Local parent Hamiltonians with near-CAT ground states and inverse-linear gap

Construct a local Hamiltonian whose ground state is close to a CAT state and whose spectral gap is \(\Omega(1/n)\), potentially by using a low-depth circuit to distill a CAT state from the critical XXZ ground state and then constructing a local parent Hamiltonian.

Background

The paper derives upper bounds on the spectral gap when a ground state has CAT-like correlations and shows that the resulting inverse-linear scaling is nearly tight using a critical XXZ Hamiltonian. However, the XXZ example is not itself close to a CAT state. The authors explicitly state that they do not know a local Hamiltonian realizing both closeness to a CAT state and an Ω(1/n)\Omega(1/n) gap, and mention low-depth distillation as a possible approach whose implementation remains unresolved.

References

Though the bound is nearly tight, we do not know of a local Hamiltonian with a ground state that is close to the CAT state itself and a spectral gap \Omega(1/n). One idea would be to develop a low-depth circuit to ``distill'' a cat state from the above CFT state and use this to construct a local parent Hamiltonian (possibly allowing for logarithmic-range interactions). However, it is unclear to us how to implement the details.

— Depth-1 expanders on the unitary group and applications  (2609.01605 - Anshu et al., 1 Sep 2026) in Section “Spectral gap versus ground state correlation,” subsection “Tightness of the bound,” final paragraph