Extending the circuit–fermion correspondence beyond the free-fermion limit

Develop the correspondence between unitary tensor-network circuits (with complex entries), specifically matchgate circuits representing free fermionic tensors, and non-Hermitian fermionic Hamiltonians, and extend this correspondence beyond the free-fermion (matchgate) limit to include interacting or non-linear fermionic terms.

Background

In the Outlook, the authors propose restricting tensors to unitary gates with complex entries to explore connections among quantum circuits, statistical mechanics, and fermionic systems. They note that for free fermionic tensors (matchgate circuits) a correspondence with non-Hermitian fermionic Hamiltonians can be established.

The authors explicitly state that extending this correspondence beyond the free-fermion limit is left for future research, indicating an unresolved task to generalize the mapping to settings with interactions or non-linearities.

References

This correspondence, which likewise affords extension beyond the free fermion limit, is left for future research.

— A minimal tensor network beyond free fermions  (2412.04216 - Wille et al., 2024) in Section 4.2 (Outlook)

In contrast to integrating out the ancilla to obtain effective non-Hermitian dynamics on the chain, one could ask whether particular non-Hermitian problems admit a unitary embedding with a hidden qubit of exactly this kind .

— Crossover from Fast Scrambling to Operator Confinement Tuned by an Auxiliary Qubit  (2608.16674 - Buechele et al., 17 Aug 2026) in Discussion and Conclusions