def Graph_t(self, step):
"""
return a state which CZ gates have been applied #step on all plus state
"""
MPS_t = []
chi = 2 # this particular graph state has bond dimension of 2
U1 = np.reshape(
np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0],
[0, 0, 0, -1]]), (2, 2, 2, 2)) # ctlr z
plus = (1.0 / np.sqrt(2.0)) * np.ones(2)
# first qubit
temp = ncon((plus, plus, U1), ([1], [2], [-1, -2, 1, 2]))
X, Y, Z = np.linalg.svd(temp, full_matrices=0)
# truncation
chi2 = np.min([np.sum(Y > 10.**(-10)), chi])
piv = np.zeros(len(Y), np.bool)
piv[(np.argsort(Y)[::-1])[:chi2]] = True
Y = Y[piv]
invsq = np.sqrt(sum(Y**2))
X = X[:, piv]
Z = Z[piv, :]
MPS_t.append(np.matmul(X, np.diag(Y)))
for i in range(step - 1):
temp = ncon((Z, plus, U1), ([-1, 1], [2], [-2, -3, 1, 2]))
s0 = temp.shape[0]
s1 = temp.shape[1]
s2 = temp.shape[2]
temp = np.reshape(temp, (s0 * s1, s2))
X, Y, Z = np.linalg.svd(temp, full_matrices=0)
# truncation
chi2 = np.min([np.sum(Y > 10.**(-10)), chi])
piv = np.zeros(len(Y), np.bool)
piv[(np.argsort(Y)[::-1])[:chi2]] = True
Y = Y[piv]
invsq = np.sqrt(sum(Y**2))
X = X[:, piv]
Z = Z[piv, :]
MPS_t.append(np.reshape(np.matmul(X, np.diag(Y)), (s0, s1, chi2)))
MPS_t.append(
np.transpose(Z)
) # transpose is to keep the indexing order consistent with my poor choice
#MPS_t.append(np.expand_dim(np.transpose(Z), axis=1)) # transpose is to keep the indexing order consistent with my poor choice
for i in range(step + 1, self.N - 1):
MPS_t.append(np.reshape(plus, [1, 2, 1]))
#if step != self.N -1:
# MPS_t.append(np.reshape(plus, [2, 1]))
MPS_t.append(np.reshape(plus, [2, 1]))
return MPS_t
def construct_Nframes(self):
# Nqubit tensor product of frame Mn and dual frame Ntn
self.Ntn = self.Nt.copy()
self.Mn = self.M.copy()
for i in range(self.N - 1):
self.Ntn = ncon((self.Ntn, self.Nt), ([-1, -3, -5], [-2, -4, -6]))
self.Mn = ncon((self.Mn, self.M), ([-1, -3, -5], [-2, -4, -6]))
self.Ntn = np.reshape(self.Ntn,
(4**(i + 2), 2**(i + 2), 2**(i + 2)))
self.Mn = np.reshape(self.Mn, (4**(i + 2), 2**(i + 2), 2**(i + 2)))
def P_gate(self, gate, mask=True):
gate_factor = int(gate.ndim / 2)
if gate_factor == 1:
g = ncon((self.M, gate, self.Nt, np.transpose(np.conj(gate))),
([-1, 4, 1], [1, 2], [-2, 2, 5], [5, 4]))
else:
g = ncon((self.M, self.M, gate, self.Nt, self.Nt, np.conj(gate)),
([-1, 9, 1], [-2, 10, 2], [1, 2, 3, 4], [-3, 3, 7],
[-4, 4, 8], [9, 10, 7, 8]))
if mask:
g_mask = np.abs(g.real) > 1e-15
g = np.multiply(g, g_mask)
return g.real.astype('float32')
def get_initial_bias(self, initial_state, as_torch_tensor=False):
