[challenge]: Convergence rates beyond even-weight 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

The first quantitative convergence results in noncommutative polynomial optimization appeared in 2026: for k-local Pauli Hamiltonians with even-weight terms, NPA-type and sum-of-Hermitian-squares (SOHS) hierarchies approximate the ground-state energy density with explicit finite-level error bounds governed by roots of Krawtchouk polynomials (arXiv:2605.29959, arXiv:2606.04940). The even-weight restriction excludes transverse fields — i.e. most physically interesting models.

Research objective

Extend the convergence-rate theory to general (odd-weight) k-local Pauli Hamiltonians, and to the fermionic/Majorana algebra — or prove a separation showing the even-weight case is genuinely faster. The even-weight proof runs through almost-reproducing kernels for the Pauli algebra; the odd-weight obstruction is a parity-sector issue that charge-symmetry decompositions (as implemented in NCTSSoS) make explicit.

Verification plan

• Success gate: a proved rate bound whose prediction is confirmed against hierarchy values computed on exactly solvable chains (transverse-field Ising, XY), where ground energies are known in closed form — the bound must hold at every accessible level and match the predicted scaling exponent. Containment against exact values is machine-checkable.
• Hope signal: numerics show a clean empirical rate that the proof technique misses by a polynomial factor — the conjecture survives with strong evidence.
• Pivot signal: level-k error does not decay at the conjectured rate for odd-weight terms — then prove the even/odd separation theorem instead, which is equally publishable.

Why this may lead to research output

This is the sharpest open theory question in the field's newest frontier: the 2026 rate papers are the first of their kind and their authors explicitly flag the even-weight restriction. Any resolution — extension or separation — is a strong mathematical-physics paper, and it fixes the a-priori accuracy story for every practical Pauli-hierarchy computation.

References

1. Quantitative semidefinite certificates for ground-state energies of Pauli Hamiltonians, arXiv:2605.29959 (2026).
2. Convergence rates of Sum-of-Hermitian-Squares hierarchies for the Pauli algebra, arXiv:2606.04940 (2026).
3. Pironio, Navascués, Acín, Convergent relaxations of polynomial optimization problems with noncommuting variables, arXiv:0903.4368.

Start by reading the cited 2026 rate papers and the NCTSSoS charge-symmetry decomposition described in the issue. Compute hierarchy values for the transverse-field Ising and XY chains, comparing them with their exact ground energies. Done means proving the proposed odd-weight or fermionic rate, or proving an even/odd separation with machine-checkable containment.