def E_s(x_, y_dim):
# (2 a_j)^2
xx_derivative = Matrix(9, 1, [0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0])
xx_outer = TensorProduct(xx_derivative, xx_derivative.T)
# (\frac{1}{2} (2 a_{j+1} x_i + b_{j+1} - 2 a_{j-1} x_i - b_{j-1}))^2
xy_derivative = Matrix(
9, 1, [-2.0 * x, -1.0, 0.0, 0.0, 0.0, 0.0, 2.0 * x, 1.0, 0.0])
xy_derivative = 0.5 * xy_derivative
xy_outer = TensorProduct(xy_derivative, xy_derivative.T)
# (a_{j+1} x_i^2 + b_{j+1} x_i + c_{j+1} - 2a_{j} x_i^2 - 2 b_{j} x_i - 2 c_{j} + a_{j-1} x_i^2 + b_{j-1} x_i + c_{j-1})^2
yy_derivative = Matrix(
9, 1, [x * x, x, 1.0, -2.0 * x * x, -2.0 * x, -2.0, x * x, x, 1.0])
yy_outer = TensorProduct(yy_derivative, yy_derivative.T)
combined = xx_outer + 2.0 * xy_outer + yy_outer
A_pattern = np.zeros((9, 9))
for i in x_:
A_pattern += np.array(combined.subs(x, i)).astype(np.float64)
# Fill along diagonal
A_dim = y_dim * 3 * 2
A = np.zeros((A_dim, A_dim))
for i in range(0, y_dim - 2):
idx_n = i * 3
A[idx_n:idx_n + 9, idx_n:idx_n + 9] += A_pattern
idx_s = i * 3 + y_dim * 3
A[idx_s:idx_s + 9, idx_s:idx_s + 9] += A_pattern
return A
def Potential(K1, K2, J, epsilon):
############ LIST OF CONSTANTS USED#################
# K1 #Spring constant of Top 1
# K2 #Spring constant of Top2
# J #Eignevalue of J^2 operator
# epsilon #Coupling strength of the two tops
N = int(
2 * J + 1
) #Dimension of the Hamiltonian of one top; Number of spin degenerate states permissible
N2 = N * N #Dimension of the Tensor Product space
####################################################
Jz = [[0 for x in range(N)]
for y in range(N)] #To calculate the representation of Jz
Identity = [[0 for x in range(N)]
for y in range(N)] #To calculate the Identity
#To calculate the Jz (Angular Momentum along z axis)
for x in range(N):
m = -x + J
Jz[x][x] = m
Identity[x][x] = 1
Identity = np.matrix(Identity)
Jz = np.matrix(Jz)
Jz_sq = Jz * Jz
J1 = (Jz + 0.5 * Identity)
J_sq = J1 * J1
Identity = np.matrix(Identity)
V = (K1 / (2 * J)) * TensorProduct(
Jz_sq, Identity) + (K2 / (2 * J)) * TensorProduct(
Identity, J_sq) + (epsilon / J) * TensorProduct(
Jz, J1) #To calculate the Fourier Coefficient of the Potential
return V
def tensorized_basis_3D(order):
total_integration_points = (order + 1) * (order + 1)
r,s,t = sym.symbols('r,s,t')
r_gll = sym.symbols('r_0:%d' % (order + 1))
s_gll = sym.symbols('s_0:%d' % (order + 1))
t_gll = sym.symbols('t_0:%d' % (order + 1))
# Get N + 1 lagrange polynomials in each direction.
generator_r = generating_polynomial_lagrange(order, 'r', r_gll)
generator_s = generating_polynomial_lagrange(order, 's', s_gll)
generator_t = generating_polynomial_lagrange(order, 't', t_gll)
# Get tensorized basis.
basis = TensorProduct(generator_t, generator_s, generator_r)
gll_coordinates, gll_weights = gauss_lobatto_legendre_quadruature_points_weights(order + 1)
basis = basis.subs([(v, c) for v, c in zip(r_gll, gll_coordinates)])
basis = basis.subs([(v, c) for v, c in zip(s_gll, gll_coordinates)])
basis = basis.subs([(v, c) for v, c in zip(t_gll, gll_coordinates)])
# Get gradient of basis functions.
basis_gradient_r = sym.Matrix([sym.diff(i, r) for i in basis])
basis_gradient_s = sym.Matrix([sym.diff(i, s) for i in basis])
basis_gradient_t = sym.Matrix([sym.diff(i, t) for i in basis])
routines = [('interpolate_order{}_hex'.format(order),basis),
('interpolate_r_derivative_order{}_hex'.format(order), basis_gradient_r),
('interpolate_s_derivative_order{}_hex'.format(order), basis_gradient_s),
('interpolate_t_derivative_order{}_hex'.format(order), basis_gradient_t)]
codegen(routines, "C", "order{}_hex".format(order), to_files=True, project="SALVUS")
# fix headers (#include "order4_hex.h" -> #include <Element/HyperCube/Autogen/order3_hex.h>)
fixHeader(order,"hex",".")
