Apply the bitflip quantum channel to a state or return the Kraus operators.
The bitflip channel is a quantum channel that flips a qubit from \(|0\rangle\) to \(|1\rangle\)
and from \(|1\rangle\) to \(|0\rangle\) with probability \(p\).
It is defined by the following operation:
\[
\mathcal{E}(\rho) = (1-p) \rho + p X \rho X
\]
where \(X\) is the Pauli-X (NOT) gate given by:
\[
X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\]
The Kraus operators for this channel are:
\[
K_0 = \sqrt{1-p} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad
K_1 = \sqrt{p} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\]
Examples:
We can generate the Kraus operators for the bitflip channel with probability 0.3:
from toqito.channels import bitflip
print(bitflip(prob=0.3))
[array([[0.83666003, 0. ],
[0. , 0.83666003]]), array([[0. , 0.54772256],
[0.54772256, 0. ]])]
We can also apply the bitflip channel to a quantum state. For the state \(|0\rangle\),
the bitflip channel with probability 0.3 produces:
import numpy as np
from toqito.channels import bitflip
rho = np.array([[1, 0], [0, 0]]) # |0><0|
print(bitflip(rho, prob=0.3))
[[0.7+0.j 0. +0.j]
[0. +0.j 0.3+0.j]]
Parameters:
-
input_mat
(ndarray | None, default:
None
)
–
A matrix or state to apply the channel to. If None, returns the Kraus operators.
-
prob
(float, default:
0
)
–
The probability of a bitflip occurring.
Returns:
-
ndarray | list[ndarray]
–
Either the Kraus operators of the bitflip channel if input_mat is None, or the result of applying the
-
ndarray | list[ndarray]
–
Source code in toqito/channels/bitflip.py
| def bitflip(
input_mat: np.ndarray | None = None,
prob: float = 0,
) -> np.ndarray | list[np.ndarray]:
r"""Apply the bitflip quantum channel to a state or return the Kraus operators.
The *bitflip channel* is a quantum channel that flips a qubit from \(|0\rangle\) to \(|1\rangle\)
and from \(|1\rangle\) to \(|0\rangle\) with probability \(p\).
It is defined by the following operation:
\[
\mathcal{E}(\rho) = (1-p) \rho + p X \rho X
\]
where \(X\) is the Pauli-X (NOT) gate given by:
\[
X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\]
The Kraus operators for this channel are:
\[
K_0 = \sqrt{1-p} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad
K_1 = \sqrt{p} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\]
Examples:
We can generate the Kraus operators for the bitflip channel with probability 0.3:
```python exec="1" source="above"
from toqito.channels import bitflip
print(bitflip(prob=0.3))
```
We can also apply the bitflip channel to a quantum state. For the state \(|0\rangle\),
the bitflip channel with probability 0.3 produces:
```python exec="1" source="above"
import numpy as np
from toqito.channels import bitflip
rho = np.array([[1, 0], [0, 0]]) # |0><0|
print(bitflip(rho, prob=0.3))
```
Args:
input_mat: A matrix or state to apply the channel to. If `None`, returns the Kraus operators.
prob: The probability of a bitflip occurring.
Returns:
Either the Kraus operators of the bitflip channel if `input_mat` is `None`, or the result of applying the
channel to `input_mat`.
"""
if not (0 <= prob <= 1):
raise ValueError("Probability must be between 0 and 1.")
k0 = np.sqrt(1 - prob) * np.eye(2)
k1 = np.sqrt(prob) * np.array([[0, 1], [1, 0]])
if input_mat is not None and input_mat.shape != (2, 2):
raise ValueError("Input matrix must be 2x2 for the bitflip channel.")
elif input_mat is None:
return [k0, k1]
input_mat = np.asarray(input_mat, dtype=complex)
return k0 @ input_mat @ k0.conj().T + k1 @ input_mat @ k1.conj().T
|