Linear optical quantum computing or linear optics quantum computation (LOQC), also photonic quantum computing (PQC), 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, or optical instruments (including reciprocal mirrors and waveplates) 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. This is because 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
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