def setUpTripartiteSystem(d):
"""
Builds desired tripartite system of 3 qudits
"""
rho = setUpNQudits(3, d)
POVM = setUpPOVMElements(d)
tau = TensorProduct(rho[0], rho[1], rho[2])
M = TensorProduct(POVM[0], POVM[1], sympy.eye(d, d))
return (M * tau).trace()
def U(t):
U00_t = U00(t)
U11_t = U11(t)
Up_t = Up(t)
Um_t = Um(t)
return TensorProduct(o0000, U00_t) + \
TensorProduct(o1111, U11_t) + \
TensorProduct(opp, Up_t) + \
TensorProduct(omm, Um_t)
def tensorized_basis_2D(order):
total_integration_points = (order + 1) * (order + 1)
eps, eta, rho = sym.symbols('epsilon eta rho')
eps_gll = sym.symbols('epsilon_0:%d' % (order + 1))
eta_gll = sym.symbols('eta_0:%d' % (order + 1))
# Get N + 1 lagrange polynomials in each direction.
generator_eps = generating_polynomial_lagrange(order, 'epsilon', eps_gll)
generator_eta = generating_polynomial_lagrange(order, 'eta', eta_gll)
# Get tensorized basis.
basis = TensorProduct(generator_eta, generator_eps)
gll_coordinates, gll_weights = gauss_lobatto_legendre_quadruature_points_weights(order + 1)
basis = basis.subs([(v, c) for v, c in zip(eps_gll, gll_coordinates)])
basis = basis.subs([(v, c) for v, c in zip(eta_gll, gll_coordinates)])
# Get gradient of basis functions.
basis_gradient_eps = sym.Matrix([sym.diff(i, eps) for i in basis])
basis_gradient_eta = sym.Matrix([sym.diff(i, eta) for i in basis])
# Get closure mapping.
closure = sym.Matrix(generate_closure_mapping(order), dtype=int)
# Write code
routines = []
autocode = CCodeGen()
routines.append(autocode.routine(
'interpolate_order{}_square'.format(order), basis,
argument_sequence=None))
routines.append(autocode.routine(
'interpolate_eps_derivative_order{}_square'.format(order), basis_gradient_eps,
argument_sequence=None))
routines.append(autocode.routine(
'interpolate_eta_derivative_order{}_square'.format(order), basis_gradient_eta,
argument_sequence=None))
routines.append(autocode.routine(
'closure_mapping_order{}_square'.format(order), closure,
argument_sequence=None))
routines.append(autocode.routine(
'gll_weights_order{}_square'.format(order), sym.Matrix(gll_weights),
argument_sequence=None))
routines.append(autocode.routine(
'gll_coordinates_order{}_square'.format(order), sym.Matrix(gll_coordinates),
argument_sequence=None))
autocode.write(routines, 'order{}_square'.format(order), to_files=True)
# reformat some code.
for code, lend in zip(['order{}_square.c', 'order{}_square.h'], [' {', ';']):
with io.open(code.format(order), 'rt') as fh:
text = fh.readlines()
text = [line.replace('double', 'int') if 'closure' in line else line for line in text]
with io.open(code.format(order), 'wt') as fh:
fh.writelines(text)
def tensorNqubits(portao, alvo, qubits): #muda o portao de 1 bit para um de N qubits no alvo a escolha
i=0
alvo=alvo-1
while i < alvo: #faz produtos tensoriais para os qubits anteriores ao alvo
portao=TensorProduct(eye(2), portao)
i = i + 1
i=0
while i < qubits - alvo - 1: #faz produtos tensoriais para os qubits posteriores ao alvo
portao=TensorProduct(portao, eye(2))
i = i + 1
return portao
def setUpN2QubitSystems(n):
r1 = sym.Symbol('r_1')
r2 = sym.Symbol('r_2')
t = sym.Symbol('theta')
rho = sym.Matrix([[1 + r1, 0], [0, 1 - r1]])
sigma = sym.Matrix([[1 + r2 * cos(t), r2 * sin(t)],[r2 * sin(t), 1 - r2 * cos(t)]])
rho_not = rho
sigma_not = sigma
for i in range(1, n):
rho = TensorProduct(rho_not, rho)
sigma = TensorProduct(sigma_not, sigma)
return (rho, sigma, r1, r2, t)
def Toom_Cook_Kronecker_Matrices(points, kernel_size, output_size, precision):
input_size = kernel_size + output_size - 1
points_number = input_size - 1
G, GT = G_Matrix(points, points_number, kernel_size)
AT, A = AT_Matrix(points, points_number, output_size)
BT, B = BT_Matrix(points, points_number)
G_Kronecker = TensorProduct(G, G)
AT_Kronecker = TensorProduct(AT, AT)
BT_Kronecker = TensorProduct(BT, BT)
G_Kronecker = np.array(G_Kronecker).astype(precision)
AT_Kronecker = np.array(AT_Kronecker).astype(precision)
BT_Kronecker = np.array(BT_Kronecker).astype(precision)
print(G_Kronecker)
return G_Kronecker, AT_Kronecker, BT_Kronecker
def tensorized_basis_2D(order):
total_integration_points = (order + 1) * (order + 1)
eps, eta, rho = sym.symbols('epsilon eta rho')
eps_gll = sym.symbols('epsilon_0:%d' % (order + 1))
eta_gll = sym.symbols('eta_0:%d' % (order + 1))
# Get N + 1 lagrange polynomials in each direction.
generator_eps = generating_polynomial_lagrange(order, 'epsilon', eps_gll)
generator_eta = generating_polynomial_lagrange(order, 'eta', eta_gll)
# Get tensorized basis.
basis = TensorProduct(generator_eta, generator_eps)
basis = basis.subs([(v, c)
for v, c in zip(eps_gll, gll_coordinates(order))])
basis = basis.subs([(v, c)
for v, c in zip(eta_gll, gll_coordinates(order))])
sym.pprint(basis)
