Linear optics implementation of symmetric matchgates

An essential requirement for any quantum computing architecture is the existence of a universal set of quantum gates. Examples of such sets are the 2-qubit CNOT [1,2] gate in combination with arbitrary 1-qubit gates, and the 3-qubit Toffoli gate [3, 4] together with the 1-qubit Hadamard. A third set, which has not been implemented so far, is Valiant's 2-qubit matchgate G(A,B) which is universal with the SWAP operation [5]. The Valiant gate has the form of equations where A and B are 1-qubit unitaries which act on the respective even and odd parity subspace of the 2-qubit state. One particularly intriguing property of the Valiant gate is that it can be efficiently simulated with classical computers [5] as long as it is restricted to act between nearest neighbors only. Once the interactions are extended to next nearest neighbors, however, the Valiant gate turns our to be universal for quantum computing. In practice, this extension can be achieved with a two-qubit SWAP operation, which is exceedingly simple in linear optics.