qiskit.quantum_info.Pauli¶
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class
Pauli(z=None, x=None, label=None)[source]¶ 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
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__init__(z=None, x=None, label=None)[source]¶ 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.
Convert to Pauli circuit instruction.
to_label()Present the pauli labels in I, X, Y, Z format.
Convert Pauli to a matrix representation.
Convert to Operator object.
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
Number of qubits.
Getter of x.
Getter of z.
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delete_qubits(indices)[source]¶ Delete pauli at the indices.
- Parameters
indices (list[int]) – the indices of to-be-deleted paulis
- Returns
self
- Return type
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classmethod
from_label(label)[source]¶ 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
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insert_paulis(indices=None, paulis=None, pauli_labels=None)[source]¶ 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
Note
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
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property
num_qubits¶ Number of qubits.
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classmethod
pauli_single(num_qubits, index, pauli_label)[source]¶ 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
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classmethod
random(num_qubits, seed=None)[source]¶ 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
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static
sgn_prod(p1, p2)[source]¶ Multiply two Paulis and track the phase.
$P_3 = P_1 otimes P_2$: X*Y
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to_label()[source]¶ Present the pauli labels in I, X, Y, Z format.
Order is $q_{n-1} …. q_0$
- Returns
pauli label
- Return type
str
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to_matrix()[source]¶ 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
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to_spmatrix()[source]¶ 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
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update_x(x, indices=None)[source]¶ 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.
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update_z(z, indices=None)[source]¶ 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.
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property
x¶ Getter of x.
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property
z¶ Getter of z.