# which initial product state?
if initial_state == '0':
s = np.array([1, 0])
elif initial_state == '1':
s = np.array([0, 1])
elif initial_state == '+':
s = (1.0 / np.sqrt(2.0)) * np.array([1, 1])
elif initial_state == '-':
s = (1.0 / np.sqrt(2.0)) * np.array([1, -1])
elif initial_state == 'r':
s = (1.0 / np.sqrt(2.0)) * np.array([1, 1j])
elif initial_state == 'l':
s = (1.0 / np.sqrt(2.0)) * np.array([1, -1j])
self.P = np.real(ncon((self.M, s, np.conj(s)), ([-1, 1, 2], [1], [2])))
# solving for bias
self.bias = np.zeros(self.K)
self.bias = np.log(self.P)
if np.sum(np.abs(self.softmax(self.bias) - self.P)) > 0.00000000001:
print("initial bias not found")
elif as_torch_tensor:
return torch.tensor(self.bias)
else:
return self.bias
def N_body_gate(self, gate, mask=True):
gn = ncon((self.Mn, gate, self.Ntn, np.transpose(np.conj(gate))),
([-1, 4, 1], [1, 2], [-2, 2, 5], [5, 4]))
if mask:
gn_mask = np.abs(gn.real) > 1e-15
gn = np.multiply(gn, gn_mask)
return gn.real.astype('float32')
def one_body_gate(self, gate, mask=True):
g1 = ncon((self.M, gate, self.Nt, np.transpose(np.conj(gate))),
([-1, 4, 1], [1, 2], [-2, 2, 5], [5, 4]))
if mask:
g1_mask = np.abs(g1.real) > 1e-15
g1 = np.multiply(g1, g1_mask)
return g1.real.astype('float32')
def two_body_gate_all(self, gate, mask=True):
g2 = ncon((self.M, self.M, gate, self.Nt, self.Nt, np.conj(gate)),
([-1, 9, 1], [-2, 10, 2], [1, 2, 3, 4], [-3, 3, 7],
[-4, 4, 8], [9, 10, 7, 8]))
#if mask:
# g2_mask = np.abs(g2.real) > 1e-15
# g2 = np.multiply(g2, g2_mask)
return g2
def cFidelity_t(self, S, logP, step):
"""
compute the classical fidelity with linear Graph state after apply CZ gates with #step
"""
Fidelity = 0.0
F2 = 0.0
Ns = S.shape[0]
L1 = 0.0
L1_2 = 0.0
MPS_t = self.Graph_t(step)
#for i in range(len(MPS_t)):
# print(MPS_t[i].shape)
plus = (1.0 / np.sqrt(2.0)) * np.ones(2)
plus_pho = np.outer(plus, np.conjugate(plus))
prob_plus = ncon((plus_pho, self.M), ([1, 2], [-1, 2, 1]))
for i in range(Ns):
P = ncon((MPS_t[0], MPS_t[0], self.M[S[i, 0], :, :]),
([1, -1], [2, -2], [1, 2]))
# contracting the entire TN for each sample S[i,:]
for j in range(1, step):
P = ncon((P, MPS_t[j], MPS_t[j], self.M[S[i, j], :, :]),
([1, 2], [1, 3, -1], [2, 4, -2], [3, 4]))
P = ncon((P, MPS_t[step], MPS_t[step], self.M[S[i, step], :, :]),
([1, 2], [3, 1], [4, 2], [3, 4]))
for j in range(step + 1, self.N):
P *= prob_plus[S[i, j]]
ratio = P / np.exp(logP[i])
ee = np.sqrt(ratio)
Fidelity = Fidelity + ee
F2 = F2 + ee**2
L1 = L1 + np.abs(1 - ratio)
L1_2 = L1_2 + np.abs(1 - ratio)**2
F2 = F2 / float(Ns)
Fidelity = np.abs(Fidelity / float(Ns))
Error = np.sqrt(np.abs(F2 - Fidelity**2) / float(Ns))
L1_2 = L1_2 / float(Ns)
L1 = np.abs(L1 / float(Ns))
L1_err = np.sqrt(np.abs(L1_2 - L1**2) / float(Ns))
return np.real(Fidelity), Error, L1, L1_err
def cFidelity(self, S, logP):
Fidelity = 0.0
F2 = 0.0
Ns = S.shape[0]
KL = 0.0
K2 = 0.0
L1 = 0.0
L1_2 = 0.0
for i in range(Ns):
P = ncon((self.MPS[0], self.MPS[0], self.M[S[i, 0], :, :]),
([1, -1], [2, -2], [1, 2]))
# contracting the entire TN for each sample S[i,:]
for j in range(1, self.N - 1):
P = ncon((P, self.MPS[j], self.MPS[j], self.M[S[i, j], :, :]),
