def eig(self,
state: State,
k=6,
which='SM',
use_initial_guess=False,
plot=False):
"""Returns a list of the eigenvalues and the corresponding valid density matrix.
Functionality only for if input is a density matrix."""
assert not state.is_ket
is_ket = state.is_ket
code = state.code
IS_subspace = state.IS_subspace
graph = state.graph
state_shape = state.shape
def f(flattened):
s = State(flattened.reshape(state_shape))
res = self.evolution_generator(s)
return res.reshape(flattened.shape)
state_flattened = state.flatten()
lindbladian = LinearOperator(shape=(len(state_flattened),
len(state_flattened)),
dtype=np.complex128,
matvec=f)
if not use_initial_guess:
v0 = None
else:
v0 = state_flattened
try:
eigvals, eigvecs = eigs(lindbladian, k=k, which=which, v0=v0)
except ArpackNoConvergence as exception_info:
eigvals = exception_info.eigenvalues
eigvecs = exception_info.eigenvectors
if eigvals.size != 0:
# Do some basic reshaping to post process the results and select for only the steady states
eigvecs = np.reshape(
eigvecs, [state.shape[0], state.shape[1], eigvecs.shape[-1]])
eigvecs = np.moveaxis(eigvecs, -1, 0)
# If there is one steady s, then normalize it because it is a valid density matrix
if plot:
eigvals_cc = eigvals.conj()
eigvals_real = np.concatenate((eigvals.real, eigvals.real))
eigvals_complex = np.concatenate(
(eigvals_cc.imag, eigvals_cc.imag))
plt.hlines(0, xmin=-1, xmax=.1)
plt.vlines(0, ymin=-100, ymax=100)
plt.scatter(eigvals_real, eigvals_complex, c='m', s=4)
plt.show()
return eigvals, eigvecs
else:
return None, None
def edg(self,
state: State,
k=6,
which='LR',
use_initial_guess=False,
tol=1e-8):
dim = state.dimension
eigval, eigvec = self.steady_state(state,
k=k,
which=which,
use_initial_guess=use_initial_guess,
plot=False)
steady_state = eigvec[np.argwhere(eigval[np.abs(eigval) <= tol])[
0, 0], :, :]
steady_state = steady_state / np.trace(steady_state)
# Diagonalize the steady state
ss_eigvals, ss_eigvecs = np.linalg.eigh(steady_state)
where_nonzero = np.abs(ss_eigvals) > tol
if np.sum(where_nonzero) < dim:
print('Steady state is not full rank.')
projected_ss_eigvals = where_nonzero * ss_eigvecs
P = projected_ss_eigvals @ projected_ss_eigvals.conj().T
Q = np.identity(dim) - P
state_shape = state.shape
def f(flattened):
s = State(flattened.reshape(state_shape))
res = P @ self.evolution_generator(P @ s @ P) @ P + Q @ self.evolution_generator(Q @ s @ P) @ P + \
P @ self.evolution_generator(P @ s @ Q) @ Q
return res.reshape(flattened.shape)
state_flattened = state.flatten()
lindbladian = LinearOperator(shape=(len(state_flattened),
len(state_flattened)),
dtype=np.complex128,
matvec=f)
if not use_initial_guess:
v0 = None
else:
v0 = state_flattened
try:
eigvals, eigvecs = eigs(lindbladian, k=k, which=which, v0=v0)
except ArpackNoConvergence as exception_info:
eigvals = exception_info.eigenvalues
if eigvals.size == 0:
return None
nonzero = np.abs(eigvals[np.abs(eigvals) > tol])
where_max = np.argmin(nonzero)
min_eigval = np.abs(nonzero[where_max])
return min_eigval
def steady_state(self,
state: State,
k=6,
which='LR',
use_initial_guess=False,
plot=False,
tol=1e-8,
verbose=False):
"""Returns a list of the eigenvalues and the corresponding valid density matrix."""
assert not state.is_ket
state_shape = state.shape
def f(flattened):
s = State(flattened.reshape(state_shape))
res = self.evolution_generator(s)
return res.reshape(flattened.shape)
state_flattened = state.flatten()
lindbladian = LinearOperator(shape=(len(state_flattened),
len(state_flattened)),
dtype=np.complex128,
matvec=f)
if not use_initial_guess:
v0 = None
else:
v0 = state_flattened
try:
eigvals, eigvecs = eigs(lindbladian, k=k, which=which, v0=v0)
except ArpackNoConvergence as exception_info:
eigvals = exception_info.eigenvalues
eigvecs = exception_info.eigenvectors
if eigvals.size != 0:
# Do some basic reshaping to post process the results and select for only the steady states
eigvecs = np.moveaxis(eigvecs, -1, 0)
eigvecs = np.reshape(
eigvecs, [eigvecs.shape[0], state_shape[0], state_shape[1]])
steady_state_indices = np.argwhere(eigvals.real > -1 * tol).T[0]
steady_state_eigvecs = eigvecs[steady_state_indices, :, :]
if verbose:
print('Number of steady states is ',
str(steady_state_eigvecs.shape[0]))
# If there is one steady s, then normalize it because it is a valid density matrix
if steady_state_eigvecs.shape[0] == 1:
steady_state_eigvecs[0, :, :] = steady_state_eigvecs[
0, :, :] / np.trace(steady_state_eigvecs[0, :, :])
if verbose:
print(
'Steady state is a valid density matrix:',
tools.is_valid_state(steady_state_eigvecs[0, :, :],
verbose=False))
steady_state_eigvals = eigvals[steady_state_indices]
if plot:
steady_state_eigvals_cc = steady_state_eigvals.conj()
steady_state_eigvals_real = np.concatenate(
(steady_state_eigvals.real, steady_state_eigvals.real))
steady_state_eigvals_complex = np.concatenate(
(steady_state_eigvals_cc.imag,
steady_state_eigvals_cc.imag))
plt.hlines(0, xmin=-1, xmax=.1)
plt.vlines(0, ymin=-100, ymax=100)
plt.scatter(steady_state_eigvals_real,
steady_state_eigvals_complex,
c='m',
s=4)
plt.show()
return steady_state_eigvals, steady_state_eigvecs
else:
return None, None
