Optical cluster states are a proposed tool to achieve quantum computational universality in linear optical quantum computing (LOQC). As direct entangling operations with photons often require nonlinear effects, probabilistic generation of entangled resource states has been proposed as an alternative path to the direct approach.
Creation of the cluster state On a silicon photonic chip, one of the most common platforms for implementing LOQC, there are two typical choices for encoding quantum information, though many more options exist. Photons have useful degrees of freedom in the spatial modes of the possible photon paths or in the polarization of the photons themselves. The way in which a cluster state is generated varies with which encoding has been chosen for implementation. Storing information in the spatial modes of the photon paths is often referred to as dual rail encoding. In a simple case, one might consider the situation where a photon has two possible paths, a horizontal path with creation operator a † {\displaystyle a^{\dagger }} and a vertical path with creation operator b † {\displaystyle b^{\dagger }} , where the logical zero and one states are then represented by
a † | 0 a , 0 b ⟩ = | 1 a , 0 b ⟩ = | 0 ⟩ L {\displaystyle a^{\dagger }|0_{a},0_{b}\rangle =|1_{a},0_{b}\rangle =|0\rangle _{L}}
and
b † | 0 a , 0 b ⟩ = | 0 a , 1 b ⟩ = | 1 ⟩ L {\displaystyle b^{\dagger }|0_{a},0_{b}\rangle =|0_{a},1_{b}\rangle =|1\rangle _{L}} . Single qubit operations are then performed by beam splitters, which allow manipulation of the relative superposition weights of the modes, and phase shifters, which allow manipulation of the relative phases of the two modes. This type of encoding lends itself to the Nielsen protocol for generating cluster states. In encoding with photon polarization, logical zero and one can be encoded via the horizontal and vertical states of a photon, e.g.
| H ⟩ = | 0 ⟩ L {\displaystyle |H\rangle =|0\rangle _{L}}
and
| V ⟩ = | 1 ⟩ L {\displaystyle |V\rangle =|1\rangle _{L}} . Given this encoding, single qubit operations can be performed using waveplates. This encoding can be used with the Browne-Rudolph protocol.
Nielsen protocol In 2004, Nielsen proposed a protocol to create cluster states, borrowing techniques from the Knill-Laflamme-Milburn protocol (KLM protocol) to probabilistically create controlled-Z connections between qubits which, when performed on a pair of | + ⟩ = | 0 ⟩ + | 1 ⟩ {\displaystyle |+\rangle =|0\rangle +|1\rangle } states (normalization being ignored), forms the basis for cluster states. While the KLM protocol requires error correction and a fairly large number of modes in order to get very high probability two-qubit gate, Nielsen's protocol only requires a success probability per gate of greater than one half. Given that the success probability for a connection using n {\displaystyle n} ancilla photons is n 2 / ( n + 1 ) 2 {\displaystyle n^{2}/(n+1)^{2}} , relaxation of the success probability from nearly one to anything over one half presents a major advantage in resources, as well as simply reducing the number of required elements in the photonic circuit. To see how Nielsen brought about this improvement, consider the photons being generated for qubits as vertices on a two dimensional grid, and the controlled-Z operations being probabilistically added edges between nearest neighbors. Using results from percolation theory, it can be shown that as long as the probability of adding edges is above a certain threshold, there will exist a complete grid as a sub-graph with near unit probability. Because of this, Nielsen's protocol doesn't rely on every individual connection being successful, just enough of them that the connections between photons allow a grid.
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