quairkit.database.representation¶
Representations of channels
- quairkit.database.representation.bit_flip_kraus(prob)¶
Kraus representation of a bit flip channel with form
\[E_0 = \sqrt{1-p} I, E_1 = \sqrt{p} X.\]- Parameters:¶
- prob : ndarray | Tensor | Iterable[float] | float¶
probability \(p\).
- dtype
data type. Defaults to be
None.
- Returns:¶
a list of Kraus operators
- Return type:¶
List[ndarray | Tensor]
prob = torch.tensor([0.5]) kraus_operators = bit_flip_kraus(prob) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[0.7071+0.j, 0.0000+0.j], [0.0000+0.j, 0.7071+0.j]], [[0.0000+0.j, 0.7071+0.j], [0.7071+0.j, 0.0000+0.j]]], dtype=torch.complex128)
- quairkit.database.representation.phase_flip_kraus(prob)¶
Kraus representation of a phase flip channel with form
\[E_0 = \sqrt{1 - p} I, E_1 = \sqrt{p} Z.\]- Parameters:¶
- prob : ndarray | Tensor | Iterable[float] | float¶
probability \(p\).
- dtype
data type. Defaults to be
None.
- Returns:¶
a list of Kraus operators
- Return type:¶
List[ndarray | Tensor]
prob = torch.tensor([0.1]) kraus_operators = phase_flip_kraus(prob) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[ 0.9487+0.j, 0.0000+0.j], [ 0.0000+0.j, 0.9487+0.j]], [[ 0.3162+0.j, 0.0000+0.j], [ 0.0000+0.j, -0.3162+0.j]]], dtype=torch.complex128)
- quairkit.database.representation.bit_phase_flip_kraus(prob)¶
Kraus representation of a bit-phase flip channel with form
\[E_0 = \sqrt{1 - p} I, E_1 = \sqrt{p} Y.\]- Parameters:¶
- prob : ndarray | Tensor | Iterable[float] | float¶
probability \(p\).
- dtype
data type. Defaults to be
None.
- Returns:¶
a list of Kraus operators
- Return type:¶
List[ndarray | Tensor]
prob = torch.tensor([0.1]) kraus_operators = bit_phase_flip_kraus(prob) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[0.9487+0.0000j, 0.0000+0.0000j], [0.0000+0.0000j, 0.9487+0.0000j]], [[0.0000+0.0000j, 0.0000-0.3162j], [0.0000+0.3162j, 0.0000+0.0000j]]], dtype=torch.complex128)
- quairkit.database.representation.amplitude_damping_kraus(gamma)¶
Kraus representation of an amplitude damping channel with form
\[\begin{split}E_0 = \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{bmatrix}, E_1 = \begin{bmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{bmatrix}.\end{split}\]- Parameters:¶
- gamma : ndarray | Tensor | Iterable[float] | float¶
coefficient \(\gamma\).
- dtype
data type. Defaults to be
None.
- Returns:¶
a list of Kraus operators
- Return type:¶
List[ndarray | Tensor]
gamma = torch.tensor(0.2) kraus_operators = amplitude_damping_kraus(gamma) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[1.0000+0.j, 0.0000+0.j], [0.0000+0.j, 0.8944+0.j]], [[0.0000+0.j, 0.4472+0.j], [0.0000+0.j, 0.0000+0.j]]], dtype=torch.complex128)
- quairkit.database.representation.generalized_amplitude_damping_kraus(gamma, prob)¶
Kraus representation of a generalized amplitude damping channel with form
\[\begin{split}E_0 = \sqrt{p} \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{bmatrix}, E_1 = \sqrt{p} \begin{bmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{bmatrix},\\ E_2 = \sqrt{1-p} \begin{bmatrix} \sqrt{1-\gamma} & 0 \\ 0 & 1 \end{bmatrix}, E_3 = \sqrt{1-p} \begin{bmatrix} 0 & 0 \\ \sqrt{\gamma} & 0 \end{bmatrix}.\end{split}\]gamma = torch.tensor(0.2) prob = torch.tensor(0.1) kraus_operators = generalized_amplitude_damping_kraus(gamma,prob) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[0.3162+0.j, 0.0000+0.j], [0.0000+0.j, 0.2828+0.j]], [[0.0000+0.j, 0.1414+0.j], [0.0000+0.j, 0.0000+0.j]], [[0.8485+0.j, 0.0000+0.j], [0.0000+0.j, 0.9487+0.j]], [[0.0000+0.j, 0.0000+0.j], [0.4243+0.j, 0.0000+0.j]]], dtype=torch.complex128)
- quairkit.database.representation.phase_damping_kraus(gamma)¶
Kraus representation of a phase damping channel with form
\[\begin{split}E_0 = \begin{bmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{bmatrix}, E_1 = \begin{bmatrix} 0 & 0 \\ 0 & \sqrt{\gamma} \end{bmatrix}.\end{split}\]- Parameters:¶
- gamma : ndarray | Tensor | Iterable[float] | float¶
coefficient \(\gamma\).
