qml.right_canonicalize_mps¶

right_canonicalize_mps(mps)[source]¶

Transform a matrix product state (MPS) into its right-canonical form.

A right-canonicalized MPS is a matrix product state in which the constituent tensors, $A^{(j)}$, satisfy the following orthonormality condition [Eq. (21) of arXiv:2310.18410]:

$$\sum_{d_{j,1}, d_{j,2}} A^{(j)}_{d_{j, 0}, d_{j, 1}, d_{j, 2}} \left( A^{(j)}_{d'_{j, 0}, d_{j, 1}, d_{j, 2}} \right)^* = \delta_{d_{j, 0}, d'_{j, 0}},$$

where $d_{i,j}$ denotes the $j$ dimension of the $i$ tensor and $\delta$ is the Kronecker delta.

Parameters

mps (list[TensorLike]) – List of tensors representing the MPS.

Returns

List of tensors representing the MPS in right-canonical form with the same dimensions as the initial MPS.

See also

MPSPrep.

Example

n_sites = 4

mps = ([np.ones((2, 4))] +
       [np.ones((4, 2, 4)) for _ in range(1, n_sites - 1)] +
       [np.ones((4, 2))])

mps_rc = qml.right_canonicalize_mps(mps)

# Check that the right-canonical definition is fulfilled
for i in range(1, n_sites - 1):
    tensor = mps_rc[i]
    contraction_matrix = np.tensordot(tensor, tensor.conj(), axes=([1, 2], [1, 2]))
    assert np.allclose(contraction_matrix, np.eye(tensor.shape[0]))

Usage Details

The input MPS must be a list of $n$ tensors $[A^{(0)}, ..., A^{(n-1)}]$ with shapes $d_0, ..., d_{n-1}$, respectively. The first and last tensors have rank $2$ while the intermediate tensors have rank $3$.

The first tensor must have the shape $d_0 = (d_{0,0}, d_{0,1})$ where $d_{0,0}$ and $d_{0,1}$ correspond to the physical dimension of the site and an auxiliary bond dimension connecting it to the next tensor, respectively.

The last tensor must have the shape $d_{n-1} = (d_{n-1,0}, d_{n-1,1})$ where $d_{n-1,0}$ and $d_{n-1,1}$ represent the auxiliary dimension from the previous site and the physical dimension of the site, respectively.

The intermediate tensors must have the shape $d_j = (d_{j,0}, d_{j,1}, d_{j,2})$, where:

• $d_{j,0}$ is the bond dimension connecting to the previous tensor

• $d_{j,1}$ is the physical dimension for the site

• $d_{j,2}$ is the bond dimension connecting to the next tensor

Note that the bond dimensions must match between adjacent tensors such that $d_{j-1,2} = d_{j,0}$.

Additionally, the physical dimension of the site should always be fixed at $2$ (since the dimension of a qubit is $2$), while the other dimensions must be powers of two.

The following example shows a valid MPS input containing four tensors with dimensions $[(2,2), (2,2,4), (4,2,2), (2,2)]$ which satisfy the criteria described above.

mps = [
    np.array([[0.0, 0.107], [0.994, 0.0]]),
    np.array(
        [
            [[0.0, 0.0, 0.0, -0.0], [1.0, 0.0, 0.0, -0.0]],
            [[0.0, 1.0, 0.0, -0.0], [0.0, 0.0, 0.0, -0.0]],
        ]
    ),
    np.array(
        [
            [[-1.0, 0.0], [0.0, 0.0]],
            [[0.0, 0.0], [0.0, 1.0]],
            [[0.0, -1.0], [0.0, 0.0]],
            [[0.0, 0.0], [1.0, 0.0]],
        ]
    ),
    np.array([[-1.0, -0.0], [-0.0, -1.0]]),
]