qiskit.quantum_info.Pauli
class Pauli(z=None, x=None, label=None)
A simple class representing Pauli Operators.
The form is P_zx = (-i)^dot(z,x) Z^z X^x where z and x are elements of Z_2^n. That is, there are 4^n elements (no phases in this group).
For example, for 1 qubit P_00 = Z^0 X^0 = I P_01 = X P_10 = Z P_11 = -iZX = (-i) iY = Y
The overload __mul__ does not track the sign: P1*P2 = Z^(z1+z2) X^(x1+x2) but sgn_prod does __mul__ and track the phase: P1*P2 = (-i)^dot(z1+z2,x1+x2) Z^(z1+z2) X^(x1+x2) where the sums are taken modulo 2.
Pauli vectors z and x are supposed to be defined as boolean numpy arrays.
Ref. Jeroen Dehaene and Bart De Moor Clifford group, stabilizer states, and linear and quadratic operations over GF(2) Phys. Rev. A 68, 042318 – Published 20 October 2003
Make the Pauli object.
Note that, for the qubit index:
- Order of z, x vectors is q_0 … q_{n-1},
- Order of pauli label is q_{n-1} … q_0
E.g.,
- z and x vectors: z = [z_0 … z_{n-1}], x = [x_0 … x_{n-1}]
- a pauli is $P_{n-1} otimes … otimes P_0$
Parameters
- z (numpy.ndarray) – boolean, z vector
- x (numpy.ndarray) – boolean, x vector
- label (str) – pauli label
__init__
__init__(z=None, x=None, label=None)
Make the Pauli object.
Note that, for the qubit index:
- Order of z, x vectors is q_0 … q_{n-1},
- Order of pauli label is q_{n-1} … q_0
E.g.,
- z and x vectors: z = [z_0 … z_{n-1}], x = [x_0 … x_{n-1}]
- a pauli is $P_{n-1} otimes … otimes P_0$
Parameters
- z (numpy.ndarray) – boolean, z vector
- x (numpy.ndarray) – boolean, x vector
- label (str) – pauli label
Methods
__init__([z, x, label]) | Make the Pauli object. |
append_paulis([paulis, pauli_labels]) | Append pauli at the end. |
delete_qubits(indices) | Delete pauli at the indices. |
from_label(label) | Take pauli string to construct pauli. |
insert_paulis([indices, paulis, pauli_labels]) | Insert or append pauli to the targeted indices. |
kron(other) | Kronecker product of two paulis. |
pauli_single(num_qubits, index, pauli_label) | Generate single qubit pauli at index with pauli_label with length num_qubits. |
random(num_qubits[, seed]) | Return a random Pauli on number of qubits. |
sgn_prod(p1, p2) | Multiply two Paulis and track the phase. |
to_instruction() | Convert to Pauli circuit instruction. |
to_label() | Present the pauli labels in I, X, Y, Z format. |
to_matrix() | Convert Pauli to a matrix representation. |
to_operator() | Convert to Operator object. |
to_spmatrix() | Convert Pauli to a sparse matrix representation (CSR format). |
update_x(x[, indices]) | Update partial or entire x. |
update_z(z[, indices]) | Update partial or entire z. |
Attributes
num_qubits | Number of qubits. |
x | Getter of x. |
z | Getter of z. |
append_paulis
append_paulis(paulis=None, pauli_labels=None)
Append pauli at the end.
Parameters
- paulis (Pauli) – the to-be-inserted or appended pauli
- pauli_labels (list[str]) – the to-be-inserted or appended pauli label
Returns
self
Return type
delete_qubits
delete_qubits(indices)
Delete pauli at the indices.
Parameters
indices (list[int]) – the indices of to-be-deleted paulis
Returns
self
Return type
from_label
classmethod from_label(label)
Take pauli string to construct pauli.
The qubit index of pauli label is q_{n-1} … q_0. E.g., a pauli is $P_{n-1} otimes … otimes P_0$
Parameters
label (str) – pauli label
Returns
the constructed pauli
Return type
Raises
QiskitError – invalid character in the label
insert_paulis
insert_paulis(indices=None, paulis=None, pauli_labels=None)
Insert or append pauli to the targeted indices.
If indices is None, it means append at the end.
Parameters
- indices (list[int]) – the qubit indices to be inserted
- paulis (Pauli) – the to-be-inserted or appended pauli
- pauli_labels (list[str]) – the to-be-inserted or appended pauli label
the indices refers to the location of original paulis, e.g. if indices = [0, 2], pauli_labels = [‘Z’, ‘I’] and original pauli = ‘ZYXI’ the pauli will be updated to ZY’I’XI’Z’ ‘Z’ and ‘I’ are inserted before the qubit at 0 and 2.
Returns
self
Return type
Raises
QiskitError – provide both paulis and pauli_labels at the same time
kron
kron(other)
Kronecker product of two paulis.
Order is $P_2 (other) otimes P_1 (self)$
Parameters
other (Pauli) – P2
Returns
self
Return type
num_qubits
Number of qubits.
pauli_single
classmethod pauli_single(num_qubits, index, pauli_label)
Generate single qubit pauli at index with pauli_label with length num_qubits.
Parameters
- num_qubits (int) – the length of pauli
- index (int) – the qubit index to insert the single qubit
- pauli_label (str) – pauli
Returns
single qubit pauli
Return type
random
classmethod random(num_qubits, seed=None)
Return a random Pauli on number of qubits.
Parameters
- num_qubits (int) – the number of qubits
- seed (int) – Optional. To set a random seed.
Returns
the random pauli
Return type
sgn_prod
static sgn_prod(p1, p2)
Multiply two Paulis and track the phase.
$P_3 = P_1 otimes P_2$: X*Y
Parameters
Returns
the multiplied pauli complex: the sign of the multiplication, 1, -1, 1j or -1j
Return type
to_instruction
to_instruction()
Convert to Pauli circuit instruction.
to_label
to_label()
Present the pauli labels in I, X, Y, Z format.
Order is $q_{n-1} …. q_0$
Returns
pauli label
Return type
str
to_matrix
to_matrix()
Convert Pauli to a matrix representation.
Order is q_{n-1} …. q_0, i.e., $P_{n-1} otimes … P_0$
Returns
a matrix that represents the pauli.
Return type
numpy.array
to_operator
to_operator()
Convert to Operator object.
to_spmatrix
to_spmatrix()
Convert Pauli to a sparse matrix representation (CSR format).
Order is q_{n-1} …. q_0, i.e., $P_{n-1} otimes … P_0$
Returns
a sparse matrix with CSR format that represents the pauli.
Return type
scipy.sparse.csr_matrix
update_x
update_x(x, indices=None)
Update partial or entire x.
Parameters
- x (numpy.ndarray or list) – to-be-updated x
- indices (numpy.ndarray or list or optional) – to-be-updated qubit indices
Returns
self
Return type
Raises
QiskitError – when updating whole x, the number of qubits must be the same.
update_z
update_z(z, indices=None)
Update partial or entire z.
Parameters
- z (numpy.ndarray or list) – to-be-updated z
- indices (numpy.ndarray or list or optional) – to-be-updated qubit indices
Returns
self
Return type
Raises
QiskitError – when updating whole z, the number of qubits must be the same.
x
Getter of x.
z
Getter of z.