Memory-Optimal Sequential Synthesis of Multimode Gaussian Transformations

Abstract: In modular quantum computing architectures, communication between hardware modules is mediated by traveling qumodes sent through transmission lines. Each output qumode interacts with the emitting module only once through a beam-splitter-type interaction and becomes inaccessible to that module after emission. Information required for subsequent outputs must therefore remain in long-lived memory qumodes. For a prescribed multimode Gaussian transformation on $N$ qumodes, this work determines the minimum memory cost for any given emission order, constructs an explicit sequential protocol attaining this minimum, and develops a greedy method for identifying memory-efficient emission orders. The transformation is represented by a symplectic matrix $S$, specified either directly or through a Gaussian gate sequence. The exact minimum memory cost is obtained from the ranks of submatrices of $S$ and further reduces to a support-based counting rule whose computational cost is linear in the size of the support data. When $S$ is specified directly, a matrix-based protocol attains the minimum memory cost. If instead $S$ is specified through a gate sequence, the original gates can be reused without additional synthesis, although the resulting memory usage need not be minimal. Gaussian transformations with local support on a $D$-dimensional cubic lattice can be realized sequentially with $O(N<sup>{(D-1)/D})$ memory qumodes. The protocols also apply to non-Gaussian inputs, including GKP and cat states, and thereby provide an explicit, resource-efficient scheme for intermodule communication in modular architectures for universal continuous-variable quantum computation.