Fast differentiable evolution of quantum states under Gaussian transformations

Understand

In a recent work we presented a recursive algorithm to compute the matrix elements of a generic Gaussian transformation in the photon-number basis.

• Its purpose was to evolve a quantum state by building the transformation matrix and subsequently computing the matrix-vector product.
• Here we present a faster algorithm that computes the final state without having to generate the full transformation matrix first.
• With this algorithm we bring the time complexity of computing the Gaussian evolution of an $N$-dimensional $M$-mode state from $O(MN^{2M})$ to $O(M(N^2/2)^M)$, which is an exponential improvement in the number of modes.