- dtype
data type. Defaults to be
None.
- Returns:¶
a list of Kraus operators
- Return type:¶
List[ndarray | Tensor]
gamma = torch.tensor(0.2) kraus_operators = phase_damping_kraus(gamma) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[1.0000+0.j, 0.0000+0.j], [0.0000+0.j, 0.8944+0.j]], [[0.0000+0.j, 0.0000+0.j], [0.0000+0.j, 0.4472+0.j]]], dtype=torch.complex128)
- quairkit.database.representation.depolarizing_kraus(prob)¶
Kraus representation of a depolarizing channel with form
\[E_0 = \sqrt{1-3p/4} I, E_1 = \sqrt{p/4} X, E_2 = \sqrt{p/4} Y, E_3 = \sqrt{p/4} Z.\]- Parameters:¶
- prob : ndarray | Tensor | Iterable[float] | float¶
probability \(p\).
- dtype
data type. Defaults to be
None.
- Returns:¶
a list of Kraus operators
- Return type:¶
List[ndarray | Tensor]
prob = torch.tensor(0.1) kraus_operators = depolarizing_kraus(prob) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[ 0.9618+0.0000j, 0.0000+0.0000j], [ 0.0000+0.0000j, 0.9618+0.0000j]], [[ 0.0000+0.0000j, 0.1581+0.0000j], [ 0.1581+0.0000j, 0.0000+0.0000j]], [[ 0.0000+0.0000j, 0.0000-0.1581j], [ 0.0000+0.1581j, 0.0000+0.0000j]], [[ 0.1581+0.0000j, 0.0000+0.0000j], [ 0.0000+0.0000j, -0.1581+0.0000j]]], dtype=torch.complex128)
-
quairkit.database.representation.generalized_depolarizing_kraus(prob, num_qubits, dtype=
torch.complex128)¶ Kraus representation of a generalized depolarizing channel with form
\[E_0 = \sqrt{1-(D - 1)p/D} I, \text{ where } D = 4^n, E_k = \sqrt{p/D} \sigma_k, \text{ for } 0 < k < D.\]prob = torch.tensor(0.1) num_qubits=1 kraus_operators = generalized_depolarizing_kraus(prob,num_qubits) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[ 1.3601+0.0000j, 0.0000+0.0000j], [ 0.0000+0.0000j, 1.3601+0.0000j]], [[ 0.0000+0.0000j, 0.2236+0.0000j], [ 0.2236+0.0000j, 0.0000+0.0000j]], [[ 0.0000+0.0000j, 0.0000-0.2236j], [ 0.0000+0.2236j, 0.0000+0.0000j]], [[ 0.2236+0.0000j, 0.0000+0.0000j], [ 0.0000+0.0000j, -0.2236+0.0000j]]])
- quairkit.database.representation.pauli_kraus(prob)¶
Kraus representation of a pauli channel
- Parameters:¶
- prob : ndarray | Tensor | Iterable[float] | float¶
a list of three probabilities corresponding to X, Y, Z gate \(p\).
- dtype
data type. Defaults to be
None.
