Computational complexity of optimal Gaussian emission-order selection

Determine whether optimizing the emission order of an arbitrary multimode Gaussian transformation to minimize the required number of memory qumodes is NP-hard.

Background

For a fixed emission order, the paper derives the exact minimum memory cost from the support pattern of the symplectic transformation. However, selecting the order itself requires optimization over all permutations of the outputs. The paper presents a deterministic greedy heuristic, but explicitly notes that it does not guarantee a globally optimal order.

The unresolved issue is the computational complexity of finding an exactly optimal emission order for an arbitrary Gaussian transformation, particularly whether the problem is NP-hard, as is known for the analogous optimization problem for qubit graph states. The result would clarify the algorithmic difficulty of minimizing memory over emission orders.

References

Whether or not optimizing the emission order of an arbitrary Gaussian transformation to minimize the required number of memory qumodes is also NP-hard remains unknown.

— Memory-Optimal Sequential Synthesis of Multimode Gaussian Transformations  (2609.05250 - Guo et al., 4 Sep 2026) in Section Conclusion and Outlook

For locally connected lattice families, characterizing how the minimum memory cost depends on the lattice geometry and emission order also remains an open problem.

— Memory-Optimal Sequential Synthesis of Multimode Gaussian Transformations  (2609.05250 - Guo et al., 4 Sep 2026) in Section Conclusion and Outlook