Linear optical quantum computing

Linear Optical Quantum Computing or Linear Optics Quantum Computation (LOQC) is a paradigm of 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 of linear optical quantum computation

Although there are many other implementations for quantum information processing (QIP) and computation, optical quantum systems are prominent candidates for QIP, since they link quantum computation and quantum communication in the same framework. Among the 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 $N\times N$ unitary matrix ($U(N)$) operations can be realized by only using mirrors, beam splitters and phase shifters (footnote: it is also a starting point of Boson sampling and computational complexity analysis for LOQC). It points out that each $U(N)$ operator with $N$ inputs and $N$ outputs can be constructed via $\mathcal{O}(N^2)$ 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 the sense 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 optics 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 optics 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 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.

DiVincenzo's criteria for quantum computation and QIP give that a system for QIP should satisfy at least the following requirements:

1. a scalable physical system with well characterized qubits,
2. the ability to initialize the state of the qubits to a simple fiducial state, such as $|000\cdots\rangle$,
3. long relevant decoherence times, much longer than the gate operation time,
4. a "universal" set of quantum gates,
5. a qubit-specific measurement capability; if the system is also aiming for quantum communication, it should also satisfy at least the following two requirements:
6. the ability to interconvert stationary and flying qubits, and
7. the ability faithfully to 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 show that LOQC is a good candidate for QIP.

Qubits and modes

A qubit is one of the fundamental QIP units. A qubit state which can be represented by $\alpha |0\rangle + \beta|1\rangle$ is a superposition state with probability $|\alpha|^2$ of being in the $|0\rangle$ state and probability $|\beta|^2$ of being in the $|1\rangle$ state, where $|\alpha|^2+|\beta|^2=1$ is the normalization condition. The states $|0\rangle$ and $|1\rangle$ could correspond to 0-photon and 1-photon in a given mode channel. In general, there could be $|n\rangle, n=0,1,2,\cdots$ photon states for existing $n$-photon cases. An optical mode is a physically 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 channels of light which can be picked out with linear optics elements, various frequency channels, or a combination of the two cases above.To avoid losing generality, the discussion below does not limit itself to a particular instance of mode representation. A state written as $|01\rangle _{VH}\equiv |0\rangle _V |1\rangle _H$ means a state with 0 photon in mode $V$ (could be the "vertical" polarization channel) and 1 photon in the mode $H$ (could be the "horizontal" polarization channel). This is a two-qubit case, as two independent modes are used.

State preparation

To prepare a desired quantum state for LOQC, usually a single-photon state, single-photon generators and some optical modules will be employed. For example, optical parametric down-conversion can be used to conditionally generate the $|1\rangle$ state in the vertical polarization channel at time $t$ (subscripts are ignored for this single qubit case). By using conditional single-photon source the output state is guaranteed, although there is a cost associated with the success rate. A joint multi-qubit state can be prepared in a similar (possibly more sophisticated) way. In general, an arbitrary quantum state can be generated for QIP with a proper set of photon sources.

A right-pointed triangle is used to represent the state preparation operator in circuit digrams in this article, following KLM's convention.

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, LOQC 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 LOQC implementation of some basic quantum gates 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 $\phi$).

File:Linear optics H gate.svg Implementation of H-gate with a beam splitter and phase shifters. Quantum circuit is on the top part.

File:Linear optics CNOT gate.svg Implementation of Controlled-NOT-gate with a beam splitter. Quantum circuit is on the top part.

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.

File:NSGate.JPG

Linear optics implementation of NS-gate. The elements framed in the box with dashed border is the linear optics implementation with three beam splitters and one phase shifter (see text for parameters). Modes 2 and 3 are 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.

File:NSCZGate.JPG

Linear optics implementation of Controlled-Z Gate with ancilla modes labelled as 2 and 3. $\theta=54.74^\circ$ and $\theta'=17.63^\circ$.

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 LOQC, 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.

Quantum circuit representation of quantum teleportation.

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 linear optical quantum computing 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.
• The Yoran-Reznik protocol.
• The Nielsen protocol.
• The Browne-Rudolph protocol.
• Circuit-based optical quantum computing revisited.
• Using one-step deterministic multipartite entanglement purification with linear optics to generate entangled photon states.

Integrated photonic circuits for LOQC

In reality, assembling a whole bunch (possibly on the order of $10^4$) 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.