Neural-Network Quantum States, String-Bond States, and Chiral Topological States (1710.04045v3)

Abstract: Neural-Network Quantum States have been recently introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between Neural-Network Quantum States in the form of Restricted Boltzmann Machines and some classes of Tensor-Network states in arbitrary dimensions. In particular we demonstrate that short-range Restricted Boltzmann Machines are Entangled Plaquette States, while fully connected Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry and low bond dimension. These results shed light on the underlying architecture of Restricted Boltzmann Machines and their efficiency at representing many-body quantum states. String-Bond States also provide a generic way of enhancing the power of Neural-Network Quantum States and a natural generalization to systems with larger local Hilbert space. We compare the advantages and drawbacks of these different classes of states and present a method to combine them together. This allows us to benefit from both the entanglement structure of Tensor Networks and the efficiency of Neural-Network Quantum States into a single Ansatz capable of targeting the wave function of strongly correlated systems. While it remains a challenge to describe states with chiral topological order using traditional Tensor Networks, we show that Neural-Network Quantum States and their String-Bond States extension can describe a lattice Fractional Quantum Hall state exactly. In addition, we provide numerical evidence that Neural-Network Quantum States can approximate a chiral spin liquid with better accuracy than Entangled Plaquette States and local String-Bond States. Our results demonstrate the efficiency of neural networks to describe complex quantum wave functions and pave the way towards the use of String-Bond States as a tool in more traditional machine-learning applications.

• The paper shows that short-range RBMs map to Entangled Plaquette States, highlighting local equivalence in quantum state representations.
• It demonstrates that fully connected RBMs correspond to a subclass of String-Bond States, offering efficient representations with reduced bond dimensions.
• The paper applies these insights to chiral topological states by exactly capturing lattice Laughlin wave functions and chiral spin liquids.

Overview of the Paper on Neural-Network Quantum States and Connections to Tensor-Network States

This paper explores the relationships between Neural-Network Quantum States (NNQS) and Tensor-Network states, focusing on the submissions of Restricted Boltzmann Machines (RBM) as wave function Ansätze for quantum many-body systems. The paper highlights that NNQS in the form of RBMs share significant connections with certain Tensor-Network states, particularly Entangled Plaquette States (EPS) and String-Bond States (SBS).

Key Results

1. Short-Range RBMs and Entangled Plaquette States:
   • RBMs with short-range connections, termed short-range RBMs, exhibit connectivity that is analogous to EPS. This mapping illustrates that EPS and RBMs can be leveraged interchangeably under certain conditions, taking advantage of their locality.
2. Fully-Connected RBMs as String-Bond States:
   • Fully connected RBMs, which allow connections between every visible and hidden unit, correspond to a subclass of SBS. This relation is due to RBMs exhibiting a non-local geometry similar to SBS. The paper shows that RBMs can essentially be expressed as SBS with reduced bond dimensions, which contributes to understanding their efficiency in representing many-body quantum states.
3. Applications to Chiral Topological States:
   • A significant application is the expression of lattice analogs of the Laughlin wave function and chiral spin liquids using RBMs. The paper demonstrates that chiral topological orders, difficult to describe with conventional local Tensor Networks, can be exactly represented by RBMs.

Implications and Future Directions

The findings hold substantial implications for both quantum state representation and machine learning in physics. By bridging NNQS and Tensor Networks, the paper bolsters the potential for combined methodological advantages: the entanglement structure from Tensor Networks and the computational efficiency from NNQS.

The presented methodology also speculates on broader applications in tackling complex quantum states which are typically challenging for traditional approaches. Furthermore, these insights pave the way to future machine-learning applications, where NNQS could introduce innovative ways of encoding quantum many-body problems.

Practical and Theoretical Implications

Practically, the results suggest that researchers can exploit non-local connections provided by RBMs to tackle problems involving complex correlations and topological orders. Theoretically, these connections refine the understanding of the expressive power of NNQS in quantum mechanics, suggesting a new framework where the geometric and architectural choices in quantum state representations might be optimized based on the mapping between these state descriptions.

Conclusion

Overall, the paper contributes significant theoretical support for using NNQS, particularly RBMs, in quantum many-body systems, offering insights into their advantageous characteristics relative to traditional methods, such as Tensor Networks. It signals fruitful research avenues to further extend these techniques to encompass more extensive and intricate quantum systems, potentially informing quantum algorithms and simulations.

These findings promise to stimulate research at the intersection of quantum physics and machine learning, where the quest to find efficient and flexible quantum state representations continues to be a pivotal challenge.