Fermionic formulation of quadratic-symmetry learning

Determine whether quadratic symmetries in the Majorana representation can be efficiently sampled and simultaneously block-diagonalised to recover a hidden tensor-product or mode decomposition up to a fermionic Gaussian transformation.

Background

The paper develops a qubit-based method that extracts hidden subsystem structure from quadratic symmetries of Bell distributions and uses simultaneous block-diagonalisation to recover a disentangling Clifford.

A proposed unresolved extension is to formulate an analogous procedure directly for fermionic systems. Such a formulation would need to account for fermionic parity and the fact that fermionic modes do not possess an ordinary subsystem tensor-product structure, potentially enabling learning and disentangling algorithms without first mapping fermions to qubits.

References

A further open question is whether the same strategy admits a genuinely fermionic formulation. In particular, can analogous quadratic symmetries in the Majorana representation be efficiently sampled and simultaneously block-diagonalised so as to recover a hidden tensor-product or mode decomposition up to a fermionic Gaussian transformation?

— Efficient learning of Clifford disentanglers and typical $t$-doped unitaries with exponentially more $T$ gates  (2609.27565 - Aguilar et al., 23 Sep 2026) in Discussion and future work, paragraph “Open questions”