([1, 2], [1, 3, -1], [2, 4, -2], [3, 4]))
P = ncon((P, self.MPS[self.N - 1], self.MPS[self.N - 1],
self.M[S[i, self.N - 1], :, :]),
([1, 2], [3, 1], [4, 2], [3, 4]))
ratio = P / np.exp(logP[i])
ee = np.sqrt(ratio)
Fidelity = Fidelity + ee
F2 = F2 + ee**2
L1 = L1 + np.abs(1 - ratio)
L1_2 = L1_2 + np.abs(1 - ratio)**2
#KL = KL + 2 * np.log(ee + 1e-9)
#K2 = K2 + 4 * (np.log(ee + 1e-9)) ** 2
F2 = F2 / float(Ns)
Fidelity = np.abs(Fidelity / float(Ns))
Error = np.sqrt(np.abs(F2 - Fidelity**2) / float(Ns))
L1_2 = L1_2 / float(Ns)
L1 = np.abs(L1 / float(Ns))
L1_err = np.sqrt(np.abs(L1_2 - L1**2) / float(Ns))
K2 = K2 / float(Ns)
KL = np.abs(KL / float(Ns))
ErrorKL = np.sqrt(np.abs(K2 - KL**2) / float(Ns))
return np.real(Fidelity), Error, L1, L1_err
def Fidelity(self, S):
Fidelity = 0.0
F2 = 0.0
Ns = S.shape[0]
for i in range(Ns):
# contracting the entire TN for each sample S[i,:]
# eT = ncon(( self.TB[0][:,:,S[i,0]], self.TB[1][:,:,:,:,S[i,1]]) ,( [1,2],[-1,-2,1,2 ]))
eT = ncon((self.it[:, S[i, 0]], self.M, self.MPS[0], self.MPS[0]),
([3], [3, 2, 1], [1, -1], [2, -2]))
for j in range(1, self.N - 1):
# eT = ncon((eT,self.TB[j][:,:,:,:,S[i,j]]),([ 1,2],[ -1,-2, 1,2 ]))
eT = ncon(
(eT, self.it[:, S[i,
j]], self.M, self.MPS[j], self.MPS[j]),
([2, 4], [1], [1, 5, 3], [2, 3, -1], [4, 5, -2]))
# eT = ncon((eT, self.TB[self.N-1][:,:,S[i,self.N-1]]),([1,2],[1,2 ]))
j = self.N - 1
eT = ncon(
(eT, self.it[:, S[i, j]], self.M, self.MPS[j], self.MPS[j]),
([2, 5], [1], [1, 4, 3], [3, 2], [4, 5]))
# print i, eT
Fidelity = Fidelity + eT
F2 = F2 + eT**2
Fest = Fidelity / float(i + 1)
F2est = F2 / float(i + 1)
Error = np.sqrt(np.abs(F2est - Fest**2) / float(i + 1))
# print i,np.real(Fest),Error
# disp([i,i/Ns, real(Fest), real(Error)])
# fflush(stdout)
F2 = F2 / float(Ns)
Fidelity = np.abs(Fidelity / float(Ns))
Error = np.sqrt(np.abs(F2 - Fidelity**2) / float(Ns))
return np.real(Fidelity), Error
def getinitialbias(self, initial_state):
self.P = np.real(
ncon((self.M, self.s, np.conj(self.s)), ([-1, 1, 2], [1], [2])))
# solving for bias
self.bias = np.zeros(self.K)
self.bias = np.log(self.P)
if np.sum(np.abs(self.softmax(self.bias) - self.P)) > 0.00000000001:
print("initial bias not found")
else:
return self.bias
def __init__(self, POVM='Trine', Number_qubits=4, MPS='GHZ'):
self.N = Number_qubits
# Hamiltonian for calculation of energy (TFIM in 1d)
# self.Jz = Jz
# self.hx = hx
# POVMs and other operators
# Pauli matrices
self.I = np.array([[1, 0], [0, 1]])
self.X = np.array([[0, 1], [1, 0]])
self.s1 = self.X
self.Z = np.array([[1, 0], [0, -1]])
self.s3 = self.Z
self.Y = np.array([[0, -1j], [1j, 0]])
self.s2 = self.Y
self.H = 1.0 / np.sqrt(2.0) * np.array([[1, 1], [1, -1]])
self.Sp = np.array([[1.0, 0.0], [0.0, -1j]])
self.oxo = np.array([[1.0, 0.0], [0.0, 0.0]])
self.IxI = np.array([[0.0, 0.0], [0.0, 1.0]])
self.Phase = np.array([[1.0, 0.0], [0.0, 1j]]) # =S = (Sp)^{\dag}
self.T = np.array([[1.0, 0], [0, np.exp(-1j * np.pi / 4.0)]])
self.U1 = np.array([[np.exp(-1j * np.pi / 3.0), 0],
[0, np.exp(1j * np.pi / 3.0)]])
# two-qubit gates
self.cy = ncon((self.oxo, self.I), ([-1, -3], [-2, -4])) + ncon(
(self.IxI, self.Y), ([-1, -3], [-2, -4]))
