Variational Autoencoder Reconstruction of Complex Many-Body Physics
Abstract
:1. Introduction
2. Transverse-Field Ising Model
3. Generative Model as a Quantum State
4. Variational Autoencoder Architecture
- We calculate the matrix of log probabilities, taking element-wise logarithm of decoder network output: ,
- We generate a matrix of samples from the standard Gumbel distribution G and sum it up element-wise with the matrix of log probabilities : ,
- Finally, we take the function of the result from the previous step: , where T is a temperature of softmax. The softmax functions is defined by the expression .
5. Results
- If one reconstructs a pure state, the VAE smooths the spectrum of the density matrix and approximates the pure state by a slightly mixed state, as illustrated with a simple example in Figure 13.
- The VAE does not account the positivity constraints, which yields negative eigenvalues for the density matrix. These negative eigenvalues even appear in the spectrum of the reduced density matrix, as shown in Figure 13.
6. Conclusions
- For a large system (32 spins), the VAE’s reliability is verified by comparing one- and two-point correlation functions.
- For small system (five spins), the VAE’s reliability is verified by direct comparison of mass functions.
- The VAE can capture a quantum phase transition.
- The response functions (magnetic differential susceptibility tensor) can be obtained using backpropagation through VAE.
- Despite the very good agreement between the VAE-based mass function and the true mass function, the VAE shows limited performance with the determination of the entangled entropy. This is point is the object of further development.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
VAE | Variational Autoencoder |
MPS | Matrix product state |
TFI | Transverse-field Ising |
IC | Informationally incomplete |
POVM | Positive-operator valued measure |
ELBO | Evidence lower bound |
NN | Neural network |
KL | Kullback–Leibler |
DMRG | Density matrix renormalization group |
Appendix A. VAE: Training and Implementation Details
Appendix B. Sampling from POVM-Induced Mass Function
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Luchnikov, I.A.; Ryzhov, A.; Stas, P.-J.; Filippov, S.N.; Ouerdane, H. Variational Autoencoder Reconstruction of Complex Many-Body Physics. Entropy 2019, 21, 1091. https://doi.org/10.3390/e21111091
Luchnikov IA, Ryzhov A, Stas P-J, Filippov SN, Ouerdane H. Variational Autoencoder Reconstruction of Complex Many-Body Physics. Entropy. 2019; 21(11):1091. https://doi.org/10.3390/e21111091
Chicago/Turabian StyleLuchnikov, Ilia A., Alexander Ryzhov, Pieter-Jan Stas, Sergey N. Filippov, and Henni Ouerdane. 2019. "Variational Autoencoder Reconstruction of Complex Many-Body Physics" Entropy 21, no. 11: 1091. https://doi.org/10.3390/e21111091