Restricted Boltzmann Machines for Quantum States with Non-Abelian or Anyonic Symmetries

Abstract

Although artificial neural networks have recently been proven to provide a promising new framework for constructing quantum many-body wave functions, the parameterization of a quantum wavefunction with nonabelian symmetries in terms of a Boltzmann machine inherently leads to biased results due to the basis dependence. We demonstrate that this problem can be overcome by sampling in the basis of irreducible representations instead of spins, for which the corresponding ansatz respects the nonabelian symmetries of the system. We apply our methodology to find the ground states of the one-dimensional antiferromagnetic Heisenberg (AFH) model with spin-1 ⁄2 and spin-1 degrees of freedom, and obtain a substantially higher accuracy than when using the sz-basis as input to the neural network. The proposed ansatz can target excited states, which is illustrated by calculating the energy gap of the AFH model. We also generalize the framework to the case of anyonic spin chains.

Dynamical Large Deviations of Two-Dimensional Kinetically Constrained Models Using a Neural-Network State Ansatz

“…x ψðxÞjxi. This ansatz has been shown recently to be capable of representing highly entangled quantum systems [28,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63], and has found use in quantum state tomography [64][65][66][67]. The expressivity of the neural-network ansatz depends on the architecture of the neural network, and typical choices include restricted Boltzmann machines, fully connected and convolutional neural networks, and autoregressive neural networks.…”

Machine learning for quantum matter

“…[144,146,150]. The possibility to directly impose symmetries on the RBM has also been studied [123], including non-Abelian or anyonic symmetries [151] In spite of the existence of heuristic algorithms to sample and compute their normalization constants, we would like to highlight that the sampling of families of neural network quantum states, in particular the RBM family, is a computationally intractable task [152]. To alleviate these issues, RBMs can be physically implemented by neuromorphic hardware which has the potential to enable better sampling.…”

Neural-Network Quantum States for Spin-1 Systems: Spin-Basis and Parameterization Effects on Compactness of Representations

“…These deficiencies show that a more general RBM energy function is needed that is sensitive to the three-valued nature of the visible units through the inclusion of terms involving the square of visible variables (a similar approach is likely to have been used in Ref. [ 47 ] already, although it was not explicitly stated). This motivates the following definition: efinition…”

“…Progress has been made in this direction in recent work [ 47 ], where multivalued RBMs were applied directly to the one-dimensional spin-1 anti-ferromagnetic Heisenberg model (AFH) and substantially enhanced by incorporating a transformation to a coupled SU(2) symmetric basis. Complementary to this, here, we propose and study a direct generalization of the RBMs to spin-1 systems that retains key properties of spin- RBM with a minimal increase in variational parameters (as we will not be examining other network architectures, we will use RBM and NQS interchangeably in this paper).…”