Pauli¶

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

Attributes

`Pauli.num_qubits`	Number of qubits.
`Pauli.numberofqubits`	Deprecated, use `num_qubits` instead.
`Pauli.x`	Getter of x.
`Pauli.z`	Getter of z.

Methods

`Pauli.__len__`()	Return number of qubits.
`Pauli.__mul__`(other)	Multiply two Paulis.
`Pauli.append_paulis`([paulis, pauli_labels])	Append pauli at the end.
`Pauli.delete_qubits`(indices)	Delete pauli at the indices.
`Pauli.from_label`(label)	Take pauli string to construct pauli.
`Pauli.insert_paulis`([indices, paulis, …])	Insert or append pauli to the targeted indices.
`Pauli.kron`(other)	Kronecker product of two paulis.
`Pauli.pauli_single`(num_qubits, index, …)	Generate single qubit pauli at index with pauli_label with length num_qubits.
`Pauli.random`(num_qubits[, seed])	Return a random Pauli on number of qubits.
`Pauli.sgn_prod`(p1, p2)	Multiply two Paulis and track the phase.
`Pauli.to_instruction`()	Convert to Pauli circuit instruction.
`Pauli.to_label`()	Present the pauli labels in I, X, Y, Z format.
`Pauli.to_matrix`()	Convert Pauli to a matrix representation.
`Pauli.to_operator`()	Convert to Operator object.
`Pauli.to_spmatrix`()	Convert Pauli to a sparse matrix representation (CSR format).
`Pauli.update_x`(x[, indices])	Update partial or entire x.
`Pauli.update_z`(z[, indices])	Update partial or entire z.