Neural-Network Representation of Quantum Many-Body States

There are many embeddings algorithm for representations. Sammon embedding is the oldest one, and we have Word2Vec, GloVe, FastText etc. for word-embedding algorithms. Embeddings are useful for dimensionality reduction.

Traditionally, quantum many-body states are represented by Fock states, which is useful when the excitations of quasi-particles are the concern. But to capture the quantum entanglement between many solitons or particles in a statistical systems, it is important not to lose the topological correlation between the states. It has been known that restricted Boltzmann machines (RBM) have been used to represent such states, but it has its limitation, which Xun Gao and Lu-Ming Duan have stated in their article published in Nature Communications:

There exist states, which can be generated by a constant-depth quantum circuit or expressed as PEPS (projected entangled pair states) or ground states of gapped Hamiltonians, but cannot be efficiently represented by any RBM unless the polynomial hierarchy collapses in the computational complexity theory.

PEPS is a generalization of matrix product states (MPS) to higher dimensions. (See this.)

However, Gao and Duan were able to prove that deep Boltzmann machine (DBM) can bridge the loophole of RBM, as stated in their article:

Any quantum state of n qubits generated by a quantum circuit of depth T can be represented exactly by a sparse DBM with O(nT) neurons.

41467_2017_705_fig3_html

(diagram adapted from Gao and Duan’s article)

  • Xun Gao, Lu-Ming Duan, “Efficient representation of quantum many-body states with deep neural networks,” Nature Communications 8:662 (2017) or arXiv:1701.05039 (2017). [NatureComm] [arXiv]
  • Kwan-Yuet Ho, “Sammon Embedding with TensorFlow,” Everything About Data Analytics, WordPress (2017). [WordPress]
  • Kwan-Yuet Ho, “Word Embedding Algorithms,” Everything About Data Analytics, WordPress (2017). [WordPress]
  • FastText. [Facebook]
  • Kwan-Yuet Ho, “Tensor Networks and Density Matrix Renormalization Group,”Everything About Data Analytics, WordPress (2016). [WordPress]
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