[challenge]: From certificates to states: quantitative rounding for Pauli Hamiltonians

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

Certified energies exist (arXiv:2310.05844, arXiv:2605.29959), but nothing certifies that the relaxation's state data — the moment-matrix correlation functions practitioners actually read off (arXiv:2601.10408) — describes the physical ground state. The stability toolkit exists in the nonlocal-game world (arXiv:2204.07084, arXiv:2203.02525, arXiv:2505.22309) but has never been transplanted to many-body physics.

Research objective

Prove a quantitative rounding theorem for the Pauli-algebra hierarchy: if the level-k SDP value is within ε of the ground energy, the optimal moment matrix's GNS state is within f(ε, k) of the true ground-state manifold in local-observable distance. Combine the Pauli reproducing-kernel bounds (arXiv:2606.04940) with approximate-Schur stability; gap assumptions enter exactly where spectral-gap certificates (Quantum, DOI 10.22331/q-2026-04-13-2065) can supply them — fully certified end-to-end.

Verification plan

• Success gate: a proved bound f(ε, k) whose predictions contain measured moment-to-exact-state distances on solvable chains (TFIM across its phase diagram, gapped and critical), plus an adversarial search for degenerate/symmetry-broken instances that must land inside the bound or expose a stated assumption — containment is binary.
• Hope signal: the theorem under a spectral-gap promise only — already upgrades all gapped-phase certifications.
• Pivot signal: counterexamples where near-optimal moment matrices are far from every ground state even with a gap — the fooling-states theorem replaces the rounding theorem and would invalidate common practice (equally important).

Why this may lead to research output

It closes the loop the field leaves open: making every published moment table trustworthy — or exposing which ones are not.

References

1. Wang et al., Certifying ground-state properties of many-body systems, arXiv:2310.05844 (PRX).
2. Almasi et al., Convergence rates of SOHS hierarchies for the Pauli algebra, arXiv:2606.04940.
3. Mortimer et al., Bounding many-body properties under partial information, arXiv:2601.10408.
4. Xu et al., Quantitative Tsirelson's theorems, arXiv:2505.22309.

Start by reading the cited Pauli-algebra hierarchy and reproducing-kernel papers, then review the approximate-Schur stability and spectral-gap references. Develop the quantitative rounding or counterexample theorem described in the objective, and validate it on TFIM chains plus degenerate or symmetry-broken instances. Done means the bound contains all measured distances under its stated assumptions, or a rigorous fooling-states result explains failure.