abp/local_cliffords.py

 #!/usr/bin/python
# -*- coding: utf-8 -*-

""" 
Generates and enumerates the 24 elements of the local Clifford group
Following the prescription of Anders (thesis pg. 26):
> Table 2.1: The 24 elements of the local Clifford group. The row index (here called the “sign symbol”) shows how the operator
> U permutes the Pauli operators σ = X, Y, Z under the conjugation σ = ±UσU† . The column index (the “permutation
> symbol”) indicates the sign obtained under the conjugation: For operators U in the I column it is the sign of the permutation
> (indicated on the left). For elements in the X, Y and Z columns, it is this sign only if the conjugated Pauli operator is the one
> indicated by the column header and the opposite sign otherwise.
""" 

# TODO:
# - check that we re-generate the table
# - sort of re-map to an ordering 
# - do conjugation
# - do times table
# - write tests

from numpy import *

def identify_pauli(m):
    """ Given a signed Pauli matrix, name it. """
    for sign in (+1, -1):
        for pauli_label, pauli in zip("xyz", paulis):
            if allclose(sign*pauli, m):
                return sign, pauli_label

def anders_sign_rule(sign, column, p):
    """ Anders' sign rule from thesis """
    return sign if (p==column or column=="i") else -sign, p

def format_action(action):
    return "".join("{}{}".format("+" if s>=0 else "-", p) for s, p in action)

# Some two-qubit matrices
i = matrix(eye(2, dtype=complex))
px = matrix([[0, 1], [1, 0]], dtype=complex)
py = matrix([[0, -1j], [1j, 0]], dtype=complex)
pz = matrix([[1, 0], [0, -1]], dtype=complex)
h = matrix([[1, 1], [1, -1]], dtype=complex) / sqrt(2)
p = matrix([[1, 0], [0, 1j]], dtype=complex)
paulis = (px, py, pz)

# More two-qubit matrices
s_rotations = [i, p, p*p, p*p*p]
s_names = ["i", "p", "pp", "ppp"]
c_rotations = [i, h, h*p, h*p*p, h*p*p*p, h*p*p*h]
c_names = ["i", "h", "hp", "hpp", "hppp", "hpph"]

# Build the table of VOPs according to Anders (verbatim from thesis)
table = (("a", "xyz", +1), ("b", "yxz", -1), ("c", "zyx", -1),
          ("d", "xzy", -1), ("e", "yxz", +1), ("f", "zxy", +1))

for label, permutation, sign in table:
    for column, operator in zip("ixyz", "i"+permutation):
        effect = [anders_sign_rule(sign, column, p) for p in "xyz"]
        print label+operator, format_action(effect)

for s, sn in zip(s_rotations, s_names):
    for c, cn in zip(c_rotations, c_names):
        u = s*c
        action = tuple(identify_pauli(u*p*u.H) for p in paulis)
        print cn, sn, format_action(action)