Large-scale simulation of�deep neural quantum states
Ao Chen
Supervisor: Markus Heyl
28.04.2025, Augsburg
Agenda
Why neural quantum states?
Image recognition
Input image
Model
0.98
0.12
Game of Go
Game status
Model
0.12
Quantum many-body problem
Model
From shallow to deep
One layer
G. Carleo & M. Troyer, Science (2017)
Restricted Boltzmann machine (RBM)
Deeper networks
More layers (Hidden)
Convolutional neural network (CNN)
K. Choo, T. Neupert & G. Carleo, Phys. Rev. B (2019)
C. Roth, A. Szabó & A. H. MacDonald, Phys. Rev. B (2023)
Group convolutional neural network (GCNN)
Method | Energy |
MPS | |
RBM | |
CNN | |
GCNN | |
Fermionic MF + Lanczos | |
Ground state | |
M. Li et al., IEEE TPDS (2022)
Optimization
Stochastic gradient descent (SGD)
Stochastic reconfiguration (SR)
A. Chen & M. Heyl, Nat. Phys. (2024)
SR and MinSR (Hidden)
Benchmark
A. Chen & M. Heyl, Nat. Phys. (2024)
Comparison (Hidden)
Method | Cost | Accuracy | Availability |
Exact diagonalization (ED) | Exponential high | Exact | Small systems |
Quantum Monte Carlo (QMC) | Medium | Very high | Sign-problem free |
Mean field (MF/HF) | Low | Low | Weak correlation |
Matrix product state (MPS) | Medium | Very high | 1D / Quasi-1D |
Tensor network (TN) | Very high | High | Most systems |
Neural quantum state (NQS) | High | High | Spin systems Fermions in develop |
NQS in�quantum spin liquids
Quantum spin liquids
Y. Nomura & M. Imada, Phys. Rev. X (2021)
Y. Iqbal, et al., Phys. Rev. B (2016)
A. Chen & M. Heyl, Nat. Phys. (2024)
Resonating valence bond (Hidden)
Structure factor
A. O. Scheie, ..., A. Chen, et al., arXiv:2406.17773 (2024)
Phase transition (Hidden)
NQS in�Fermi-Hubbard model
What’s the difficulty?
Determinant
Pfaffian
Hidden fermion
Determinant state
Hidden fermion
Pfaffian state
J. R. Moreno, et al., PNAS (2022)
A. Chen, et al., in preparation
Fermionic NQS
Hidden fermion determinant state (HFDS)
J. R. Moreno, et al., PNAS (2022)
Hidden fermion pfaffian state (HFPS)
A. Chen, et al., in preparation
Complexity (Hidden)
Mean field
Hidden fermion
Backflow
Order parameter
Superconducting order (Hidden)
d-wave pairing
pair-pair correlation
Summary
Publication
Ao Chen and Markus Heyl
Nat. Ph “Empowering deep neural quantum states through efficient optimization”
ys. 20, 1476 (2024)
Preprints
Ao Chen*, Vighnesh Naik*, and Markus Heyl
“Convolutional transformer wave functions”
arXiv:2503.10462 (2025)
AO Scheie, Minseong Lee, Kevin Wang, P Laurell, ES Choi, D Pajerowski, Qingming Zhang, Jie Ma, HD Zhou, Sangyun Lee, SM Thomas, MO Ajeesh, PFS Rosa, Ao Chen, Vivien S Zapf, M Heyl, CD Batista, E Dagotto,
JE Moore, and D Alan Tennant
“Spectrum and low-energy gap in triangular quantum spin liquid NaYbSe2”
arXiv:2406.17773 (2024)
Hongzheng Zhao, Ao Chen, Shu-Wei Liu, Marin Bukov, Markus Heyl, and Roderich Moessner
“Learning effective Hamiltonians for adaptive time-evolution quantum algorithms”
arXiv:2406.06198 (2024)
In preparation
Ao Chen, Anirvan Sengupta, Antoine Georges, and Christopher Roth
“Hidden fermion pfaffian state”
Vighnesh Naik, Ao Chen, and Markus Heyl
“Neural quantum dynamics assisted by basis rotation”
Thank you.