self.cz = ncon((self.oxo, self.I), ([-1, -3], [-2, -4])) + ncon(
(self.IxI, self.Z), ([-1, -3], [-2, -4]))
# Tetra POVM
if POVM == '4Pauli':
self.K = 4
self.M = np.zeros((self.K, 2, 2), dtype=complex)
self.M[0, :, :] = 1.0 / 3.0 * np.array([[1, 0], [0, 0]])
self.M[1, :, :] = 1.0 / 6.0 * np.array([[1, 1], [1, 1]])
self.M[2, :, :] = 1.0 / 6.0 * np.array([[1, -1j], [1j, 1]])
self.M[3, :, :] = 1.0 / 3.0 * (np.array([[0, 0], [0, 1]]) +
0.5 * np.array([[1, -1], [-1, 1]]) +
0.5 * np.array([[1, 1j], [-1j, 1]]))
if POVM == 'Tetra':
self.K = 4
self.M = np.zeros((self.K, 2, 2), dtype=complex)
self.v1 = np.array([0, 0, 1.0])
self.M[0, :, :] = 1.0 / 4.0 * (self.I + self.v1[0] * self.s1 +
self.v1[1] * self.s2 +
self.v1[2] * self.s3)
self.v2 = np.array([2.0 * np.sqrt(2.0) / 3.0, 0.0, -1.0 / 3.0])
self.M[1, :, :] = 1.0 / 4.0 * (self.I + self.v2[0] * self.s1 +
self.v2[1] * self.s2 +
self.v2[2] * self.s3)
self.v3 = np.array(
[-np.sqrt(2.0) / 3.0,
np.sqrt(2.0 / 3.0), -1.0 / 3.0])
self.M[2, :, :] = 1.0 / 4.0 * (self.I + self.v3[0] * self.s1 +
self.v3[1] * self.s2 +
self.v3[2] * self.s3)
self.v4 = np.array(
[-np.sqrt(2.0) / 3.0, -np.sqrt(2.0 / 3.0), -1.0 / 3.0])
self.M[3, :, :] = 1.0 / 4.0 * (self.I + self.v4[0] * self.s1 +
self.v4[1] * self.s2 +
self.v4[2] * self.s3)
if POVM == 'Tetra_pos':
self.K = 4
self.M = np.zeros((self.K, 2, 2), dtype=complex)
self.v1 = np.array([1.0, 1.0, 1.0]) / np.sqrt(3)
self.M[0, :, :] = 1.0 / 4.0 * (self.I + self.v1[0] * self.s1 +
self.v1[1] * self.s2 +
self.v1[2] * self.s3)
self.v2 = np.array([1.0, -1.0, -1.0]) / np.sqrt(3)
self.M[1, :, :] = 1.0 / 4.0 * (self.I + self.v2[0] * self.s1 +
self.v2[1] * self.s2 +
self.v2[2] * self.s3)
self.v3 = np.array([-1.0, 1.0, -1.0]) / np.sqrt(3)
self.M[2, :, :] = 1.0 / 4.0 * (self.I + self.v3[0] * self.s1 +
self.v3[1] * self.s2 +
self.v3[2] * self.s3)
self.v4 = np.array([-1.0, -1.0, 1.0]) / np.sqrt(3)
self.M[3, :, :] = 1.0 / 4.0 * (self.I + self.v4[0] * self.s1 +
self.v4[1] * self.s2 +
self.v4[2] * self.s3)
elif POVM == 'Trine':
self.K = 3
self.M = np.zeros((self.K, 2, 2), dtype=complex)
phi0 = 0.0
for k in range(self.K):
phi = phi0 + k * 2 * np.pi / 3.0
self.M[k, :, :] = 0.5 * (self.I + np.cos(phi) * self.Z +
np.sin(phi) * self.X) * 2 / 3.0
# % T matrix and its inverse
self.t = ncon((self.M, self.M), ([-1, 1, 2], [-2, 2, 1]))
self.it = np.linalg.inv(self.t)
# Tensor for expectation value
# self.Trsx = np.zeros((self.N,self.K),dtype=complex)
# self.Trsy = np.zeros((self.N,self.K),dtype=complex)
# self.Trsz = np.zeros((self.N,self.K),dtype=complex)
# self.Trrho = np.zeros((self.N,self.K),dtype=complex)
# self.Trrho2 = np.zeros((self.N,self.K,self.K),dtype=complex)
# self.T2 = np.zeros((self.N,self.K,self.K),dtype=complex)
self.Trsx = np.zeros(self.K, dtype=complex)
self.Trsy = np.zeros(self.K, dtype=complex)
self.Trsz = np.zeros(self.K, dtype=complex)
self.Trrho = np.zeros((self.N, self.K), dtype=complex)
self.Trrho2 = np.zeros((self.N, self.K, self.K), dtype=complex)
self.T2 = np.zeros((self.N, self.K, self.K), dtype=complex)
self.Trsx = ncon((self.M, self.it, self.X),
([3, 2, 1], [3, -1], [2, 1]))
self.Trsy = ncon((self.M, self.it, self.Y),
