Basis#

Set of functions related to basis and basis transformations.

Pauli basis#

qibo.quantum_info.pauli_basis(nqubits, normalize=False, vectorize=False, sparse=False, order=None, pauli_order='IXYZ', backend=None)[source]#

Creates the nqubits-qubit Pauli basis.

Parameters:

  • nqubits (int) – number of qubits.

  • normalize (bool, optional) – If True, normalized basis is returned. Defaults to False.

  • vectorize (bool, optional) – If False, returns a nested array with all Pauli matrices. If True, returns an array where every row is a vectorized Pauli matrix. Defaults to False.

  • sparse (bool, optional) – If True, returns Pauli basis in a sparse representation. Defaults to False.

  • order (str, optional) – If "row", vectorization of Pauli basis is performed row-wise. If "column", vectorization is performed column-wise. If "system", system-wise vectorization is performed. Required when vectorize=True and ignored when vectorize=False. Defaults to None.

  • pauli_order (str, optional) – corresponds to the order of 4 single-qubit Pauli elements. Defaults to "IXYZ".

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

all Pauli matrices forming the basis. If sparse=True

and vectorize=True, tuple is composed of an array of non-zero elements and an array with their row-wise indexes.

Return type:

ArrayLike or tuple

Computational basis to Pauli basis#

qibo.quantum_info.comp_basis_to_pauli(nqubits, normalize=False, sparse=False, order='row', pauli_order='IXYZ', backend=None)[source]#

Matrix \(U\) that converts operators from the Liouville representation in the computational basis to the Pauli-Liouville representation.

The matrix \(U\) is given by

\[U = \sum_{k = 0}^{d^{2} - 1} \, |b_{k})(P_{k}| \, ,\]

where \(|P_{k})\) is the vectorization of the \(k\)-th Pauli operator \(P_{k}\), and \(|b_{k})\) is the vectorization of the \(k\)-th computational basis element. \(U\) is unitary only if the Pauli basis is normalized, i.e. normalize=True. For a definition of vectorization, see qibo.quantum_info.vectorization().

Example

from qibo.quantum_info import random_density_matrix, vectorization, comp_basis_to_pauli
nqubits = 2
d = 2**nqubits
rho = random_density_matrix(d)
U_c2p = comp_basis_to_pauli(nqubits, normalize=True, order="system")
rho_liouville = vectorization(rho, order="system")
rho_pauli_liouville = U_c2p @ rho_liouville
Parameters:

  • nqubits (int) – number of qubits.

  • normalize (bool, optional) – If True, converts to the normalized Pauli basis. Defaults to False.

  • sparse (bool, optional) – If True, returns unitary matrix in sparse representation. Defaults to False.

  • order (str, optional) – If "row", vectorization of Pauli basis is performed row-wise. If "column", vectorization is performed column-wise. If "system", system-wise vectorization is performed. Defaults to "row".

  • pauli_order (str, optional) – corresponds to the order of 4 single-qubit Pauli elements. Defaults to "IXYZ".

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

Matrix \(U\). If sparse=True,

tuple is composed of array of non-zero elements and an array with their row-wise indexes.

Return type:

ArrayLike or tuple

Pauli basis to computational basis#

qibo.quantum_info.pauli_to_comp_basis(nqubits, normalize=False, sparse=False, order='row', pauli_order='IXYZ', backend=None)[source]#

Matrix \(U\) that converts operators from the Pauli-Liouville representation to the Liouville representation in the computational basis.

The matrix \(U\) is given by

\[U = \sum_{k = 0}^{d^{2} - 1} \, |P_{k})(b_{k}| \, ,\]

where \(|P_{k})\) is the vectorization of the \(k\)-th Pauli operator \(P_{k}\), and \(|b_{k})\) is the vectorization of the \(k\)-th computational basis element. \(U\) is unitary only if the Pauli basis is normalized, i.e. normalize=True. For a definition of vectorization, see qibo.quantum_info.vectorization().

Parameters:

  • nqubits (int) – number of qubits.

  • normalize (bool, optional) – If True, converts to the normalized Pauli basis. Defaults to False.

  • sparse (bool, optional) – If True, returns unitary matrix in sparse representation. Defaults to False.

  • order (str, optional) – If "row", vectorization of Pauli basis is performed row-wise. If "column", vectorization is performed column-wise. If "system", system-wise vectorization is performed. Defaults to "row".

  • pauli_order (str, optional) – corresponds to the order of 4 single-qubit Pauli elements. Defaults to "IXYZ".

  • backend (qibo.backends.abstract.Backend, optional) – backend to be used in the execution. If None, it uses the current backend. Defaults to None.

Returns:

Matrix \(U\). If sparse=True,

tuple is composed of array of non-zero elements and an array with their row-wise indexes.

Return type:

ArrayLike or tuple