toqito.channels.pauli_channel

Generates and applies Pauli Channel to a matrix.

Module Contents

toqito.channels.pauli_channel.pauli_channel(prob, return_kraus_ops=False, input_mat=None)[source]

Generate and apply a Pauli channel to a matrix.

Generates the Choi matrix of a Pauli channel with given probabilities and optionally applies it to an input matrix. The Pauli channel is defined by the set of Pauli operators weighted by the probability vector. For a given probability vector ((p_0, ldots, p_{4^q -1 })), the channel is defined as shown below. Where, $q$ is the number of qubits.

[

Phi(rho) = sum_{i=0}^{4^q - 1} p_i P_i rho P_i^*

]

where (P_i) are Pauli operators generated by a lexographically increasing sequence of pauli operators of length strictly equal to (q), and (p_i) is the corresponding probability of that operator. For example, when (q = 2),

(P_{0} = I otimes I), (P_{1} = I otimes X), (P_{2} = I otimes Y), (P_{3} = I otimes Z), (P_{4} = X otimes I), (P_{5} = X otimes Y , ldots P_{15} = Z otimes Z)

If prob is a scalar, it generates a random prob-qubit Pauli channel. The length of the probability vector (if provided) must be (4^q) for some integer (q) (number of qubits).

Examples

Generate a random single-qubit Pauli channel:

```python exec=”1” source=”above” from toqito.channels import pauli_channel

print(pauli_channel(prob=1)) ```

Apply a specific two-qubit Pauli channel to an input matrix:

```python exec=”1” source=”above” import numpy as np from toqito.channels import pauli_channel

_, output = pauli_channel(

prob=np.array([0.1, 0.2, 0.3, 0.4]), input_mat=np.eye(2)

) print(output) ```

Raises:

  • ValueError – If probabilities are negative or don’t sum to 1.

  • ValueError – If length of probability vector is not 4^q for some integer q.

Parameters:

  • prob (int | numpy.ndarray) – Probability vector for Pauli operators. If scalar, generates random probabilities for (q =) prob

  • return_kraus_ops (bool)

  • input_mat (numpy.ndarray | None)

Return type:

numpy.ndarray | scipy.sparse.csc_matrix | tuple

:param qubits. The probabilities correspond to Pauli operators in lexographical order of length strictly equal to : ,when prob is a vector. :type qubits. The probabilities correspond to Pauli operators in lexographical order of length strictly equal to : q:param return_kraus_ops: Flag to return Kraus operators. Default is False. :param input_mat: Optional input matrix to apply the channel to. Default is None.

Returns:

The Choi matrix of the channel. If input_mat is provided, also returns the output matrix. If return_kraus_ops is True, returns Kraus operators as well.

Parameters:

  • prob (int | numpy.ndarray)

  • return_kraus_ops (bool)

  • input_mat (numpy.ndarray | None)

Return type:

numpy.ndarray | scipy.sparse.csc_matrix | tuple