def cyperm_op(nbits):
"""
Constructing a single shift cyclic permuation operator
"""
if nbits == 1:
return qo.Qoperator(1, np.asmatrix([[0, 1], [1, 0]]))
else:
S = qo.Qoperator(1, np.asmatrix([[0, 1], [1, 0]]))
N = np.power(2, nbits)
M1 = np.asmatrix(np.zeros((N / 2, N / 2)))
M1[0, 0] = 1
M2 = np.asmatrix(np.eye(N / 2) - M1)
control0 = qo.Qoperator(nbits - 1, M1)
control1 = qo.Qoperator(nbits - 1, M2)
temp1 = qo.directopprod(S, control0)
temp2 = qo.directopprod(identityop(1), control1)
U2 = qo.Qoperator(nbits, temp1.matrix + temp2.matrix)
U1 = qo.directopprod(identityop(1), cyperm_op(nbits - 1))
U = qo.Qoperator(nbits, U2.matrix * U1.matrix)
return U
def expH_2order_2D(nxbits, nybits, dt, hbar, c, expV):
"""
Simulate exp(-i*dt*H/hbar)
error = O(dt^2)
"""
nbits = nxbits + nybits
M = c * np.asmatrix([[1, -1], [-1, 1]])
expM = qo.Qoperator(1, qo.matpowerh(np.power(np.e, -1j), (dt / hbar) * M))
expTeven = qo.directopprod(identityop(nbits - 1), expM)
P = cyperm_op(nxbits)
P1 = qo.directopprod(identityop(nybits), P)
P1inv = qo.Qoperator(nbits, P1.matrix.H)
expTodd = qo.Qoperator(nbits, P1inv.matrix * expTeven.matrix * P1.matrix)
#TODO Find a linear algebra realization of the switching opeartor
S = switchop(nxbits, nybits)
expTup = qo.Qoperator(nbits, S.matrix.H * expTeven.matrix * S.matrix)
expTdown = qo.Qoperator(nbits, S.matrix.H * expTodd.matrix * S.matrix)
U1 = expTeven.matrix * expTodd.matrix * expTup.matrix * expTdown.matrix
U = U1 * expV
return qo.Qoperator(nbits, U)
def cyperm_op(nbits):
"""
Constructing a single shift cyclic permuation operator
"""
if nbits == 1:
return qo.Qoperator(1,np.asmatrix([[0,1],[1,0]]))
else :
S = qo.Qoperator(1,np.asmatrix([[0,1],[1,0]]))
N = np.power(2,nbits)
M1 = np.asmatrix(np.zeros((N/2,N/2)))
M1[0,0] = 1
M2 = np.asmatrix(np.eye(N/2) - M1)
control0 = qo.Qoperator(nbits-1,M1)
control1 = qo.Qoperator(nbits-1,M2)
temp1 = qo.directopprod(S,control0)
temp2 = qo.directopprod(identityop(1),control1)
U2 = qo.Qoperator(nbits, temp1.matrix+temp2.matrix)
U1 = qo.directopprod(identityop(1), cyperm_op(nbits-1))
U = qo.Qoperator(nbits,U2.matrix*U1.matrix)
return U
def expH_2order_2D(nxbits,nybits,dt,hbar,c,expV): """ Simulate exp(-i*dt*H/hbar) error = O(dt^2) """ nbits = nxbits + nybits M = c*np.asmatrix([[1,-1],[-1,1]]) expM = qo.Qoperator(1,qo.matpowerh(np.power(np.e,-1j), (dt/hbar)*M)) expTeven = qo.directopprod(identityop(nbits-1),expM) P = cyperm_op(nxbits) P1 = qo.directopprod(identityop(nybits),P) P1inv = qo.Qoperator(nbits,P1.matrix.H) expTodd = qo.Qoperator(nbits,P1inv.matrix * expTeven.matrix * P1.matrix) #TODO Find a linear algebra realization of the switching opeartor S = switchop(nxbits,nybits) expTup = qo.Qoperator(nbits,S.matrix.H * expTeven.matrix * S.matrix) expTdown = qo.Qoperator(nbits,S.matrix.H * expTodd.matrix * S.matrix) U1 = expTeven.matrix * expTodd.matrix * expTup.matrix * expTdown.matrix U = U1 * expV return qo.Qoperator(nbits, U)
def expT_2order_1D(nbits, dt, hbar, c):
"""
Simulate exp(-i*dt*T/hbar)
error = O(dt^2)
"""
M = c * np.asmatrix([[1, -1], [-1, 1]])
expM = qo.Qoperator(1, qo.matpowerh(np.power(np.e, -1j), (dt / hbar) * M))
expTeven = qo.directopprod(identityop(nbits - 1), expM)
P = cyperm_op(nbits)
Pinv = qo.Qoperator(nbits, P.matrix.H)
U = expTeven.matrix * P.matrix * expTeven.matrix * Pinv.matrix
return qo.Qoperator(nbits, U)
def expT_2order_1D(nbits,dt,hbar,c): """ Simulate exp(-i*dt*T/hbar) error = O(dt^2) """ M = c*np.asmatrix([[1,-1],[-1,1]]) expM = qo.Qoperator(1,qo.matpowerh(np.power(np.e,-1j), (dt/hbar)*M)) expTeven = qo.directopprod(identityop(nbits-1),expM) P = cyperm_op(nbits) Pinv = qo.Qoperator(nbits,P.matrix.H) U = expTeven.matrix * P.matrix * expTeven.matrix * Pinv.matrix return qo.Qoperator(nbits,U)
def expH_3order_1D(nbits,dt,hbar,c,expV): """ Simulate exp(-i*dt*T/hbar) error = O(dt^3) """ M = c*np.asmatrix([[1,-1],[-1,1]]) expM = qo.Qoperator(1,qo.matpowerh(np.power(np.e,-1j), (0.5*dt/hbar)*M)) expTeven = qo.directopprod(identityop(nbits-1),expM) P = cyperm_op(nbits) Pinv = qo.Qoperator(nbits,P.matrix.H) U1 = expTeven.matrix * Pinv.matrix * expTeven.matrix * P.matrix U2 = P.matrix * expTeven.matrix * Pinv.matrix * expTeven.matrix #XXX Check correctness of U1 and U2 U = U1 * expV * U2 return qo.Qoperator(nbits,U)
def expH_3order_1D(nbits, dt, hbar, c, expV):
"""
Simulate exp(-i*dt*T/hbar)
error = O(dt^3)
"""
M = c * np.asmatrix([[1, -1], [-1, 1]])
expM = qo.Qoperator(
1, qo.matpowerh(np.power(np.e, -1j), (0.5 * dt / hbar) * M))
expTeven = qo.directopprod(identityop(nbits - 1), expM)
P = cyperm_op(nbits)
Pinv = qo.Qoperator(nbits, P.matrix.H)
U1 = expTeven.matrix * Pinv.matrix * expTeven.matrix * P.matrix
U2 = P.matrix * expTeven.matrix * Pinv.matrix * expTeven.matrix #XXX Check correctness of U1 and U2
U = U1 * expV * U2
return qo.Qoperator(nbits, U)