# Get gradient of basis functions.
basis_gradient_eps = sym.Matrix([sym.diff(i, eps) for i in basis])
basis_gradient_eta = sym.Matrix([sym.diff(i, eta) for i in basis])
# Get diagonal mass matrix
mass_matrix = rho * basis * basis.T
mass_matrix_diagonal = sym.Matrix(
[mass_matrix[i, i] for i in range(total_integration_points)])
# Write code
routines = []
autocode = CCodeGen()
routines.append(
autocode.routine('interpolate_order{}_square'.format(order),
basis,
argument_sequence=None))
routines.append(
autocode.routine(
'interpolate_eps_derivative_order{}_square'.format(order),
basis_gradient_eps,
argument_sequence=None))
routines.append(
autocode.routine(
'interpolate_eta_derivative_order{}_square'.format(order),
basis_gradient_eta,
argument_sequence=None))
routines.append(
autocode.routine('diagonal_mass_matrix_order{}_square'.format(order),
mass_matrix_diagonal,
argument_sequence=None))
autocode.write(routines, 'order{}_square'.format(order), to_files=True)
def E_d(x_shadow, x_nonshadow, pen_inter, y_dim, patch):
############################
# Compute A
############################
# Compute quadratic outer product via sympy
quadratic = Matrix(3, 1, [x * x, x, 1.0])
quadratic_outer = TensorProduct(quadratic, quadratic.T)
A_nonshadow = np.zeros((3, 3))
# Non-shadow pattern
for x_ in x_nonshadow:
A_nonshadow += np.array(quadratic_outer.subs(x, x_)).astype(np.float64)
# Shadow pattern
A_shadow = np.zeros((3, 3))
for x_ in x_shadow:
A_shadow += np.array(quadratic_outer.subs(x, x_)).astype(np.float64)
# Fill A along diagonal with the two patterns
# First half non_shadow, second half shadow pattern
A_dim = y_dim * 3 * 2
A = np.zeros((A_dim, A_dim))
for i in np.arange(0, A_dim / 2, 3, dtype=np.int16):
A[i:i + 3, i:i + 3] = A_nonshadow
for i in np.arange(A_dim / 2, A_dim, 3, dtype=np.int16):
A[i:i + 3, i:i + 3] = A_shadow
###########################
# Compute b
###########################
b = np.zeros(y_dim * 3 * 2)
for i in range(0, y_dim):
# Cut out nonshadow patch
patch_nonshadow = patch[i, int(np.ceil(pen_inter[1])):]
# Cut out shadow patch
patch_shadow = patch[i, 0:int(np.floor(pen_inter[0])) + 1]
# sum_j I_{i,j} * x_i^2
b[3 * i] = np.sum(np.multiply(patch_nonshadow, np.square(x_nonshadow)))
# sum_j I_{i,j} * x_i
b[3 * i + 1] = np.sum(np.multiply(patch_nonshadow, x_nonshadow))
# sum_j I_{i,j}
b[3 * i + 2] = np.sum(patch_nonshadow)
b[3 * i + y_dim * 3] = np.sum(
np.multiply(patch_shadow, np.square(x_shadow)))
b[3 * i + y_dim * 3 + 1] = np.sum(np.multiply(patch_shadow, x_shadow))
b[3 * i + y_dim * 3 + 2] = np.sum(patch_shadow)
return A, b
def enlarge_single_opt_sympy(self, opt, qubit, number_of_qubits):
"""Enlarge single operator to n qubits.
It is exponential in the number of qubits.
Args:
opt (object): the single-qubit opt.
qubit (int): the qubit to apply it on counts from 0 and order
is q_{n-1} ... otimes q_1 otimes q_0.
number_of_qubits (int): the number of qubits in the system.
Returns:
Matrix: the enlarged matrix that operates on all qubits in the system.
"""
temp_1 = eye(2**(number_of_qubits - qubit - 1))
temp_2 = eye(2**(qubit))
enlarge_opt = TensorProduct(temp_1, TensorProduct(opt, temp_2))
return enlarge_opt
def test_evaluate_pauli_product():
from sympy.physics.paulialgebra import evaluate_pauli_product
assert evaluate_pauli_product(I * sigma2 * sigma3) == -sigma1
# Check issue 6471
assert evaluate_pauli_product(-I * 4 * sigma1 * sigma2) == 4 * sigma3
assert evaluate_pauli_product(
1 + I*sigma1*sigma2*sigma1*sigma2 + \
I*sigma1*sigma2*tau1*sigma1*sigma3 + \
((tau1**2).subs(tau1, I*sigma1)) + \
sigma3*((tau1**2).subs(tau1, I*sigma1)) + \
TensorProduct(I*sigma1*sigma2*sigma1*sigma2, 1)
) == 1 -I + I*sigma3*tau1*sigma2 - 1 - sigma3 - I*TensorProduct(1,1)
def __init__(self, N, s, kind):
self.__N = N
self.__s = s
self.__kind = kind
# First, initialize single-qubit density matrix depending on 'kind'
if s * 2 % 1 != 0:
raise ValueError('s must be either an integer or a half-integer.')
if s < 0:
raise ValueError('s cannot be negative.')
single_dim = int(2 * s + 1)
if kind == 'up':
vec = sym.Matrix(basis(single_dim, 0))
dens = sym.Matrix(
sym.Matrix(basis(single_dim, 0)) *
sym.conjugate(sym.Matrix(basis(single_dim, 0)).T))
#self.__vector = TensorProduct(sym.Matrix(basis(single_dim, 0)), sym.Matrix(basis(single_dim, 0)))
#dens = self.__vector * sym.conjugate(self.__vector.T)#np.outer(basis(single_dim, 0), basis(single_dim, 0))
self.__pure = True
elif kind == 'down':
vec = sym.Matrix(basis(single_dim, single_dim - 1))
dens = sym.Matrix(
sym.Matrix(basis(single_dim, single_dim - 1)) *
sym.conjugate(sym.Matrix(basis(single_dim, single_dim - 1)).T))
#self.__vector = TensorProduct(sym.Matrix(basis(single_dim, single_dim-1)), sym.Matrix(basis(single_dim, single_dim-1)))
#dens = self.__vector * sym.conjugate(self.__vector.T)#np.outer(basis(single_dim, 0), basis(single_dim, 0))
self.__pure = True
elif kind == 'mixed':
self.__vector = None
dens = sym.Matrix(np.identity(single_dim) /
single_dim) #qu.maximally_mixed_dm(single_dim)
self.__pure = False
else:
raise ValueError('State kind not recognized.')