([3, 2, 1], [3, -1], [2, 1]))
self.Trsz = ncon((self.M, self.it, self.Z),
([3, 2, 1], [3, -1], [2, 1]))
self.stab_ops = [self.Trsx, self.Trsz]
if MPS == "GHZ":
# Copy tensors used to construct GHZ as an MPS. The procedure below should work for any other MPS
cc = np.zeros((2, 2)) # corner
cc[0, 0] = 2**(-1.0 / (2 * self.N))
cc[1, 1] = 2**(-1.0 / (2 * self.N))
cb = np.zeros((2, 2, 2)) # bulk
cb[0, 0, 0] = 2**(-1.0 / (2 * self.N))
cb[1, 1, 1] = 2**(-1.0 / (2 * self.N))
self.MPS = []
self.MPS.append(cc)
for i in range(self.N - 2):
self.MPS.append(cb)
self.MPS.append(cc)
elif MPS == "Graph":
MPS = []
chi = 2 # this particular graph state has bond dimension of 2
U1 = np.reshape(
np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0],
[0, 0, 0, -1]]), (2, 2, 2, 2)) # ctlr z
plus = (1.0 / np.sqrt(2.0)) * np.ones(2)
# first qubit
temp = ncon((plus, plus, U1), ([1], [2], [-1, -2, 1, 2]))
X, Y, Z = np.linalg.svd(temp, full_matrices=0)
# truncation
chi2 = np.min([np.sum(Y > 10.**(-10)), chi])
piv = np.zeros(len(Y), np.bool)
piv[(np.argsort(Y)[::-1])[:chi2]] = True
Y = Y[piv]
invsq = np.sqrt(sum(Y**2))
X = X[:, piv]
Z = Z[piv, :]
MPS.append(np.matmul(X, np.diag(Y)))
for i in range(1, self.N - 1):
temp = ncon((Z, plus, U1), ([-1, 1], [2], [-2, -3, 1, 2]))
s0 = temp.shape[0]
s1 = temp.shape[1]
s2 = temp.shape[2]
temp = np.reshape(temp, (s0 * s1, s2))
X, Y, Z = np.linalg.svd(temp, full_matrices=0)
# truncation
chi2 = np.min([np.sum(Y > 10.**(-10)), chi])
piv = np.zeros(len(Y), np.bool)
piv[(np.argsort(Y)[::-1])[:chi2]] = True
Y = Y[piv]
invsq = np.sqrt(sum(Y**2))
X = X[:, piv]
Z = Z[piv, :]
MPS.append(np.reshape(np.matmul(X, np.diag(Y)),
(s0, s1, chi2)))
# last qubit
MPS.append(
np.transpose(Z)
) # transpose is to keep the indexing order consistent with my poor choice
# testing
# GS = ncon((MPS[0],MPS[1],MPS[2],MPS[3]),([-1,1],[1,-2,2],[2,-3,3],[3,-4]))
# GS = np.reshape(GS,(2**4))
self.MPS = MPS
elif MPS == "plus":
print(MPS, "squi")
plus = (1.0 / np.sqrt(2.0)) * np.ones(2)
self.MPS = []
self.MPS.append(np.reshape(plus, [2, 1]))
for i in range(1, self.N - 1):
self.MPS.append(np.reshape(plus, [1, 2, 1]))
self.MPS.append(np.reshape(plus, [2, 1]))
def stabilizers(self, i, si, j, sj, k, sk):
MPS = deepcopy(self.MPS)
if i == 0:
MPS[i] = ncon((MPS[i], si), ([1, -2], [-1, 1]))
elif i == self.N - 1:
MPS[i] = ncon((MPS[i], si), ([-2, 1], [-1, 1]))
else:
MPS[i] = ncon((MPS[i], si), ([-1, 1, -3], [-2, 1]))
if j == 0:
MPS[j] = ncon((MPS[j], sj), ([1, -2], [-1, 1]))
elif j == self.N - 1:
MPS[j] = ncon((MPS[j], sj), ([-2, 1], [-1, 1]))
else:
MPS[j] = ncon((MPS[j], sj), ([-1, 1, -3], [-2, 1]))
if k == 0:
MPS[k] = ncon((MPS[k], sk), ([1, -2], [-1, 1]))
elif k == self.N - 1:
MPS[k] = ncon((MPS[k], sk), ([-2, 1], [-1, 1]))
else:
MPS[k] = ncon((MPS[k], sk), ([-1, 1, -3], [-2, 1]))
C = ncon((MPS[0], np.conj(self.MPS[0])), ([1, -1], [1, -2]))
for ii in range(1, self.N - 1):
C = ncon((C, MPS[ii], np.conj(self.MPS[ii])),
([1, 2], [1, 3, -1], [2, 3, -2]))
ii = self.N - 1
C = ncon((C, MPS[ii], np.conj(self.MPS[ii])), ([1, 2], [3, 1], [3, 2]))
return C
def __init__(self,
POVM='4Pauli',
Number_qubits=4,
initial_state='+',
Jz=1.0,
hx=1.0,
eps=1e-4):
