def unitary_to_axis(matrix_U, basis): (matrix_V, matrix_W) = diagonalize(matrix_U, basis) #print "V= " + str(matrix_V) #print "W= " + str(matrix_W) matrix_ln = get_matrix_logarithm(matrix_W) #print "matrix_ln= " + str(matrix_ln) # Reconjugate to transform into iH matrix_iH = matrix_V * matrix_ln * matrix_V.I # Factor out -i (since we used -i in exp_hermitian_to_unitary) matrix_H = (-1.0/1j) * matrix_iH #print "matrix_H= " + str(matrix_H) trace_norm = numpy.trace(matrix_H * matrix_H.H) #print "trace_norm(H)= " + str(trace_norm) # Compare the calculated components with our original (components2, K) = get_basis_components(matrix_H, basis) #print "K= " + str(K) #angle = K/2.0 # Scale matrix by our calculated angle #matrix_H = matrix_H / angle return (components2, K, matrix_H)
def unitary_to_axis(matrix_U, basis):
(matrix_V, matrix_W) = diagonalize(matrix_U, basis)
#print "V= " + str(matrix_V)
#print "W= " + str(matrix_W)
matrix_ln = get_matrix_logarithm(matrix_W)
#print "matrix_ln= " + str(matrix_ln)
# Reconjugate to transform into iH
matrix_iH = matrix_V * matrix_ln * matrix_V.I
# Factor out -i (since we used -i in exp_hermitian_to_unitary)
matrix_H = (-1.0 / 1j) * matrix_iH
#print "matrix_H= " + str(matrix_H)
trace_norm = numpy.trace(matrix_H * matrix_H.H)
#print "trace_norm(H)= " + str(trace_norm)
# Compare the calculated components with our original
(components2, K) = get_basis_components(matrix_H, basis)
#print "K= " + str(K)
#angle = K/2.0
# Scale matrix by our calculated angle
#matrix_H = matrix_H / angle
return (components2, K, matrix_H)
def exp_hermitian_to_unitary(matrix_H, angle, basis):
#print "exp_hermitian_to_unitary angle= " + str(angle)
(matrix_V, matrix_W) = diagonalize(matrix_H, basis)
Udiag = matrix_exp_diag(-1j * angle * matrix_W)
assert_matrix_unitary(Udiag)
# Now translate it back to its non-diagonal form
U = matrix_V * Udiag * matrix_V.I
assert_matrix_unitary(U)
return U
def exp_hermitian_to_unitary(matrix_H, angle, basis): #print "exp_hermitian_to_unitary angle= " + str(angle) (matrix_V, matrix_W) = diagonalize(matrix_H, basis) Udiag = matrix_exp_diag(-1j*angle*matrix_W) assert_matrix_unitary(Udiag) # Now translate it back to its non-diagonal form U = matrix_V * Udiag * matrix_V.I assert_matrix_unitary(U) return U
def test_decomposing_unitary(d):
#print "*******************************************************************"
#print "TESTING DECOMPOSITION OF UNITARY IN SU("+str(d)+")"
B = get_hermitian_basis(d)
(matrix_U, components, angle) = get_random_unitary(B)
#print "U= " + str(matrix_U)
(matrix_V, matrix_W) = diagonalize(matrix_U, B)
#print "V= " + str(matrix_V)
#print "W= " + str(matrix_W)
matrix_ln = get_matrix_logarithm(matrix_W)
#print "matrix_ln= " + str(matrix_ln)
# Reconjugate to transform into iH
matrix_iH = matrix_V * matrix_ln * matrix_V.I
# Factor out -i (since we used -i in exp_hermitian_to_unitary)
matrix_H = (-1.0/1j) * matrix_iH
#print "matrix_H= " + str(matrix_H)
# Compare the calculated components with our original
(components2, K) = get_basis_components(matrix_H, B)
#print "K= " + str(K)
#print "angle= " + str(angle)
assert_approx_equals_tolerance(numpy.abs(K), numpy.abs(2*angle), TOLERANCE4)
#print "Renormalizing... "
# Renormalize components
#for key,value in components2.items():
# components2[key] = value / K
# print "("+str(key)+")= " + str(components2[key])
# Assert that the components are now normalized
norm = scipy.linalg.norm(components.values())
norm2= scipy.linalg.norm(components2.values())
#print "norm= " + str(norm)
#print "norm2= " + str(norm2)
assert_approx_equals(norm2, 1)
for key in components2.keys():
#print str(key)
#print " actual= " + str(components[key])
#print " comput= " + str(components2[key])
ratio = abs(components[key]) / abs(components2[key])
#print " ratio= " + str(ratio)
assert_approx_equals_tolerance(ratio, 1, TOLERANCE6)
assert_matrix_hermitian(SX.matrix) RX2 = matrix_exp(-1j * (math.pi / 2) * SX.matrix, 50) print "RX2= " + str(RX2) # We can't assert unitarity here, b/c our matrix_exp misformats the matrix # somehow so that the elements aren't complex, or something. # Just visually inspect the print str above. #assert_matrix_unitary(RX2) # Verify that exponentiating a Hermitian (via its diagonalized form) # gives a unitary B2 = get_hermitian_basis(d=2) (matrix_V, matrix_W) = diagonalize(SX.matrix, B2) RX3 = matrix_exp_diag(-1j * (math.pi / 2) * matrix_W) print "RX3= " + str(RX3) # Success! Hopefully assert_matrix_unitary(RX3) # Now translate it back to its non-diagonal form RX4 = matrix_V * RX3 * matrix_V.I print "RX4= " + str(RX4) # Should still be unitary assert_matrix_unitary(RX3)