- Returns:¶
a list of Kraus operators
- Return type:¶
List[ndarray | Tensor]
prob_list = torch.tensor([0.1, 0.2, 0.3]) kraus_operators = pauli_kraus(prob_list) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[ 0.6325+0.0000j, 0.0000+0.0000j], [ 0.0000+0.0000j, 0.6325+0.0000j]], [[ 0.0000+0.0000j, 0.3162+0.0000j], [ 0.3162+0.0000j, 0.0000+0.0000j]], [[ 0.0000+0.0000j, 0.0000-0.4472j], [ 0.0000+0.4472j, 0.0000+0.0000j]], [[ 0.5477+0.0000j, 0.0000+0.0000j], [ 0.0000+0.0000j, -0.5477+0.0000j]]], dtype=torch.complex128)
- quairkit.database.representation.reset_kraus(prob)¶
Kraus representation of a reset channel with form
\[\begin{split}E_0 = \begin{bmatrix} \sqrt{p} & 0 \\ 0 & 0 \end{bmatrix}, E_1 = \begin{bmatrix} 0 & \sqrt{p} \\ 0 & 0 \end{bmatrix},\\ E_2 = \begin{bmatrix} 0 & 0 \\ \sqrt{q} & 0 \end{bmatrix}, E_3 = \begin{bmatrix} 0 & 0 \\ 0 & \sqrt{q} \end{bmatrix},\\ E_4 = \sqrt{1-p-q} I.\end{split}\]- Parameters:¶
- prob : ndarray | Tensor | Iterable[float] | float¶
list of two probabilities of resetting to state \(|0\rangle\) and \(|1\rangle\).
- dtype
data type. Defaults to be
None.
- Returns:¶
a list of Kraus operators
- Return type:¶
List[ndarray | Tensor]
prob_list = torch.tensor([0.1, 0.2]) kraus_operators = reset_kraus(prob_list) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[0.3162+0.j, 0.0000+0.j], [0.0000+0.j, 0.0000+0.j]], [[0.0000+0.j, 0.3162+0.j], [0.0000+0.j, 0.0000+0.j]], [[0.0000+0.j, 0.0000+0.j], [0.4472+0.j, 0.0000+0.j]], [[0.0000+0.j, 0.0000+0.j], [0.0000+0.j, 0.4472+0.j]], [[0.8367+0.j, 0.0000+0.j], [0.0000+0.j, 0.8367+0.j]]], dtype=torch.complex128)
- quairkit.database.representation.thermal_relaxation_kraus(const_t, exec_time)¶
Kraus representation of a thermal relaxation channel
const_t = torch.tensor([50, 30]) exec_time = torch.tensor([100]) kraus_operators = thermal_relaxation_kraus(const_t, exec_time) print(f'The Kraus operators are:\n{kraus_operators}')The Kraus operators are: tensor([[[ 0.9987+0.j, 0.0000+0.j], [ 0.0000+0.j, 0.9987+0.j]], [[ 0.0258+0.j, 0.0000+0.j], [ 0.0000+0.j, -0.0258+0.j]], [[ 0.0447+0.j, 0.0000+0.j], [ 0.0000+0.j, 0.0000+0.j]], [[ 0.0000+0.j, 0.0447+0.j], [ 0.0000+0.j, 0.0000+0.j]]], dtype=torch.complex128)
- quairkit.database.representation.replacement_choi(sigma)¶
Choi representation of a replacement channel
- Parameters:¶
- sigma : ndarray | Tensor | State¶
output state of this channel.
- dtype
data type. Defaults to be
None.
- Returns:¶
a Choi operator.
- Return type:¶
ndarray | Tensor | State
sigma= torch.tensor([[0.8, 0.0], [0.0, 0.2]]) choi_operator = replacement_choi(sigma) print(f'The Choi operator is :\n{choi_operator}')The Choi operator is : tensor([[0.8000+0.j, 0.0000+0.j, 0.0000+0.j, 0.0000+0.j], [0.0000+0.j, 0.2000+0.j, 0.0000+0.j, 0.0000+0.j], [0.0000+0.j, 0.0000+0.j, 0.8000+0.j, 0.0000+0.j], [0.0000+0.j, 0.0000+0.j, 0.0000+0.j, 0.2000+0.j]], dtype=torch.complex128)