Compute the Kronecker tensor product 1.
Tensor two matrices or vectors together using the standard Kronecker
operation provided from numpy.
Given two matrices \(A\) and \(B\), computes \(A \otimes B\).
The same concept also applies to two vectors \(v\) and \(w\) which
computes \(v \otimes w\).
One may also compute the tensor product one matrix \(n\) times with itself.
For a matrix, \(A\) and an integer \(n\), the result of this
function computes \(A^{\otimes n}\).
Similarly for a vector \(v\) and an integer \(n\), the result of
of this function computes \(v^{\otimes n}\).
One may also perform the tensor product on a list of matrices.
Given a list of \(n\) matrices \(A_1, A_2, \ldots, A_n\) the result
of this function computes
\[
A_1 \otimes A_2 \otimes \cdots \otimes A_n.
\]
Similarly, for a list of \(n\) vectors \(v_1, v_2, \ldots, v_n\),
the result of this function computes
\[
v_1 \otimes v_2 \otimes \cdots \otimes v_n.
\]
Parameters:
-
args
(ndarray | int | list[ndarray], default:
()
)
–
Input to the tensor function is expected to be either:
- list[np.ndarray]: List of numpy matrices,
- np.ndarray, ... , np.ndarray: An arbitrary number of numpy arrays,
- np.ndarray, int: A numpy array and an integer.
Returns:
-
ndarray | None
–
The computed tensor product.
Raises:
-
ValueError
–
Input must be a vector or matrix.
Examples:
Tensor product two matrices or vectors
Consider the following ket vector
\[
e_0 = \left[1, 0 \right]^{\text{T}}.
\]
Computing the following tensor product
\[
e_0 \otimes e_0 = [1, 0, 0, 0]^{\text{T}}.
\]
This can be accomplished in |toqito⟩ as follows.
from toqito.states import basis
from toqito.matrix_ops import tensor
e_0 = basis(2, 0)
print(tensor(e_0, e_0))
Tensor product one matrix \(n\) times with itself.
We may also tensor some element with itself some integer number of times.
For instance we can compute
\[
e_0^{\otimes 3} = \left[1, 0, 0, 0, 0, 0, 0, 0 \right]^{\text{T}}
\]
in |toqito⟩ as follows.
from toqito.states import basis
from toqito.matrix_ops import tensor
e_0 = basis(2, 0)
print(tensor(e_0, 3))
[[1]
[0]
[0]
[0]
[0]
[0]
[0]
[0]]
Perform the tensor product on a list of vectors or matrices.
If we wish to compute the tensor product against more than two matrices or
vectors, we can feed them in as a list. For instance, if we wish to
compute \(e_0 \otimes e_1 \otimes e_0\), we can do
so as follows.
from toqito.states import basis
from toqito.matrix_ops import tensor
e_0, e_1 = basis(2, 0), basis(2, 1)
print(tensor([e_0, e_1, e_0]))
[[0]
[0]
[1]
[0]
[0]
[0]
[0]
[0]]
References
1 Wikipedia. Tensor product. link.
Source code in toqito/matrix_ops/tensor.py
| def tensor(*args: np.ndarray | int | list[np.ndarray]) -> np.ndarray | None:
r"""Compute the Kronecker tensor product [@wikipediatensor].
Tensor two matrices or vectors together using the standard Kronecker
operation provided from numpy.
Given two matrices \(A\) and \(B\), computes \(A \otimes B\).
The same concept also applies to two vectors \(v\) and \(w\) which
computes \(v \otimes w\).
One may also compute the tensor product one matrix \(n\) times with itself.
For a matrix, \(A\) and an integer \(n\), the result of this
function computes \(A^{\otimes n}\).
Similarly for a vector \(v\) and an integer \(n\), the result of
of this function computes \(v^{\otimes n}\).
One may also perform the tensor product on a list of matrices.
Given a list of \(n\) matrices \(A_1, A_2, \ldots, A_n\) the result
of this function computes
\[
A_1 \otimes A_2 \otimes \cdots \otimes A_n.
\]
Similarly, for a list of \(n\) vectors \(v_1, v_2, \ldots, v_n\),
the result of this function computes
\[
v_1 \otimes v_2 \otimes \cdots \otimes v_n.
\]
Args:
args: Input to the tensor function is expected to be either:
- list[np.ndarray]: List of numpy matrices,
- np.ndarray, ... , np.ndarray: An arbitrary number of numpy arrays,
- np.ndarray, int: A numpy array and an integer.
Returns:
The computed tensor product.
Raises:
ValueError: Input must be a vector or matrix.
Examples:
Tensor product two matrices or vectors
Consider the following ket vector
\[
e_0 = \left[1, 0 \right]^{\text{T}}.
\]
Computing the following tensor product
\[
e_0 \otimes e_0 = [1, 0, 0, 0]^{\text{T}}.
\]
This can be accomplished in `|toqito⟩` as follows.
```python exec="1" source="above" result="text"
from toqito.states import basis
from toqito.matrix_ops import tensor
e_0 = basis(2, 0)
print(tensor(e_0, e_0))
```
Tensor product one matrix \(n\) times with itself.
We may also tensor some element with itself some integer number of times.
For instance we can compute
\[
e_0^{\otimes 3} = \left[1, 0, 0, 0, 0, 0, 0, 0 \right]^{\text{T}}
\]
in `|toqito⟩` as follows.
```python exec="1" source="above" result="text"
from toqito.states import basis
from toqito.matrix_ops import tensor
e_0 = basis(2, 0)
print(tensor(e_0, 3))
```
Perform the tensor product on a list of vectors or matrices.
If we wish to compute the tensor product against more than two matrices or
vectors, we can feed them in as a `list`. For instance, if we wish to
compute \(e_0 \otimes e_1 \otimes e_0\), we can do
so as follows.
```python exec="1" source="above" result="text"
from toqito.states import basis
from toqito.matrix_ops import tensor
e_0, e_1 = basis(2, 0), basis(2, 1)
print(tensor([e_0, e_1, e_0]))
```
"""
def fast_exp(matrix: np.ndarray, q: int) -> np.ndarray:
"""Efficient exponentiation by squaring."""
if q == 1:
return matrix
tmp = fast_exp(matrix, q >> 1)
tmp = np.kron(tmp, tmp)
if q & 1: # If q is odd
tmp = np.kron(matrix, tmp)
return tmp
result = None
# Input is provided as a list of numpy matrices.
if (len(args) == 1 and isinstance(args[0], list)) or (len(args) == 1 and isinstance(args[0], np.ndarray)):
if len(args[0]) == 1:
return args[0][0]
if len(args[0]) == 2:
return np.kron(args[0][0], args[0][1])
if len(args[0]) >= 3:
result = args[0][0]
for i in range(1, len(args[0])):
result = np.kron(result, args[0][i])
return result
# Tensor product one matrix `n` times with itself.
if len(args) == 2 and isinstance(args[1], int):
num_tensor = args[1]
if num_tensor == 0:
return np.eye(1, dtype=args[0].dtype)
if num_tensor == 1:
return args[0]
return fast_exp(args[0], num_tensor)
# Tensor product between two or more matrices.
if len(args) == 2:
return np.kron(args[0], args[1])
if len(args) >= 3:
result = args[0]
for i in range(1, len(args)):
result = np.kron(result, args[i])
return result
raise ValueError("The `tensor` function must take either a matrix or vector.")
|