Learning states sparse in an unknown Gaussian-rotated Fock basis

Determine whether a fermionic state promised to be k-sparse in an unknown Gaussian-rotated Fock basis can be learned using polynomially many samples and polynomial time in m, k, and 1/ε, and determine the corresponding bosonic guarantee with an appropriate energy-constraint dependence.

Background

The fixed-basis question is extended to an unknown basis obtained by a Gaussian rotation. The proposed target is polynomial sample and time complexity in the number of modes, sparsity, and inverse accuracy for fermions, with an additional dependence on an energy constraint for bosons. The question remains unresolved because the cited optimal-sample algorithms are not known to be efficiently implementable and the efficient alternatives are not sample optimal.

References

We then ask the further question -- if a state is promised to be $k$-sparse in some unknown Gaussian rotated Fock basis, can we learn it with $poly(m,k,1/\eps)$ samples and time for fermions, possibly with an energy constraint dependence for bosons?

— Learning Sparse Quantum States  (2609.12219 - Sen, 10 Sep 2026) in Section "Open Questions"

The gap to our upper bounds remains open: neither optimal dependence on $n$ nor optimal classical runtime is claimed.

— Efficient Learning of Structured Fermionic States under General Quadratic Evolution  (2610.08723 - Amidi et al., 6 Oct 2026) in Section V, “Scope and interpretation”