User:Yaniv.N/sandbox/KLM model

KLM protocol is a paradigm of Linear optical quantum computing, allowing universal quantum computation

Overview of the KLM protocol
An intrinsic problem in using photons as information carriers is that photons hardly interact with each other. This potentially causes the scalability problem of LOQC, since nonlinear operations are hard to implement which can increase the complexity of operators and hence can reduce the resources required to realize a given computational function. There are basically two ways to solve this problem. One is to bring in 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. 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 optics 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 optics elements. The detailed descriptions below will follow the KLM scheme and subsequent improvements upon the KLM scheme.

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.

Elements of LOQC
The basic building blocks for LOQC are introduced below. As discussed above, the KLM scheme will mainly be followed at first, and the next section will introduce improvements for LOQC that have been studied after KLM's proposal.

State measurement/readout of KLM protocol
In the KLM protocol, a quantum state can be readout 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.

Implementations of elementary quantum gates
To achieve universal quantum computing, KLM should be capable of realizing a complete set of universal gates (please refer to the quantum gate article for the universality of quantum gates). The implementation of some basic quantum gates in the KLM protocol are shown here.

Ignoring error correction and other issues, implementations of elementary quantum gates using only mirrors, beam splitters and phase shifters have been summarized in some early publications. See, for example, Ref. The basic principle is that using these linear optics elements, one can construct an arbitrary (at least) 2-qubit unitary operation which links 2 or dual-rail qubits; in other words, those linear optical elements support a complete set of $$SU(2)$$ operators. For example, the unitary matrix associated with a beam splitter $$\mathbf{B}_{\theta,\phi}$$ is


 * $$ U(\mathbf{B}_{\theta,\phi})

=\begin{bmatrix} \cos \theta & -e^{i\phi}\sin \theta \\ e^{-i\phi} \sin \theta & \cos \theta \end{bmatrix}\,,$$

where $$\theta$$ and $$\phi$$ are determined by the reflection amplitude $$r$$ and the transmission amplitude $$t$$ (relationship will be given later for a simpler case). For a symmetric beam splitter, which has a phase shift $$\phi=\frac{\pi}{2}$$ under the unitary transformation condition $$|t|^2+|r|^2=1$$ and $$t^*r+tr^*=0$$, one can show that


 * $$ 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 dual-rail qubit about the $$x$$-axis by $$2\theta=2\cos^{-1}(|t|)$$ in the Bloch sphere with Pauli operator $$\hat{\sigma}_x$$.

A mirror is a special case that 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}\,. $$

For most cases of mirrors used in QIP, the incident angle $$\theta=45^\circ$$. Similarly, a phase shifter operator $$\mathbf{P}_\phi$$ associates with a unitary operator described by $$U(\mathbf{P}_\phi)=e^{i\phi}$$, or, if written in a 2-qubit format


 * $$ 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 $$-\phi$$ about the $$z$$-axis. 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 phase shifters to realize an arbitrary $$SU(2)$$ operators for QIP. The figures below are examples of making an equivalent H-gate and CNOT-gate using beam splitters (illustrated as rectangles connecting two sets of crossing lines with parameters $$\theta$$ and $$\phi$$) and phase shifters (illustrated as rectangles on a line with parameter $$\phi$$).

In the pictures showing the implementations of the quantum gates, each qubit is encoded using two mode channels (horizontal lines), such that $$\left\vert0\right\rangle$$ represents a photon in the top mode, and $$\left\vert1\right\rangle$$ represents the photon in the bottom mode. The control line is not shown in the optical realization of the CNOT gate.

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.

