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. 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.
Overview 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. 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.
Elements of LOQC in the KLM scheme
Qubits and modes To avoid losing generality, the discussion below does not limit itself to a particular instance of mode representation. A state written as | 0 , 1 ⟩ V H {\displaystyle |0,1\rangle _{VH}} means a state with zero photons in mode V {\displaystyle V} (could be the "vertical" polarization channel) and one photon in the mode H {\displaystyle H} (could be the "horizontal" polarization channel). 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. 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 ⟩ V H {\displaystyle |0,0\rangle _{VH}} containing zero photons in those two modes. As another example, when having two photons in two separated modes (e.g. two time bins or two arms of an interferometer), it is easy to describe an entangled state of the two photons. The singlet state (two linked photons with overall spin quantum number s = 0 {\displaystyle s=0} ) can be described as follows: if | 1 , 0 ⟩ V H a , | 0 , 1 ⟩ V H a {\displaystyle |1,0\rangle _{VH}^{a},|0,1\rangle _{VH}^{a}} and | 1 , 0 ⟩ V H b , | 0 , 1 ⟩ V H b {\displaystyle |1,0\rangle _{VH}^{b},|0,1\rangle _{VH}^{b}} describe the basis states of the two separated modes, then the singlet state is ( | 1 , 0 ⟩ V H a | 0 , 1 ⟩ V H b − | 0 , 1 ⟩ V H a | 1 , 0 ⟩ V H b ) / 2 . {\displaystyle (|1,0\rangle _{VH}^{a}|0,1\rangle _{VH}^{b}-|0,1\rangle _{VH}^{a}|1,0\rangle _{VH}^{b})/{\sqrt {2}}.}
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