for ii in range(N):
if ii == 0:
if self.__pure:
vector = vec
rho = dens
else:
if self.__pure:
vector = TensorProduct(vector, vec)
rho = TensorProduct(rho, dens) #qu.tensor(rho, dens)
self.__vector = vector
self.__rho = rho
def _setup(self):
self._F0_N_ssro, self._F1_N_ssro, self._F0_e_ssro, self._F1_e_ssro, self._F0_RO_pulse, self._F1_RO_pulse, self._P_min1, self._P_0 = \
sympy.symbols('F0_N_ssro F1_N_ssro F0_e_ssro F1_e_ssro F0_RO_pulse F1_RO_pulse P_min1 P_0')
self._p_correlations = sympy.DeferredVector('p_correlations')
# The total error matrix is the tensor product of the nitrogen and electron error matrices.
# The nitrogen error matrix is built up of three matrices RO_err, CNOT_err and Init_err.
# RO_err reflects fidelities of the electron RO,
# CNOT_err reflects the fidelities in the pi (F0) and 2pi (F1) parts of the pi-2pi pulse (as CNOT gate).
# Init_err includes populations in the three nitrogen lines after initialization in -1.
#The inverse matrices are calculated before the tensorproduct, since this is much faster.
self.error_matrix_N = (sympy.Matrix([[self._F0_N_ssro, 1.-self._F1_N_ssro],[1.-self._F0_N_ssro, self._F1_N_ssro]]) * \
sympy.Matrix([[self._F0_RO_pulse, 1.-self._F1_RO_pulse, 0.],[1.-self._F0_RO_pulse, self._F1_RO_pulse, 1.]]) * \
sympy.Matrix([[self._P_min1,self._P_0],[self._P_0,self._P_min1],[1.-self._P_0-self._P_min1,1.-self._P_0-self._P_min1]]))
self.error_matrix_e = (sympy.Matrix(
[[self._F0_e_ssro, 1. - self._F1_e_ssro],
[1. - self._F0_e_ssro, self._F1_e_ssro]]))
self.correction_matrix_N = self.error_matrix_N.inv()
self.correction_matrix_e = self.error_matrix_e.inv()
self.correction_matrix = TensorProduct(self.correction_matrix_N,
self.correction_matrix_e)
corr_vec = self.correction_matrix * \
sympy.Matrix([self._p_correlations[0], self._p_correlations[1], self._p_correlations[2], self._p_correlations[3]])
corr_p_correlations = corr_vec
self.p0_formula = error.Formula()
self.p0_formula.formula = corr_p_correlations
def __init__(self,
C0=sp.Symbol('C0', real=True),
C2=sp.Symbol('C2', real=True),
A=sp.Symbol('A', real=True),
R=sp.Symbol('R', real=True),
kcut=0,
mz=0,
n_sheets=sp.Symbol('n_sheets', int=True)):
so = sp.Matrix([[1, 0], [0, 1]])
sx = sp.Matrix([[0, 1], [1, 0]])
sy = sp.Matrix([[0, -sp.I], [sp.I, 0]])
sz = sp.Matrix([[1, 0], [0, -1]])
ho = C0 + C2 * (self.kx**2 + self.ky**2)
hx = A * self.ky
hy = -A * self.kx
hz = 2 * R * (self.kx**3 - 3 * self.kx * self.ky**2) + mz
if (not np.isclose(kcut, 0)):
ratio = (self.kx**2 + self.ky**2) / kcut**2
cutfactor = 1 / (1 + (ratio))
hz *= cutfactor
hdiag = ho * so + hx * sx + hy * sy + hz * sz
#build hamiltonian with nz sheets
diag = sp.eye(n_sheets)
h = TensorProduct(diag, hdiag)
super().__init__(h, n_sheets)
def solution_space_basis(oper,
matrix_M,
expr_list,
base_expr,
pool="",
debug=False):
'''solve (matrix_F otimes matrix_M)*alpha=alpha for a certain symmetry
####return (list of Matrix): a list of vectors (single column Matrix object)
return (list of list of number): a list of lists (each sublist represent a vector)
oper (dict): a dictionary with four keys "op3x3", "rep", "is_au" and "name"
matrix_M (Matrix): matrix_M matrix
expr_list (list of Symbol): a list of expression basis (f_i(k))
base_expr (list of Symbol): three symbols used in expr_list
pool (pool): if pool object passed in, then use it
debug (boolean): if True, print the progress infomation
'''
debug_print(oper["name"], "start", do_print=debug)
matrix_F = get_matrix_F(oper["op3x3"], expr_list, base_expr, pool=pool)
debug_print(oper["name"],
"got matrix_F, calculating (FxM)*alpha=alpha",
do_print=debug)
matrix_FM = TensorProduct(matrix_F, matrix_M)
temp = matrix_FM - sp.eye(matrix_FM.shape[0])
# curr_basis = np.array(mat.nullspace(simplify=sp.nsimplify)).tolist()
# solution_basis = [m.T.tolist()[0] for m in nullspace(mat, simplify=sp.nsimplify, pool=pool)]
solution_basis = nullspace(temp, pool=pool, debug=debug)
debug_print(oper["name"], "end", do_print=debug)
return solution_basis
def E_c_hardcoder(y_dim):
A_pattern_dim = y_dim * 3 + 9
# (2 a_{j, n} x_i + b_{j, n} - 2 a_{j, s} x_i - b_{j, s})^2
x_derivative = Matrix.zeros(A_pattern_dim, 1)
x_derivative[3] = 2.0 * x # 2 a_{j, n}
x_derivative[4] = 1.0 # b_{j, n}
x_derivative[-6] = -2.0 * x # - 2 a_{j, s} x_i
x_derivative[-5] = -1.0 # - b_{j, s}
x_outer = TensorProduct(x_derivative, x_derivative.T)
# \frac{1}{4} (a_{j+1,n} x_i^2 + b_{j+1,n} x_i + c_{j+1,n}
# - a_{j+1,s} x_i^2 - b_{j+1,s} x_i - c_{j+1,s} - a_{j-1,n} x_i^2
# - b_{j-1,n} x_i - c_{j-1,n} + a_{j-1,s} x_i^2 + b_{j-1,s} x_i + c_{j-1,s})^2
y_derivative = Matrix.zeros(A_pattern_dim, 1)