self.type = POVM
self.N = Number_qubits
# Hamiltonian for calculation of energy (TFIM in 1d)
self.Jz = Jz
self.hx = hx
self.eps = eps
# POVMs and other operators
# Pauli matrices,gates,simple states
self.I = np.array([[1, 0], [0, 1]])
self.X = np.array([[0, 1], [1, 0]])
self.s1 = self.X
self.Z = np.array([[1, 0], [0, -1]])
self.s3 = self.Z
self.Y = np.array([[0, -1j], [1j, 0]])
self.s2 = self.Y
self.H = 1.0 / np.sqrt(2.0) * np.array([[1, 1], [1, -1]])
self.Sp = np.array([[1.0, 0.0], [0.0, -1j]])
self.oxo = np.array([[1.0, 0.0], [0.0, 0.0]])
self.IxI = np.array([[0.0, 0.0], [0.0, 1.0]])
self.Phase = np.array([[1.0, 0.0], [0.0, 1j]]) # =S = (Sp)^{\dag}
aa = 8.0
self.R8 = np.array([[np.cos(np.pi / aa), -np.sin(np.pi / aa)],
[np.sin(np.pi / aa),
np.cos(np.pi / aa)]])
self.T = np.array([[1.0, 0], [0, np.exp(-1j * np.pi / 4.0)]])
self.U1 = np.array([[np.exp(-1j * np.pi / 3.0), 0],
[0, np.exp(1j * np.pi / 3.0)]])
#two-qubit gates
self.cx = ncon((self.oxo, self.I), ([-1, -3], [-2, -4])) + ncon(
(self.IxI, self.X), ([-1, -3], [-2, -4]))
self.cy = ncon((self.oxo, self.I), ([-1, -3], [-2, -4])) + ncon(
(self.IxI, self.Y), ([-1, -3], [-2, -4]))
self.cz = ncon((self.oxo, self.I), ([-1, -3], [-2, -4])) + ncon(
(self.IxI, self.Z), ([-1, -3], [-2, -4]))
self.cnot = ncon((self.oxo, self.I), ([-1, -3], [-2, -4])) + ncon(
(self.IxI, self.X), ([-1, -3], [-2, -4]))
self.cu1 = ncon((self.oxo, self.I), ([-1, -3], [-2, -4])) + ncon(
(self.IxI, self.U1), ([-1, -3], [-2, -4]))
self.HI = ncon((self.H, self.I), ([-1, -3], [-2, -4]))
self.PhaseI = ncon((self.Phase, self.I), ([-1, -3], [-2, -4]))
self.R8I = ncon((self.R8, self.I), ([-1, -3], [-2, -4]))
self.TI = ncon((self.T, self.I), ([-1, -3], [-2, -4]))
self.single_qubit = [self.H, self.Phase, self.T, self.U1]
self.two_qubit = [
self.cnot, self.cz, self.cy, self.cu1, self.HI, self.PhaseI,
self.R8I, self.TI, self.cx
]
if POVM == '4Pauli':
self.K = 4
self.M = np.zeros((self.K, 2, 2), dtype=complex)
self.M[0, :, :] = 1.0 / 3.0 * np.array([[1, 0], [0, 0]])
self.M[1, :, :] = 1.0 / 6.0 * np.array([[1, 1], [1, 1]])
self.M[2, :, :] = 1.0 / 6.0 * np.array([[1, -1j], [1j, 1]])
self.M[3,:,:] = 1.0/3.0*(np.array([[0, 0],[0, 1]]) + \
0.5*np.array([[1, -1],[-1, 1]]) \
+ 0.5*np.array([[1, 1j],[-1j, 1]]) )
if POVM == 'Tetra': ## symmetric
self.K = 4
self.M = np.zeros((self.K, 2, 2), dtype=complex)
self.v1 = np.array([0, 0, 1.0])
self.M[0, :, :] = 1.0 / 4.0 * (self.I + self.v1[0] * self.s1 +
self.v1[1] * self.s2 +
self.v1[2] * self.s3)
self.v2 = np.array([2.0 * np.sqrt(2.0) / 3.0, 0.0, -1.0 / 3.0])
self.M[1, :, :] = 1.0 / 4.0 * (self.I + self.v2[0] * self.s1 +
self.v2[1] * self.s2 +
self.v2[2] * self.s3)
self.v3 = np.array(
[-np.sqrt(2.0) / 3.0,
np.sqrt(2.0 / 3.0), -1.0 / 3.0])
self.M[2, :, :] = 1.0 / 4.0 * (self.I + self.v3[0] * self.s1 +
self.v3[1] * self.s2 +
self.v3[2] * self.s3)
self.v4 = np.array(
[-np.sqrt(2.0) / 3.0, -np.sqrt(2.0 / 3.0), -1.0 / 3.0])
self.M[3, :, :] = 1.0 / 4.0 * (self.I + self.v4[0] * self.s1 +
self.v4[1] * self.s2 +
self.v4[2] * self.s3)
if POVM == 'Tetra_pos':
self.K = 4
self.M = np.zeros((self.K, 2, 2), dtype=complex)
self.v1 = np.array([1.0, 1.0, 1.0]) / np.sqrt(3)
self.M[0, :, :] = 1.0 / 4.0 * (self.I + self.v1[0] * self.s1 +
self.v1[1] * self.s2 +
self.v1[2] * self.s3)
self.v2 = np.array([1.0, -1.0, -1.0]) / np.sqrt(3)
self.M[1, :, :] = 1.0 / 4.0 * (self.I + self.v2[0] * self.s1 +
self.v2[1] * self.s2 +
self.v2[2] * self.s3)
self.v3 = np.array([-1.0, 1.0, -1.0]) / np.sqrt(3)
self.M[2, :, :] = 1.0 / 4.0 * (self.I + self.v3[0] * self.s1 +
self.v3[1] * self.s2 +
self.v3[2] * self.s3)
self.v4 = np.array([-1.0, -1.0, 1.0]) / np.sqrt(3)
self.M[3, :, :] = 1.0 / 4.0 * (self.I + self.v4[0] * self.s1 +
self.v4[1] * self.s2 +
self.v4[2] * self.s3)
elif POVM == 'Trine':
self.K = 3
self.M = np.zeros((self.K, 2, 2), dtype=complex)
phi0 = 0.0
for k in range(self.K):
phi = phi0 + (k) * 2 * np.pi / 3.0
self.M[k, :, :] = 0.5 * (self.I + np.cos(phi) * self.Z +
np.sin(phi) * self.X) * 2 / 3.0
#% T matrix and its inverse
self.t = ncon((self.M, self.M), ([-1, 1, 2], [-2, 2, 1])).real
self.it = np.linalg.inv(self.t)
# dual frame of M
self.Nt = ncon((self.it, self.M), ([-1, 1], [1, -2, -3]))
self.Nt_norm = []
for i in range(self.Nt.shape[0]):
self.Nt_norm.append(np.linalg.norm(self.Nt[i]))
self.Nt_norm = np.array(self.Nt_norm)
# Tensor for expectation value
self.Trsx = np.zeros((self.N, self.K), dtype=complex)
self.Trsy = np.zeros((self.N, self.K), dtype=complex)
self.Trsz = np.zeros((self.N, self.K), dtype=complex)
self.Trrho = np.zeros((self.N, self.K), dtype=complex)
self.Trrho2 = np.zeros((self.N, self.K, self.K), dtype=complex)
self.T2 = np.zeros((self.N, self.K, self.K), dtype=complex)
# probability gate set single qubit
self.p_single_qubit = []
for i in range(len(self.single_qubit)):
#mat = ncon((self.M,self.single_qubit[i],self.M,self.it,np.transpose(np.conj(self.single_qubit[i]))),([-1,4,1],[1,2],[3,2,5],[3,-2],[5,4]))
mat = self.one_body_gate(self.single_qubit[i])
self.p_single_qubit.append(mat)
# probability gate set two qubit
self.p_two_qubit = []
for i in range(len(self.two_qubit)):
#mat = ncon((self.M,self.M,self.two_qubit[i],self.M,self.M,self.it,self.it,np.conj(self.two_qubit[i])),([-1,9,1],[-2,10,2],[1,2,3,4],[5,3,7],[6,4,8],[5,-3],[6,-4],[9,10,7,8]))
mat = self.two_body_gate(self.two_qubit[i])
self.p_two_qubit.append(mat)
#print(np.real(np.sum(np.reshape(self.p_two_qubit[i],(16,16)),1)),np.real(np.sum(np.reshape(self.p_two_qubit[i],(16,16)),0)))
# set initial wavefunction
if initial_state == '0':
self.s = np.array([1, 0])
elif initial_state == '1':
self.s = np.array([0, 1])
elif initial_state == '+':
self.s = (1.0 / np.sqrt(2.0)) * np.array([1, 1])
elif initial_state == '-':
self.s = (1.0 / np.sqrt(2.0)) * np.array([1, -1])
elif initial_state == 'r':
self.s = (1.0 / np.sqrt(2.0)) * np.array([1, 1j])
elif initial_state == 'l':
self.s = (1.0 / np.sqrt(2.0)) * np.array([1, -1j])
# time evolution gate
"""
hl = ZZ + XI
hlx = ZZ + XI + IX
"""
self.ZZ = np.kron(self.Z, self.Z)
self.XI = np.kron(self.X, self.I)
self.XI2 = np.kron(self.X, self.I) + np.kron(self.I, self.X)
self.hl = self.Jz * np.kron(self.Z, self.Z) + self.hx * np.kron(
self.X, self.I)
self.hl = -np.reshape(self.hl, (2, 2, 2, 2))
self.hlx = self.Jz * np.kron(self.Z, self.Z) + self.hx * (
np.kron(self.X, self.I) + np.kron(self.I, self.X))