Implementation of 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 $$|10\rangle_{2,3}$$ state. The subscript $$x$$ is the phase shift of the output, and is determined by the parameters of inner optical elements chosen. For $$x=-1$$ case, the following parameters are used: $$\theta_1=22.5^\circ$$, $$\phi_1=0^\circ$$, $$\theta_2=65.5302^\circ$$, $$\phi_2=0^\circ$$, $$\theta_3=-22.5^\circ$$, $$\phi_3=0^\circ$$, and $$\phi_4=180^\circ$$. For the $$x=e^{i\pi/2}$$ case, the parameters can be chosen as $$\theta_1=36.53^\circ$$, $$\phi_1=88.24^\circ$$, $$\theta_2=62.25^\circ$$, $$\phi_2=-66.53^\circ$$, $$ \theta_3=-36.53^\circ$$, $$\phi_3=-11.25^\circ$$, and $$\phi_4=102.24^\circ$$. Similarly, by changing the parameters of beam splitters and phase shifters, or by combining multiple NS gates, one can create various quantum gates. By sharing two ancilla modes, Knill invented the following controlled-Z gate (see the figure on the right) with success rate of 2/27.

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 $$x=-1$$ NS gate is $$1/4$$. To further improve successful rate and solve the scalability problem, one needs to use gate teleportation, described next.

Gates teleportation and near-deterministic gates
Given the use of non-deterministic quantum gates for KLM, there may be only a very small probability $$p^N$$ that a circuit with $$N$$ gates with a single-gate success possibility of $$p$$ will work perfectly by running the circuit once. Therefore, the operations must on average be repeated on the order of $$p^{-N}$$ times or $$p^{-N}$$ such systems must be run in parallel. Either way, the required time or circuit resources scale exponentially. In 1999, Gottesman and Chuang pointed out that one can prepare the probabilistic gates offline from the quantum circuit by using quantum teleportation. The basic idea is that each probabilistic gate is prepared offline, and the successful event signal is teleported back to the quantum circuit. An illustration of quantum teleportation is given in the figure on the right. As can be seen, the quantum state in mode 1 is teleported to mode 3 through a Bell measurement and an entangled resource Bell state $$|\Phi^+\rangle$$, where the state 1 may be regarded as prepared offline.



By using teleportation, many probabilistic gates may be prepared in parallel with $$n$$-photon entangled states, sending a control signal to the output mode. Through using $$n$$ probabilistic gates in parallel offline, a success rate of $$\frac{n^2}{(n+1)^2}$$ can be obtained, which is close to 1 as $$n$$ becomes large. The number of gates needed to realize a certain accuracy scales polynomially rather than exponentially. In this sense, the KLM protocol is resource-efficient. One experiment using the KLM originally proposed controlled-NOT gate with four-photon input was demonstrated in 2011, and gave an average fidelity of $$F=0.82\pm 0.01$$. This result shows that the KLM proposal is feasible for quantum computing tasks.

Error detection and correction
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 $$n$$. A more efficient approach is to encode against gate failure (error) based on the well-defined failure mode of the teleporters. In the KLM protocol, the teleporter's failure can be diagnosed if zero or $$n+1$$ photons are detected. If the computing device can be encoded against accidental measurements of some certain number of photons, then it will be possible to correct gate failures and the probability of eventually successfully applying the gate will increase.

Many experimental trials using this idea has been carried out (see, for example, Refs ). 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.

Improvements of KLM protocol
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. and other subsequent articles:


 * Using cluster states in optical quantum computing.
 * Circuit-based optical quantum computing revisited.
 * Using one-step deterministic multipartite entanglement purification with linear optics to generate entangled photon states.

there are several protocols for using cluster states to improve the KLM protocol:
 * The Yoran-Reznik protocol - this protocol make use of cluster-chains in order to increase the success probability of teleportation.
 * The Nielsen protocol - this protocol improves The Yoran-Reznik protocol by first using teleportation to add qubits to cluster-chains and then uses the enlarged cluster chains to further increase the success probability of teleportation.
 * The Browne-Rudolph protocol - this protocol improves The Nielsen protocol by using teleportation not only to add qubits to cluster-chains but also to fuse them.