y_derivative[6] = x * x # a_{j+1,n} x_i^2
y_derivative[7] = x # b_{j+1,n} x_i
y_derivative[8] = 1.0 # c_{j+1,n}
y_derivative[-3] = -x * x # - a_{j+1,s} x_i^2
y_derivative[-2] = -x # - b_{j+1,s} x_i
y_derivative[-1] = -1.0 # - c_{j+1,s}
y_derivative[0] = -x * x #- a_{j-1,n} x_i^2
y_derivative[1] = -x #- b_{j-1,n} x_i
y_derivative[2] = -1.0 #- c_{j-1,n}
y_derivative[-9] = x * x # a_{j-1,s} x_i^2
y_derivative[-8] = x # b_{j-1,s} x_i
y_derivative[-7] = 1.0 # c_{j-1,s}
y_derivative = y_derivative * 0.5
y_outer = TensorProduct(y_derivative, y_derivative.T)
# Combine x and y derivatives
combined = x_outer + y_outer
# Convert to numpy array
combined_np = np.array(combined.tolist())
# Find indices of nonzero matrix entries
idx = np.nonzero(combined_np)
# Build sympy array of nonzero entries
combined_reduced = Array(combined_np[idx[0], idx[1]])
# Now we have an index list and an sympy expression Array, where
# combined_reduced[i] corresponds to matrix position (idx[0,i], idx[1, i])
# Save these two data structures as files
file_Ec_sympy = open('Ec_sympy.obj', 'wb')
pickle.dump(combined_reduced, file_Ec_sympy)
file_Ec_idx = open('Ec_idx.obj', 'wb')
pickle.dump(idx, file_Ec_idx)
def tensorized_basis_2D(order):
total_integration_points = (order + 1) * (order + 1)
eps, eta, rho = sym.symbols("epsilon eta rho")
eps_gll = sym.symbols("epsilon_0:%d" % (order + 1))
eta_gll = sym.symbols("eta_0:%d" % (order + 1))
# Get N + 1 lagrange polynomials in each direction.
generator_eps = generating_polynomial_lagrange(order, "epsilon", eps_gll)
generator_eta = generating_polynomial_lagrange(order, "eta", eta_gll)
# Get tensorized basis.
basis = TensorProduct(generator_eta, generator_eps)
basis = basis.subs([(v, c) for v, c in zip(eps_gll, gll_coordinates(order))])
basis = basis.subs([(v, c) for v, c in zip(eta_gll, gll_coordinates(order))])
sym.pprint(basis)
# Get gradient of basis functions.
basis_gradient_eps = sym.Matrix([sym.diff(i, eps) for i in basis])
basis_gradient_eta = sym.Matrix([sym.diff(i, eta) for i in basis])
# Get diagonal mass matrix
mass_matrix = rho * basis * basis.T
mass_matrix_diagonal = sym.Matrix([mass_matrix[i, i] for i in range(total_integration_points)])
# Write code
routines = []
autocode = CCodeGen()
routines.append(autocode.routine("interpolate_order{}_square".format(order), basis, argument_sequence=None))
routines.append(
autocode.routine(
"interpolate_eps_derivative_order{}_square".format(order), basis_gradient_eps, argument_sequence=None
)
)
routines.append(
autocode.routine(
"interpolate_eta_derivative_order{}_square".format(order), basis_gradient_eta, argument_sequence=None
)
)
routines.append(
autocode.routine(
"diagonal_mass_matrix_order{}_square".format(order), mass_matrix_diagonal, argument_sequence=None
)
)
autocode.write(routines, "order{}_square".format(order), to_files=True)
def kron(self, other): # Kronecker product
if self.__cuda:
if cupy_is_available:
result = cp.kron(self.__matrix, other.get())
else:
result = np.kron(self.__matrix, other.get())
else:
result = TensorProduct(self.__matrix, other.get())
return Matrix(result, self.__cuda)
def apply_measurement(self, op, input_register, moutputs):
# Find the operator of the measurement instance we're running
measurement = self.find_measurement(op['operator_id'])
# Using the same principle as in apply_gate, obtain a matrix we can
# run on all lines
before = op['lines'][0]
after = self.register_size - op['lines'][-1] - 1
matrix = TensorProduct(eye(2**before), measurement.matrix,
eye(2**after))
# To contain the possible states the system could fall into after the
# measurement
states = []
# For each triple in the eigensystem
# Eigenvalue, duplicity (not needed here), eigenvectors
for val, dup, vecs in matrix.eigenvects():
# For each vector in the eigenvector basis
for vec in vecs:
# Obtain a projection operator P = x*x^T
p = vec * vec.H
# Probability of this outcome Pr = |phi>^T * P * |phi>
# [0] because we want the value of a 1x1 matrix
probability = (input_register.H * p * input_register)[0]
# Don't need to consider this outcome anymore if it
# won't happen
if probability == 0: continue
# Calculate the new register |phi'> = P*|phi> / sqrt(Pr)
register = p * input_register / sqrt(probability)
# Copy the measurement outputs as we're branching so don't
# want to affect other branches
moutputs = dict(moutputs)
# Store the outcome that's just happened in the measurement
# outputs
moutputs[op['oid']] = val