self.hlx = -np.reshape(self.hlx, (2, 2, 2, 2))
self.zz2 = np.reshape(self.ZZ, (2, 2, 2, 2))
#self.sx = np.reshape(np.kron(self.X,self.I)+ np.kron(self.I,self.X),(2,2,2,2))
self.sx = np.reshape(np.kron(self.X, self.I), (2, 2, 2, 2))
self.exp_hl = np.reshape(-self.eps * self.hl, (4, 4))
self.exp_hl = expm(self.exp_hl)
self.exp_hl_norm = np.linalg.norm(self.exp_hl)
self.exp_hl2 = self.exp_hl / self.exp_hl_norm
self.mat = np.reshape(self.exp_hl, (2, 2, 2, 2))
self.mat2 = np.reshape(self.exp_hl2, (2, 2, 2, 2))
self.Up = self.two_body_gate(self.mat)
self.Up2 = self.two_body_gate(self.mat2)
#self.hlp = ncon((self.it,self.M,self.hl,self.it,self.M),([1,-1],[1,3,2],[2,5,3,6],[4,-2],[4,6,5])).real
#self.sxp = ncon((self.it,self.M,self.sx,self.it,self.M),([1,-1],[1,3,2],[2,5,3,6],[4,-2],[4,6,5])).real
# Hamiltonian observable list
self.hl_ob = ncon(
(self.hl, self.Nt, self.Nt),
([1, 2, 3, 4], [-1, 3, 1], [-2, 4, 2])).real.astype(np.float32)
self.hlx_ob = ncon(
(self.hlx, self.Nt, self.Nt),
([1, 2, 3, 4], [-1, 3, 1], [-2, 4, 2])).real.astype(np.float32)
self.x_ob = ncon((-self.hx * self.X, self.Nt),
([1, 2], [-1, 2, 1])).real.astype(np.float32)
self.zz_ob = ncon(
(self.zz2, self.Nt, self.Nt),
([1, 2, 3, 4], [-1, 3, 1], [-2, 4, 2])).real.astype(np.float32)
# commuting and anti_computing operator
hl_Nt = ncon((self.hl, self.Nt, self.Nt),
([-1, -2, 1, 2], [-3, 1, -5], [-4, 2, -6]))
Nt_hl = ncon((self.Nt, self.Nt, self.hl),
([-3, -1, 1], [-4, -2, 2], [1, 2, -5, -6]))
hlx_Nt = ncon((self.hlx, self.Nt, self.Nt),
([-1, -2, 1, 2], [-3, 1, -5], [-4, 2, -6]))
Nt_hlx = ncon((self.Nt, self.Nt, self.hlx),
([-3, -1, 1], [-4, -2, 2], [1, 2, -5, -6]))
x_Nt = ncon((-self.hx * self.X, self.Nt), ([-1, 1], [-2, 1, -3]))
Nt_x = ncon((self.Nt, -self.hx * self.X), ([-2, -1, 1], [1, -3]))
self.hl_anti = ncon(
(hl_Nt + Nt_hl, self.M, self.M),
([1, 2, -3, -4, 3, 4], [-1, 3, 1], [-2, 4, 2])).real.astype(
np.float32)
self.hl_com = ncon(
(-hl_Nt + Nt_hl, self.M, self.M),
([1, 2, -3, -4, 3, 4], [-1, 3, 1], [-2, 4, 2])).imag.astype(
np.float32)
self.hlx_anti = ncon(
(hlx_Nt + Nt_hlx, self.M, self.M),
([1, 2, -3, -4, 3, 4], [-1, 3, 1], [-2, 4, 2])).real.astype(
np.float32)
self.hlx_com = ncon(
(-hlx_Nt + Nt_hlx, self.M, self.M),
([1, 2, -3, -4, 3, 4], [-1, 3, 1], [-2, 4, 2])).imag.astype(
np.float32)
self.x_anti = ncon((x_Nt + Nt_x, self.M),
([1, -2, 2], [-1, 2, 1])).real.astype(np.float32)
self.x_com = ncon((-x_Nt + Nt_x, self.M),
([1, -2, 2], [-1, 2, 1])).imag.astype(np.float32)
# MPO H
self.Ham = []
mat = np.zeros((3, 3, 2, 2))
mat[0, 0] = self.I
mat[1, 0] = -self.Z
mat[2, 0] = -self.X * self.hx
mat[2, 1] = self.Z
mat[2, 2] = self.I
self.Ham.append(mat[2])
for i in range(1, self.N - 1):
self.Ham.append(mat)
self.Ham.append(mat[:, 0, :, :])
# MPS for Hamiltonian in probability space
self.Hp = []
mat = ncon((self.Ham[0], self.M, self.it),
([-2, 3, 1], [2, 1, 3], [2, -1]))
self.Hp.append(mat)
for i in range(1, self.N - 1):
mat = ncon((self.Ham[i], self.M, self.it),
([-1, -3, 3, 1], [2, 1, 3], [2, -2]))
self.Hp.append(mat)
mat = ncon((self.Ham[self.N - 1], self.M, self.it),
([-1, 3, 1], [2, 1, 3], [2, -2]))
self.Hp.append(mat)