[challenge]: Certified spectral gaps for blockaded Rydberg chains

Released by

Jie Wang (AMSS, Chinese Academy of Sciences) & Jin-Guo Liu (Hong Kong University of Science and Technology (Guangzhou))

Contact email

cacate0129@gmail.com

Method

Other

Challenge issue

Difficulty: ★ (rated by Jie Wang)

Background

SDP hierarchies now certify spectral gaps — not just energies — for frustration-free spin systems (Quantum 10.22331/q-2026-04-13-2065). PXP-type blockade-constrained Rydberg chains, central to neutral-atom quantum simulation and QEC, sit just outside the frustration-free assumptions; no certified gap statements exist for them.

Research objective

Adapt the gap-certificate hierarchy to the Pauli/projector algebra of blockade-constrained chains: express blockade constraints as localizing constraints, exploit translation invariance (arXiv:2301.06142, arXiv:2212.03014), and produce the first certified gap/gaplessness windows for experimentally relevant neutral-atom Hamiltonians across their parameter space.

Verification plan

• Success gate: certified gap lower bounds that (i) contain the exact-diagonalization gap for all system sizes n ≤ 20 (containment against ground truth) and (ii) remain bounded away from zero in the thermodynamic-limit relaxation over a stated parameter window, with exportable, independently checkable certificates.
• Hope signal: certificates work deep in the blockade regime but degrade near criticality — the degradation curve is itself physics.
• Pivot signal: gap certificates never beat trivial bounds at reachable levels — document the constraint-algebra obstruction and stop.

Why this may lead to research output

Certified gap bounds for PXP-class models would be directly cited by the neutral-atom community (adiabatic protocols, scar physics, QEC proposals) — a proof-grade contribution to an experimentally hot platform.

References

1. A hierarchy of spectral gap certificates for frustration-free spin systems, Quantum (2026), DOI 10.22331/q-2026-04-13-2065.
2. Eisert et al., Lower bounds to variational problems with guarantees, arXiv:2301.06142.
3. Kull et al., RG lower bounds, arXiv:2212.03014.

No files, tests, or entry points are named. Start by reading the cited spectral-gap and variational-bound references, then locate the repository's SDP hierarchy and exact-diagonalization components before determining how blockade constraints could be represented. Done means producing independently checkable certificates that meet the stated n ≤ 20 containment and thermodynamic-limit verification gates, or documenting the obstruction if the pivot signal occurs.