[challenge]: Convergence-rate theory for fermionic hierarchies

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 2026 Pauli-algebra convergence rates (arXiv:2605.29959, arXiv:2606.04940) have no fermionic counterpart. Fermionic SDP lower bounds are the mathematical core of variational electronic structure (SDP relaxations for condensed matter: arXiv:1106.5275, arXiv:1106.4966), which has never had an a priori accuracy statement for a polynomial-cost method.

Research objective

Establish the CAR-algebra analogue: explicit finite-level error bounds for NC-SOS relaxations of interacting-fermion Hamiltonians (Hubbard-class) in the Majorana monomial algebra. The Pauli proof's almost-reproducing-kernel construction has a natural Clifford/Majorana counterpart (Krawtchouk → weight-graded analogues); parity superselection plays the role of the even-weight restriction, suggesting the fermionic case may be cleaner than odd-weight Pauli.

Verification plan

• Success gate: a proved rate whose predictions contain the computed hierarchy errors on free-fermion and integrable Hubbard instances (exact reference values) across all accessible levels and sizes — containment failures falsify instantly.
• Hope signal: a rate for quadratic-plus-weak-interaction regimes only — already the first fermionic guarantee.
• Pivot signal: Majorana kernels provably lack the spectral gap driving the Pauli argument — the obstruction theorem redirects to weighted/twisted kernels.

Why this may lead to research output

A rate theorem would give quantum chemistry an a-priori accuracy statement for SDP lower bounds — and would fix which relaxation level any quantum-advantage claim in electronic structure must beat.

References

1. arXiv:2605.29959; arXiv:2606.04940 (Pauli-algebra rates).
2. Baumgratz, Plenio, Lower bounds for ground states of condensed matter systems, arXiv:1106.5275.
3. Barthel, Hübener, Solving condensed-matter ground-state problems by semidefinite relaxations, arXiv:1106.4966.

No repository files, tests, or entry points are named. Start with the cited Pauli-rate papers and fermionic SDP references, then formulate the CAR/Majorana rate and compare it with free-fermion and integrable Hubbard hierarchy errors at accessible levels. Done means a proved bound containing those errors, or a documented obstruction and pivot.