# Add the state tuple into states
state = (register, probability, moutputs)
states.append(state)
return states
def apply_measurement(self, op, input_register, moutputs):
# Find the operator of the measurement instance we're running
measurement = self.find_measurement(op['operator_id'])
# Using the same principle as in apply_gate, obtain a matrix we can
# run on all lines
before = op['lines'][0]
after = self.register_size - op['lines'][-1] - 1
matrix = TensorProduct(eye(2**before), measurement.matrix, eye(2**after))
# To contain the possible states the system could fall into after the
# measurement
states = []
# For each triple in the eigensystem
# Eigenvalue, duplicity (not needed here), eigenvectors
for val, dup, vecs in matrix.eigenvects():
# For each vector in the eigenvector basis
for vec in vecs:
# Obtain a projection operator P = x*x^T
p = vec * vec.H
# Probability of this outcome Pr = |phi>^T * P * |phi>
# [0] because we want the value of a 1x1 matrix
probability = (input_register.H * p * input_register)[0]
# Don't need to consider this outcome anymore if it
# won't happen
if probability == 0: continue
# Calculate the new register |phi'> = P*|phi> / sqrt(Pr)
register = p * input_register / sqrt(probability)
# Copy the measurement outputs as we're branching so don't
# want to affect other branches
moutputs = dict(moutputs)
# Store the outcome that's just happened in the measurement
# outputs
moutputs[op['oid']] = val
# Add the state tuple into states
state = (register, probability, moutputs)
states.append(state)
return states
def build_hurwitz_radon_general(n):
""" The function gets as input a number 'n' and returns a set of matrices of size n x n
that meets the Hurwitz-Radon constraints. Here no assumption is made on 'b', where b is a Hurwitz-Radon parameter"""
a, b, c, d = find_hurwitz_radon_parameters(n)
mat_set = build_hurwitz_radon(n)
if not mat_set: # if the list is empty
return mat_set
if b > 1:
for ind, mat in enumerate(mat_set):
mat_set[ind] = TensorProduct(mat, eye(b))
return mat_set
def tensorized_basis_3D(order):
total_integration_points = (order + 1) * (order + 1)
r, s, t = sym.symbols('r,s,t')
r_gll = sym.symbols('r_0:%d' % (order + 1))
s_gll = sym.symbols('s_0:%d' % (order + 1))
t_gll = sym.symbols('t_0:%d' % (order + 1))
# Get N + 1 lagrange polynomials in each direction.
generator_r = generating_polynomial_lagrange(order, 'r', r_gll)
generator_s = generating_polynomial_lagrange(order, 's', s_gll)
generator_t = generating_polynomial_lagrange(order, 't', t_gll)
# Get tensorized basis.
basis = TensorProduct(generator_t, generator_s, generator_r)
gll_coordinates, gll_weights = gauss_lobatto_legendre_quadruature_points_weights(
order + 1)
basis = basis.subs([(v, c) for v, c in zip(r_gll, gll_coordinates)])
basis = basis.subs([(v, c) for v, c in zip(s_gll, gll_coordinates)])
basis = basis.subs([(v, c) for v, c in zip(t_gll, gll_coordinates)])
# Get gradient of basis functions.
basis_gradient_r = sym.Matrix([sym.diff(i, r) for i in basis])
basis_gradient_s = sym.Matrix([sym.diff(i, s) for i in basis])
basis_gradient_t = sym.Matrix([sym.diff(i, t) for i in basis])
routines = [('interpolate_order{}_hex'.format(order), basis),
('interpolate_r_derivative_order{}_hex'.format(order),
basis_gradient_r),
('interpolate_s_derivative_order{}_hex'.format(order),
basis_gradient_s),
('interpolate_t_derivative_order{}_hex'.format(order),
basis_gradient_t)]
codegen(routines,
"C",
"order{}_hex".format(order),
to_files=True,
project="SALVUS")
# fix headers (#include "order4_hex.h" -> #include <Element/HyperCube/Autogen/order3_hex.h>)
fixHeader(order, "hex", ".")
def cnot3(psi, qubit_negado):
#matriz e produto tensorial se o bit negado for o 1
if qubit_negado == 1:
cnotm2= Matrix([[1, 0, 0, 0], #matriz de 2 qubits onde primeiro controla o posterior
[0, 0, 0, 1],
[0, 0, 1, 0],
[0, 1, 0, 0]])
cnot=TensorProduct(cnotm2, eye(2))#produto tensorial para operar 3 qubits
#matriz e produto tensorial se o bit negado for o 3
else:
cnotm2= Matrix([[1, 0, 0, 0], #matriz de 2 qubits onde um qubit controla o anterior
[0, 1, 0, 0],
[0, 0, 0, 1],
[0, 0, 1, 0]])
cnot=TensorProduct(eye(2), cnotm2)#produto tensorial para operar 3 qubits
psi=np.matrix(psi) #para poder multiplicar o vetor de estado pelo operador
psi=psi*cnot #opera a multiplicacao de matrizes
return psi.tolist()[0] #transforma a matriz de volta em lista
def unitary_partial_swap(angles, step, n_qubits):
inter = int(n_qubits / 2 - 1)
if step > inter :
raise NameError("Le qubit selectionne est incorrect.")
