.5302 |
| \circ | |
| \phi | |
| 2=0 |
| \circ | |
| \theta | |
| 3=-22.5 |
| \circ | |
| \phi | |
| 3=0 |
| \circ | |
| \phi | |
| 4=180 |
x=ei\pi/2
| \circ | |
| \theta | |
| 1=36.53 |
| \circ | |
| \phi | |
| 1=88.24 |
| \circ | |
| \theta | |
| 2=62.25 |
| \circ | |
| \phi | |
| 2=-66.53 |
| \circ | |
| \theta | |
| 3=-36.53 |
| \circ | |
| \phi | |
| 3=-11.25 |
| \circ | |
| \phi | |
| 4=102.24 |
The advantage of using NS gates is that the output can be guaranteed conditionally processed with some success rate which can be improved to nearly 1. Using the configuration as shown in the figure above on the right, the success rate of an
x=-1
1/4
Given the use of non-deterministic quantum gates for KLM, there may be only a very small probability
pN
N
p
p-N
p-N
|\Phi+\rangle=
| 1 | |
| \sqrt{2 |
|\Phi+\rangle
|10\rangle
\theta=
| \pi | |
| 4 |
.
By using teleportation, many probabilistic gates may be prepared in parallel with
n
n
| n2 | |
| (n+1)2 |
n
F=0.82\pm0.01
As discussed above, the success probability of teleportation gates can be made arbitrarily close to 1 by preparing larger entangled states. However, the asymptotic approach to the probability of 1 is quite slow with respect to the photon number
n
n+1
Many experimental trials using this idea have been carried out (see, for example, Refs[7] [8] [9]). However, a large number of operations are still needed to achieve a success probability very close to 1. In order to promote the KLM protocol as a viable technology, more efficient quantum gates are needed. This is the subject of the next part.
This section discusses the improvements of the KLM protocol that have been studied after the initial proposal.
There are many ways to improve the KLM protocol for LOQC and to make LOQC more promising. Below are some proposals from the review article Ref.[10] and other subsequent articles:
There are several protocols for using cluster states to improve the KLM protocol, the computation model with those protocols is an LOQC implementation of the one-way quantum computer: