Generation of universal linear optics by any beam splitter

Abstract

In 1994, Reck et al. showed how to realize any unitary transformation on a single photon using a product of beam splitters and phase shifters. Here we show that any single beam splitter that nontrivially mixes two modes also densely generates the set of unitary transformations (or orthogonal transformations, in the real case) on the single-photon subspace with m 3 modes. (We prove the same result for any two-mode real optical gate, and for any two-mode optical gate combined with a generic phase shifter.) Experimentally, this means that one does not need tunable beam splitters or phase shifters for universality: any nontrivial beam splitter is universal for linear optics. Theoretically, it means that one cannot produce "intermediate" models of linear optical computation (analogous to the Clifford group for qubits) by restricting the allowed beam splitters and phase shifters: there is a dichotomy; one either gets a trivial set or else a universal set. No similar classification theorem for gates acting on qubits is currently known. We leave open the problem of classifying optical gates that act on three or more modes.

Coincidence landscapes for three-channel linear optical networks

“…States (27) are, respectively, the = 1, m = 0 (triplet) state and the = 0, m = 0 (singlet) state, which can be obtained from the usual theory of two-mode systems in terms of angular momentum. The interferometric input and output states can be expanded in terms of |Ψ ± , and the effect of permuting frequencies of the output state is determined from the permutation of frequencies on |Ψ ± .…”

Robust Architecture for Programmable Universal Unitaries

“…which is the core requirement for the network designs of [15,16] to decompose any arbitrary unitary. It should be noted that in principle any fixed beamsplitter transformation mixing a pair of optical modes densely generates SU(N ) [31]. However, for practical purposes it is more efficient to implement a network scheme allowing to program the required unitary.…”

Sampling arbitrary photon-added or photon-subtracted squeezed states is in the same complexity class as boson sampling

“…The exact permanent case is known to be #P complete even for binary entries, Utj e {0,1} [35], There is also a known algorithm for efficiently approximating a permanent if the matrix has entries consisting of only non negative real numbers. In the same work, it is shown that for a matrix with even a single negative entry, an efficient approximation algorithm would allow one to compute an exact {0,1} permanent efficiently [36], Although having to compute a difficult permanent is a necessary but not sufficient condition for computational hardness, since SO(w) is considered to be universal for linear optics [37], there is no such complexity gap between unitary and orthogonal matrices.…”