[challenge]: Certifying ground-state properties of quantum 1/2-spin systems via the coarse-grained NPA hierarchy

Released by

王杰（Jie Wang）, AMSS-CAS

Contact email

wangjie212@amss.ac.cn

Method

Noncommutative polynomial optimization/Quantum bootstrap

Challenge: Certifying ground-state properties of quantum 1/2-spin systems via the coarse-grained NPA hierarchy

Summary
The problem of certifying ground-state properties (i.e., putting rigorous bounds on expectation values) of quantum spin systems could be formulated as a noncommutative polynomial optimization problem which can be then solved with the NPA hierarchy of SDP relaxations. The accuracy of computational results increases as the relaxation order grows. However, this approach suffers from the scalability bottleneck due to the explosion of SDP sizes. The recent work Phys. Rev. X 14, 031006 and arXiv:2602.21468 significantly improves the scalability of this approach by systematically exploiting various structures of the system. Moreover, the coarse-graining maps of a renormalization scheme can be employed to reduce SDP relaxations for quantum spin systems as shown in Phys. Rev. X 14, 021008.

The goal is to combine the NPA hierarchy with the coarse-graining technique for quantum spin systems (outlined in Sec. III-D-2, Phys. Rev. X 14, 021008), and furthermore, to apply the structure-exploiting techniques of arXiv:2602.21468 to further improve computational scalability and accuracy. This would lead to a new and powerful numerical approach for quantum spin systems.

Targets

• 1D Heisenberg model: Lower bounding ground-state energies up to 200 spins with $10^{-5}$ accuracy;
• 1D J1-J2 Heisenberg model: Lower bounding ground-state energies up to 100 spins with $10^{-3}$ accuracy;
• 2D Heisenberg model: Lower bounding ground-state energies up to 16×16 spins with $10^{-3}$ accuracy;
• 2D J1-J2 Heisenberg model: Lower bounding ground-state energies up to 10×10 spins with $10^{-2}$ accuracy, and trying to address the controversy discussed in arXiv:2602.21468.

Related packages

• QMBCertify
• NCTSSoS.jl
• NCTSSOS

References
Most relevant

• 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

Background material

• Certifying ground-state properties of 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)
• Burgdorf, Klep, Povh, Optimization of Polynomials in Non-Commuting Variables, Springer (2016)

Start by reviewing the related QMBCertify, NCTSSoS.jl, and NCTSSOS packages alongside Sec. III-D-2 of Phys. Rev. X 14, 021008. The work is complete when the combined coarse-grained NPA and structure-exploiting approach reaches the stated accuracy and system-size targets for the four Heisenberg models.