[challenge]: Certified ground-state energies of Heisenberg spin systems via coarse-grained noncommutative polynomial optimization

[fliingelephant]

Released by

Jie Wang, AMSS-CAS

Contact email

wangjie212@amss.ac.cn

Method

Other

Challenge issue

Challenge: Certified ground-state energies of Heisenberg spin systems via coarse-grained noncommutative polynomial optimization

Summary

Combine the NPA hierarchy with the coarse-graining technique for quantum
spin-1/2 systems of Sec. III-D-2, Phys. Rev. X 14, 021008,
and apply the structure-exploiting techniques of arXiv:2604.01555
to improve the computation scale and precision.

Targets

• 1D Heisenberg model: ground-state energy up to 200 spins, precision $10^{-5}$.
• 2D Heisenberg model: ground-state energy up to $16 \times 16$ spins, precision $10^{-3}$; address the controversy in arXiv:2602.21468.

References

• Lower Bounds on Ground-State Energies of Local Hamiltonians through the Renormalization Group, Phys. Rev. X 14, 021008
• Scalable ground-state certification of quantum spin systems via structured noncommutative polynomial optimization, arXiv:2604.01555
• Coarse-grained bootstrap of quantum many-body systems, JHEP 02 (2026) 222
• Certifying ground-state properties of quantum many-body systems, Phys. Rev. X 14, 031006
• Pironio, Navascués, Acín, Convergent relaxations of polynomial optimization problems with noncommuting variables, SIAM J. Optim. 20, 2157 (2010)
• Background: Burgdorf, Klep, Povh, Optimization of Polynomials in Non-Commuting Variables, Springer (2016)