Gate-minimal symplectic completions and decompositions

Determine symplectic completions and corresponding Gaussian gate decompositions that minimize the total number of beam splitters and single-qumode squeezers when a multimode Gaussian transformation is specified by its symplectic matrix.

Background

The matrix-based sequential protocol fixes the symplectic row pair corresponding to the output to be emitted, then completes that pair to a full symplectic transformation acting on the memory register. The completion is generally nonunique, and different completions can produce different elementary Gaussian circuits after Bloch–Messiah decomposition.

The protocol is memory-optimal regardless of this freedom, but the paper does not solve the separate compilation problem of choosing completions and decompositions that minimize the total number of beam splitters and single-qumode squeezers. This is relevant when elementary-gate count, rather than memory alone, is the principal resource.