168
11 Quantum Manybody Systems, Tensor Networks and Neural Networks
by optimizing the wave function of the conventional method. And it can be seen that
the more hidden units α, the better the neural network wave function is.
11.2 Tensor Networks and Neural Networks
Tensor networks provide a way to efficiently represent wave functions of quantum
many-body systems, that is, a way to construct subspaces in Hilbert space. So let us
discuss the relationship with neural networks here.
11.2.1 Tensor Network
The state represented by the simplest tensor network is the case with N = 2,
i,j =0,1
A ij |i .
(11.6)
Here, i, j = 0, 1 is an index for the spin, and A ij is called a tensor. The tensor
A has 2 × 2 = 4 components, so it has four complex degrees of freedom. In
the following, we will use Einstein’s convention 3 and omit the
symbol. Next,
consider the following state of N = 4:
B mn A mij A nkl |i .
(11.7)
Tensor A has three indices and tensor B has two indices. They are contracted by
indices m, n. In the N = 2 example (11.6), the tensor parameter A can represent
all of the Hilbert space, but in the N = 4 example (11.7), it is not possible. This is
because if N = 4, the entire Hilbert space is spanned by 2 4 = 16 complex numbers,
but the degree of freedom of the tensor is 2 3 = 8 for A and 2 2 = 4 for B, thus only
12 in total. In other words, a subspace of the Hilbert space is parameterized.
It is standard to use a graph to represent the wave function by the tensor A or
B. The typical notation is that a tensor has legs whose number is that of the indices
(see Fig. 11.2). It appears to be similar to the neural network notation, but note that
the meaning of the line is completely different in the following ways: In a tensor
network, tensors are represented by squares, triangles, and circles. The line (leg)
extending from it has the meaning of subscript i = 0, 1, and so, the line means
an input or an output. Since a tensor with three indices has three legs, some of
the three are inputs and the remaining lines are outputs. In neural networks, on
3 Einstein’s convention is the understanding that indices that appear more than once will be summed
over.
11 Quantum Manybody Systems, Tensor Networks and Neural Networks
by optimizing the wave function of the conventional method. And it can be seen that
the more hidden units α, the better the neural network wave function is.
11.2 Tensor Networks and Neural Networks
Tensor networks provide a way to efficiently represent wave functions of quantum
many-body systems, that is, a way to construct subspaces in Hilbert space. So let us
discuss the relationship with neural networks here.
11.2.1 Tensor Network
The state represented by the simplest tensor network is the case with N = 2,
i,j =0,1
A ij |i .
(11.6)
Here, i, j = 0, 1 is an index for the spin, and A ij is called a tensor. The tensor
A has 2 × 2 = 4 components, so it has four complex degrees of freedom. In
the following, we will use Einstein’s convention 3 and omit the
symbol. Next,
consider the following state of N = 4:
B mn A mij A nkl |i .
(11.7)
Tensor A has three indices and tensor B has two indices. They are contracted by
indices m, n. In the N = 2 example (11.6), the tensor parameter A can represent
all of the Hilbert space, but in the N = 4 example (11.7), it is not possible. This is
because if N = 4, the entire Hilbert space is spanned by 2 4 = 16 complex numbers,
but the degree of freedom of the tensor is 2 3 = 8 for A and 2 2 = 4 for B, thus only
12 in total. In other words, a subspace of the Hilbert space is parameterized.
It is standard to use a graph to represent the wave function by the tensor A or
B. The typical notation is that a tensor has legs whose number is that of the indices
(see Fig. 11.2). It appears to be similar to the neural network notation, but note that
the meaning of the line is completely different in the following ways: In a tensor
network, tensors are represented by squares, triangles, and circles. The line (leg)
extending from it has the meaning of subscript i = 0, 1, and so, the line means
an input or an output. Since a tensor with three indices has three legs, some of
the three are inputs and the remaining lines are outputs. In neural networks, on
3 Einstein’s convention is the understanding that indices that appear more than once will be summed
over.