166
11 Quantum Manybody Systems, Tensor Networks and Neural Networks
be approximated by increasing the number of units sufficiently. Therefore, we can
use neural networks as a method of constructing quantum states.
In the following, we will introduce the methods proposed in literature [122] and
the results, and also look at the differences between tensor networks and neural
networks.
11.1 Neural Network Wave Function
First, let us consider how the wave function of a quantum system can be represented
by a neural network. A wave function of an N-qubit system is the coefficient
ψ(s 1 , · · · , s N ) in the state |ψ
|ψ =
s 1 ,··· ,s N
ψ(s 1 , · · · , s N )|s 1 · · · |s N .
(11.1)
Here, s a = 0, 1 is the basis of the qubit, and ψ is a complex function
using it as a variable. That is, the nonlinear function ψ transforms the input
such as (s 1 , · · · , s N ) = (0, 0, 1, 0, 1, · · · ) into a complex-valued output
ψ(0, 0, 1, 0, 1, · · · ). The nonlinear function ψ defines the quantum state. When
this nonlinear function is determined so that the energy of the system
E =
|ψ
(11.2)
is minimized, we call the state a ground state.
Therefore, if we recapture this problem as machine learning, we represent the
nonlinear function ψ by a neural network and consider the error function as the
system energy E. Hence, this is not supervised learning. In supervised machine
learning, the error function is defined as the difference from the correct output,
because there is a correct combination of an input and an output, and training is
performed so that a function provides the correct output when the input is given. On
the other hand, in the problem of finding the ground-state wave function, the energy
of the system is adopted as the error function, and the wave function is trained so
that the energy becomes small.
Now, according to the literature [122], let us express the wave function by a
restricted Boltzmann machine. 2 As seen in Chap. 6, the restricted Boltzmann
machine provides a Boltzmann weight factor as an output. The relation between
the hidden layer unit and the input unit is given with a spin Hamiltonian. If the unit
2 See also [123]. A physical interpretation of the neural network is described in [124].
11 Quantum Manybody Systems, Tensor Networks and Neural Networks
be approximated by increasing the number of units sufficiently. Therefore, we can
use neural networks as a method of constructing quantum states.
In the following, we will introduce the methods proposed in literature [122] and
the results, and also look at the differences between tensor networks and neural
networks.
11.1 Neural Network Wave Function
First, let us consider how the wave function of a quantum system can be represented
by a neural network. A wave function of an N-qubit system is the coefficient
ψ(s 1 , · · · , s N ) in the state |ψ
|ψ =
s 1 ,··· ,s N
ψ(s 1 , · · · , s N )|s 1 · · · |s N .
(11.1)
Here, s a = 0, 1 is the basis of the qubit, and ψ is a complex function
using it as a variable. That is, the nonlinear function ψ transforms the input
such as (s 1 , · · · , s N ) = (0, 0, 1, 0, 1, · · · ) into a complex-valued output
ψ(0, 0, 1, 0, 1, · · · ). The nonlinear function ψ defines the quantum state. When
this nonlinear function is determined so that the energy of the system
E =
|ψ
(11.2)
is minimized, we call the state a ground state.
Therefore, if we recapture this problem as machine learning, we represent the
nonlinear function ψ by a neural network and consider the error function as the
system energy E. Hence, this is not supervised learning. In supervised machine
learning, the error function is defined as the difference from the correct output,
because there is a correct combination of an input and an output, and training is
performed so that a function provides the correct output when the input is given. On
the other hand, in the problem of finding the ground-state wave function, the energy
of the system is adopted as the error function, and the wave function is trained so
that the energy becomes small.
Now, according to the literature [122], let us express the wave function by a
restricted Boltzmann machine. 2 As seen in Chap. 6, the restricted Boltzmann
machine provides a Boltzmann weight factor as an output. The relation between
the hidden layer unit and the input unit is given with a spin Hamiltonian. If the unit
2 See also [123]. A physical interpretation of the neural network is described in [124].