elif step == 0 :
return TensorProduct(TensorProduct(eye(2 ** (inter + step)), partial_swap(angles[inter - step])), eye(2 ** (inter - step)))
else :
U = TensorProduct(TensorProduct(eye(2 ** (inter + step)), partial_swap(angles[inter - step])), eye(2 ** (inter - step)))
return TensorProduct(TensorProduct(eye(2 ** (inter - step)), partial_swap(angles[inter + step])), eye(2 ** (inter + step))) * U
def __init__(self,
A=sp.Symbol('A', real=True),
t=sp.Symbol('t', real=True),
tt=sp.Symbol('tt', real=True),
m5=sp.Symbol('m5', real=True),
n_sheets=sp.Symbol('n_sheets', int=True)):
#set of hermitian matrices
Gamma1 = sp.Matrix([[0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0],
[1, 0, 0, 0]])
Gamma2 = sp.Matrix([[0, 0, 0, -sp.I], [0, 0, sp.I, 0],
[0, -sp.I, 0, 0], [sp.I, 0, 0, 0]])
Gamma3 = sp.Matrix([[0, 0, 1, 0], [0, 0, 0, -1], [1, 0, 0, 0],
[0, -1, 0, 0]])
Gamma5 = sp.Matrix([[0, 0, -sp.I, 0], [0, 0, 0, -sp.I],
[sp.I, 0, 0, 0], [0, sp.I, 0, 0]])
#diagonal part of hamiltonian
hdiag = A * ( m5 + t * ( sp.cos(self.kx) + sp.cos(self.ky) ) ) * Gamma5 \
+ A * tt * ( sp.sin(self.kx) * Gamma1 + sp.sin(self.ky) * Gamma2 )
#hopping between sheets
hoffdiag = A * (1 / 2 * t * Gamma5 + 1 /
(2 * sp.I) * tt * Gamma3) #lower offdiag blocks
hoffdiag_c = A * (1 / 2 * t * Gamma5 - 1 /
(2 * sp.I) * tt * Gamma3) #upper offdiag blocks
#build hamiltonian with nz sheets
diag = sp.eye(n_sheets)
lower = sp.Matrix.jordan_block(n_sheets, 0, band='lower')
upper = sp.Matrix.jordan_block(n_sheets, 0, band='upper')
h = TensorProduct(diag, hdiag) + TensorProduct(
lower, hoffdiag) + TensorProduct(upper, hoffdiag_c)
super().__init__(h, n_sheets, degenerate_eigenvalues=True)
def apply_gate(self, op, input_register, moutputs):
# Find the operator of the gate instance we're running
gate = self.find_gate(op['operator_id'])
# Number of lines before the lines we're running on
before = op['lines'][0]
# Number of lines after the lines we're running on
after = self.register_size - op['lines'][-1] - 1
# Obtain an overall matrix by Kronecking together (matrix tensor
# product) an identity on (before) lines, the matrix we want to run
# and an identity on (after) lines
matrix = TensorProduct(eye(2**before), gate.matrix, eye(2**after))
# The only thing running the gate affects is the state, which is just
# |phi'> = M|phi>
return [(matrix * input_register, 1, moutputs)]
def test_tensor_product():
a11, a12, a21, a22, b11, b12, b21, b22 = sp.symbols(
'a11 a12 a21 a22 b11 b12 b21 b22')
A = Matrix([
[a11, a12],
[a21, a22],
])
B = Matrix([
[b11, b12],
[b21, b22],
])
actual = TensorProduct(A, B)
expected = Matrix([
[a11 * b11, a11 * b12, a12 * b11, a12 * b12],
[a11 * b21, a11 * b22, a12 * b21, a12 * b22],
[a21 * b11, a21 * b12, a22 * b11, a22 * b12],
[a21 * b21, a21 * b22, a22 * b21, a22 * b22],
])
assert actual == expected
def apply_controlled(self, op, input_register, moutputs):
# Find the operator of the controlled gate instance we're running
gate = self.find_controlled_gate(op['operator_id'])
# Find the value that was measured by the measurement we're
# conditional upon
# We need the second line because the sympified value may be different
# from the moutputs value. This will affect array indexing, but ==
# will still find them equal
value = moutputs[op['measurement_id']]
value = filter(lambda val: val == value, gate.values)[0]
# Find the matrix corresponding to that value
# TODO Do we need both the parse_expr.str and the filter expression?
small_matrix = gate.matrices[custom_parse_expr(str(value))]
# Use the same logic as in apply_gate and apply_measurement to tensor
# together a large matrix we can apply to all lines
before = op['lines'][0]
after = self.register_size - op['lines'][-1] - 1
matrix = TensorProduct(eye(2**before), small_matrix, eye(2**after))
# Now just apply the gate like in apply_gate
return ([(matrix * input_register, 1, moutputs)])
R = (1 / sqrt(2)) * Matrix([[I, -I], [1, 1]])
# %%
R.adjoint() * T * R
# %%
Omega_T_d = (pi / ((pi / (2 * omega))**2)) * F * T_d * F.adjoint()
# %%
Omega_T_d
# %%
Hs = I * hbar * omega * Matrix([[0, 1], [-1, 0]])
# %%
J = TensorProduct(hbar * Omega_T_d, eye(2)) + TensorProduct(eye(2), Hs)
# %%
J
# %%
J.eigenvects()
# %% [markdown]
# ## Ordinary quantum theory
# %%
t = Symbol('t')
t0 = Symbol('t_0')
# %%
def qsimplify_pauli(e):
"""
Simplify an expression that includes products of pauli operators.
Parameters
==========
e : expression
An expression that contains products of Pauli operators that is
to be simplified.
