Depth-independent random-spin-glass NLTS ensemble

Determine whether there exists a single random bounded-degree Hamiltonian ensemble, together with a positive energy-density gap independent of circuit depth, that rules out near-ground-state preparation by circuits of every constant depth, thereby establishing an NLTS-type result for random spin glasses.

Background

The paper studies state-preparation complexity for dense random quantum p-spin Hamiltonians and obtains depth and width tradeoffs, but its cited earlier bounded-degree result allows the required degree prefactor to depend on the circuit depth.

The unresolved problem is to choose one bounded-degree random Hamiltonian ensemble and one positive energy-density separation that work uniformly across all constant depths. A solution would provide an NLTS-style obstruction for random spin-glass Hamiltonians.

References

Finally, can one fixed random bounded-degree Hamiltonian ensemble rule out near-ground-state preparation at every constant circuit depth? An earlier result of~\ allows the required degree prefactor to depend on the depth. Choosing both the ensemble and a positive energy-density gap independently of depth would yield an NLTS-type result for random spin glasses~.

— Beyond Light Cones: State Preparation Complexity in Quantum Spin Glasses  (2610.02166 - Al-Ghattas et al., 1 Oct 2026) in Section 'Open questions'