Chapter 11
Quantum Manybody Systems, Tensor
Networks and Neural Networks
Abstract In condensed matter physics, finding the wave function of a quantum
many-body system is the most important issue. Theoretical development in recent
years includes a wave function approximation using a tensor network. At first
glance, the tensor network looks very similar to neural network diagrams, but how
are they actually related? In this chapter we will see the relation and the mapping,
and that the restricted Boltzmann machine is closely related to tensor networks.
Microscopically, physics is a quantum system, and all isolated quantum systems are
governed by wave functions. For example, for simplicity, let us consider a situation
with zero temperature. Given a Hamiltonian of a system, the quantum phase is
determined by its lowest energy state (the ground state). To obtain a concrete wave
function is one of the most important ways to characterize quantum systems.
Hilbert space, which builds states of quantum many-body systems, suffers
from enormous combinatorial possibilities. For example, consider a system with N
qubits. A qubit is a system with two states, |0 and |1, and is quantum mechanically
the same as a system with a spin ¯
h/2. In this case, the Hilbert space has 2 N
dimensions, so the number of states increases exponentially with the number of
degrees of freedom. Since the problem of finding the ground state of a quantum
many-body system is the task of selecting only one state from this huge Hilbert
space, some “physical sense” is needed. That is where machine learning approaches
come into play.
Until now, methods have been developed that minimize the energy in a subspace,
considering only the states described by a relatively small number of parameters
in the Hilbert space. A method of constructing a subspace particularly efficiently
is called a tensor network. 1 There are various types of tensor networks depending
on their physical meaning and their targets. On the other hand, in this book we
have described various neural networks and how to use them. According to the
neural network universal approximation theorem (see Chap. 3), any function can
1 The tensor network is not a physical network arranged spatially. It is a graph specifying how spins
etc. are intertwined with each other.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
A. Tanaka et al., Deep Learning and Physics, Mathematical Physics Studies,
https://doi.org/10.1007/978-981-33-6108-9_11
165
Quantum Manybody Systems, Tensor
Networks and Neural Networks
Abstract In condensed matter physics, finding the wave function of a quantum
many-body system is the most important issue. Theoretical development in recent
years includes a wave function approximation using a tensor network. At first
glance, the tensor network looks very similar to neural network diagrams, but how
are they actually related? In this chapter we will see the relation and the mapping,
and that the restricted Boltzmann machine is closely related to tensor networks.
Microscopically, physics is a quantum system, and all isolated quantum systems are
governed by wave functions. For example, for simplicity, let us consider a situation
with zero temperature. Given a Hamiltonian of a system, the quantum phase is
determined by its lowest energy state (the ground state). To obtain a concrete wave
function is one of the most important ways to characterize quantum systems.
Hilbert space, which builds states of quantum many-body systems, suffers
from enormous combinatorial possibilities. For example, consider a system with N
qubits. A qubit is a system with two states, |0 and |1, and is quantum mechanically
the same as a system with a spin ¯
h/2. In this case, the Hilbert space has 2 N
dimensions, so the number of states increases exponentially with the number of
degrees of freedom. Since the problem of finding the ground state of a quantum
many-body system is the task of selecting only one state from this huge Hilbert
space, some “physical sense” is needed. That is where machine learning approaches
come into play.
Until now, methods have been developed that minimize the energy in a subspace,
considering only the states described by a relatively small number of parameters
in the Hilbert space. A method of constructing a subspace particularly efficiently
is called a tensor network. 1 There are various types of tensor networks depending
on their physical meaning and their targets. On the other hand, in this book we
have described various neural networks and how to use them. According to the
neural network universal approximation theorem (see Chap. 3), any function can
1 The tensor network is not a physical network arranged spatially. It is a graph specifying how spins
etc. are intertwined with each other.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
A. Tanaka et al., Deep Learning and Physics, Mathematical Physics Studies,
https://doi.org/10.1007/978-981-33-6108-9_11
165