Examples
========
>>> from sympy.physics.quantum.pauli import SigmaX, SigmaY
>>> from sympy.physics.quantum.pauli import qsimplify_pauli
>>> sx, sy = SigmaX(), SigmaY()
>>> sx * sy
SigmaX()*SigmaY()
>>> qsimplify_pauli(sx * sy)
I*SigmaZ()
"""
if isinstance(e, Operator):
return e
if isinstance(e, (Add, Pow, exp, TensorProduct)):
t = type(e)
return t(*[qsimplify_pauli(arg) for arg in e.args])
if isinstance(e, Mul):
c, nc = e.args_cnc()
nc_s = []
while nc:
if not isinstance(nc[0], SigmaOpBase):
curr = nc.pop(0)
if isinstance(curr, TensorProduct):
curr = TensorProduct(*(qsimplify_pauli(arg)
for arg in curr.args))
nc_s.append(curr)
continue
curr = [nc.pop(0)]
names = [curr[0].name]
while (len(nc) and isinstance(nc[0], SigmaOpBase)):
x = nc.pop(0)
if x.name in names:
idx = names.index(x.name)
y = _qsimplify_pauli_product(curr[idx], x)
c1, nc1 = y.args_cnc()
curr[idx] = Mul(*nc1)
c = c + c1
else:
curr.append(x)
names.append(x.name)
# TODO: Don't make assumptions about what the names are.
# TODO: This sorted order is properly part of a canonical form which
# should be enforced and preserved elsewhere in the module.
curr = sorted(curr,
key=lambda e: e.name if hasattr(e, 'name') else 0)
nc_s.extend(curr)
return Mul(*c) * Mul(*nc_s)
return e
def mueller_matrix(J):
u"""The Mueller matrix corresponding to Jones matrix `J`.
Parameters
----------
``J`` : sympy Matrix
A Jones matrix.
Returns
-------
sympy Matrix
The corresponding Mueller matrix.
Examples
--------
Generic optical components.
>>> from sympy import pprint, symbols, pi, simplify
>>> from sympy.physics.optics.polarization import (mueller_matrix,
... linear_polarizer, half_wave_retarder, quarter_wave_retarder)
>>> theta = symbols("theta", real=True)
A linear_polarizer
>>> pprint(mueller_matrix(linear_polarizer(theta)), use_unicode=True)
⎡ cos(2⋅θ) sin(2⋅θ) ⎤
⎢ 1/2 ──────── ──────── 0⎥
⎢ 2 2 ⎥
⎢ ⎥
⎢cos(2⋅θ) cos(4⋅θ) 1 sin(4⋅θ) ⎥
⎢──────── ──────── + ─ ──────── 0⎥
⎢ 2 4 4 4 ⎥
⎢ ⎥
⎢sin(2⋅θ) sin(4⋅θ) 1 cos(4⋅θ) ⎥
⎢──────── ──────── ─ - ──────── 0⎥
⎢ 2 4 4 4 ⎥
⎢ ⎥
⎣ 0 0 0 0⎦
A half-wave plate
>>> pprint(mueller_matrix(half_wave_retarder(theta)), use_unicode=True)
⎡1 0 0 0 ⎤
⎢ ⎥
⎢ 4 2 ⎥
⎢0 8⋅sin (θ) - 8⋅sin (θ) + 1 sin(4⋅θ) 0 ⎥
⎢ ⎥
⎢ 4 2 ⎥
⎢0 sin(4⋅θ) - 8⋅sin (θ) + 8⋅sin (θ) - 1 0 ⎥
⎢ ⎥
⎣0 0 0 -1⎦
A quarter-wave plate
>>> pprint(mueller_matrix(quarter_wave_retarder(theta)), use_unicode=True)
⎡1 0 0 0 ⎤
⎢ ⎥
⎢ cos(4⋅θ) 1 sin(4⋅θ) ⎥
⎢0 ──────── + ─ ──────── -sin(2⋅θ)⎥
⎢ 2 2 2 ⎥
⎢ ⎥
⎢ sin(4⋅θ) 1 cos(4⋅θ) ⎥
⎢0 ──────── ─ - ──────── cos(2⋅θ) ⎥
⎢ 2 2 2 ⎥
⎢ ⎥
⎣0 sin(2⋅θ) -cos(2⋅θ) 0 ⎦
"""
A = Matrix([[1, 0, 0, 1], [1, 0, 0, -1], [0, 1, 1, 0], [0, -I, I, 0]])
return simplify(A * TensorProduct(J, J.conjugate()) * A.inv())
l = np.count_nonzero(v)
m = np.reshape(v,(3,3))
print (l)
print (m)
print(m.T)
print(inv(m))
print(np.dot(m,inv(m)))
print("=============")
print(m)
swap_col(m,0,2)
print(m)
print(swap_col_r(m,0,2))
print("===Tensor Product == ")
id = Matrix([[1,0],[0,1]])
i = Matrix([[1,0],[-1,0]])
id_i = TensorProduct(id,i)
print(id_i)
print(np.reshape(id_i,(2,8)))
print("===Fun - operation on matrix elemnets")
f_id_i = [ fun(x) for x in id_i]
print(f_id_i)
print("==Reed_Muller Transform==")
w= Matrix([[1,0],[1,1]])
w_1 = w.inv()
R3 = TensorProduct(w,TensorProduct(w,w))
print(R3)
pprint(R3)
f = Matrix([1,0,0,1,1,0,1,1])
pprint(f)
print("===")
s_f = [x%2 for x in R3*f]
The cnot performs the following transformation:
|00> -> |00>
|01> -> |01>
|10> -> |11>
|11> -> |10>
"""
theta = Symbol('theta')
X = np.array([[0, 1j], [-1j, 0]])
CNOT = Matrix(
np.array([[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0]]))
I = Matrix(np.eye(2))
sigmax = Matrix(1j * X * theta)
sigmax_inv = Matrix(-1j * X * theta)
R = simplify(exp(sigmax))
R_ = simplify(exp(sigmax_inv))
IX = simplify(TensorProduct(I, R))
IX_ = simplify(TensorProduct(I, R_))
U = IX_ * CNOT * IX
M_ij = Matrix([[U[0, 0], U[0, 2]], [U[2, 0], U[2, 2]]])
pprint(simplify(M_ij))
