KLM protocol
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.
[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.
For example, it is very easy to consider a loss of a single photon using these notations, simply by adding the vacuum state
The singlet state (two linked photons with overall spin quantum number
In the KLM protocol, a quantum state can be read out or measured using photon detectors along selected modes.
As discussed in KLM's proposal,[1] photon loss and detection efficiency dramatically influence the reliability of the measurement results.
A left-pointed triangle will be used in circuit diagrams to represent the state readout operator in this article.
[1] 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.
) and mirrors (illustrated as rectangles connecting two sets of crossing lines with parameter
In the above figures, a qubit is encoded using two mode channels (horizontal lines):
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.
is the phase shift of the output, and is determined by the parameters of inner optical elements chosen.
By sharing two ancilla modes, Knill invented the following controlled-Z gate (see the figure on the right) with success rate of 2/27.
[5] 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.
To further improve successful rate and solve the scalability problem, one needs to use gate teleportation, described next.
[4] The basic idea is that each probabilistic gate is prepared offline, and the successful event signal is teleported back to the quantum circuit.
The number of gates needed to realize a certain accuracy scales polynomially rather than exponentially.
One experiment using the KLM originally proposed controlled-NOT gate with four-photon input was demonstrated in 2011,[6] and gave an average fidelity of
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
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 section discusses the improvements of the KLM protocol that have been studied after the initial proposal.
