def fredkin(N=None, control=0, targets=[1, 2]):
"""Quantum object representing the Fredkin gate.
Returns
-------
fredkin_gate : qobj
Quantum object representation of Fredkin gate.
Examples
--------
>>> fredkin()
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], \
shape = [8, 8], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]
"""
if [control, targets[0], targets[1]] != [0, 1, 2] and N is None:
N = 3
if N is not None:
return gate_expand_3toN(fredkin(), N,
[control, targets[0]], targets[1])
else:
return Qobj([[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1]],
dims=[[2, 2, 2], [2, 2, 2]])
def num(N, offset=0):
"""Quantum object for number operator.
Parameters
----------
N : int
The dimension of the Hilbert space.
offset : int (default 0)
The lowest number state that is included in the finite number state
representation of the operator.
Returns
-------
oper: qobj
Qobj for number operator.
Examples
--------
>>> num(4) # doctest: +SKIP
Quantum object: dims = [[4], [4]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[0 0 0 0]
[0 1 0 0]
[0 0 2 0]
[0 0 0 3]]
"""
if offset == 0:
data = np.arange(1, N, dtype=complex)
ind = np.arange(1, N, dtype=np.int32)
ptr = np.array([0] + list(range(0, N)), dtype=np.int32)
ptr[-1] = N - 1
else:
data = np.arange(offset, offset + N, dtype=complex)
ind = np.arange(N, dtype=np.int32)
ptr = np.arange(N + 1, dtype=np.int32)
ptr[-1] = N
return Qobj(fast_csr_matrix((data, ind, ptr), shape=(N, N)), isherm=True)
def enr_fock(dims, excitations, state):
"""
Generate the Fock state representation in a excitation-number restricted
state space. The `dims` argument is a list of integers that define the
number of quantums states of each component of a composite quantum system,
and the `excitations` specifies the maximum number of excitations for
the basis states that are to be included in the state space. The `state`
argument is a tuple of integers that specifies the state (in the number
basis representation) for which to generate the Fock state representation.
Parameters
----------
dims : list
A list of the dimensions of each subsystem of a composite quantum
system.
excitations : integer
The maximum number of excitations that are to be included in the
state space.
state : list of integers
The state in the number basis representation.
Returns
-------
ket : Qobj
A Qobj instance that represent a Fock state in the exication-number-
restricted state space defined by `dims` and `exciations`.
"""
nstates, state2idx, idx2state = enr_state_dictionaries(dims, excitations)
data = sp.lil_matrix((nstates, 1), dtype=np.complex)
try:
data[state2idx[tuple(state)], 0] = 1
except:
raise ValueError("The state tuple %s is not in the restricted "
"state space" % str(tuple(state)))
return Qobj(data, dims=[dims, 1])
def _compute_prop_grad(self, k, j, compute_prop=True):
"""
Calculate the gradient of propagator wrt the control amplitude
in the timeslot.
Returns:
[prop], prop_grad
"""
dyn = self.parent
dyn._ensure_decomp_curr(k)
if compute_prop:
prop = self._compute_propagator(k)
if dyn.oper_dtype == Qobj:
# put control dyn_gen in combined dg diagonal basis
cdg = (dyn._get_dyn_gen_eigenvectors_adj(k) *
dyn._get_phased_ctrl_dyn_gen(k, j) *
dyn._dyn_gen_eigenvectors[k])
# multiply (elementwise) by timeslice and factor matrix
cdg = Qobj(np.multiply(cdg.full() * dyn.tau[k],
dyn._dyn_gen_factormatrix[k]),
dims=dyn.dyn_dims)
# Return to canonical basis
prop_grad = (dyn._dyn_gen_eigenvectors[k] * cdg *
dyn._get_dyn_gen_eigenvectors_adj(k))
else:
# put control dyn_gen in combined dg diagonal basis
cdg = dyn._get_dyn_gen_eigenvectors_adj(k).dot(
dyn._get_phased_ctrl_dyn_gen(k, j)).dot(
dyn._dyn_gen_eigenvectors[k])
# multiply (elementwise) by timeslice and factor matrix
cdg = np.multiply(cdg * dyn.tau[k], dyn._dyn_gen_factormatrix[k])
# Return to canonical basis
prop_grad = dyn._dyn_gen_eigenvectors[k].dot(cdg).dot(
dyn._get_dyn_gen_eigenvectors_adj(k))
if compute_prop:
return prop, prop_grad
else:
return prop_grad
def qeye(dimensions):
"""
Identity operator.
Parameters
----------
dimensions : (int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the
dimension is the product over this list, but the ``dims`` property of
the new Qobj are set to this list. This can produce either `oper` or
`super` depending on the passed `dimensions`.
Returns
-------
oper : qobj
Identity operator Qobj.
Examples
--------
>>> qeye(3) # doctest: +SKIP
Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, \
isherm = True
Qobj data =
[[ 1. 0. 0.]
[ 0. 1. 0.]
[ 0. 0. 1.]]
>>> qeye([2,2]) # doctest: +SKIP
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, \
isherm = True
Qobj data =
[[1. 0. 0. 0.]
[0. 1. 0. 0.]
[0. 0. 1. 0.]
[0. 0. 0. 1.]]
"""
size, dimensions = _implicit_tensor_dimensions(dimensions)
return Qobj(fast_identity(size),
dims=dimensions,
isherm=True,
isunitary=True)
def fredkin(N=None, control1=None, control2=None, target=None):
"""Quantum object representing the Fredkin gate.
Returns
-------
fred_gate : qobj
Quantum object representation of Fredkin gate.
Examples
--------
>>> fredkin()
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], \
shape = [8, 8], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]
"""
if (not N is None and not control1 is None and not control2 is None
and not target is None):
return gate_expand_3toN(fredkin(), N, control1, control2, target)
else:
uuu = qstate('uuu')
uud = qstate('uud')
udu = qstate('udu')
udd = qstate('udd')
duu = qstate('duu')
dud = qstate('dud')
ddu = qstate('ddu')
ddd = qstate('ddd')
Q = ddd * ddd.dag() + ddu * ddu.dag() + dud * dud.dag() + \
duu * duu.dag() + udd * udd.dag() + uud * udu.dag() + \
udu * uud.dag() + uuu * uuu.dag()
return Qobj(Q)
def enr_thermal_dm(dims, excitations, n):
"""
Generate the density operator for a thermal state in the excitation-number-
restricted state space defined by the `dims` and `exciations` arguments.
See the documentation for enr_fock for a more detailed description of
these arguments. The temperature of each mode in dims is specified by
the average number of excitatons `n`.
Parameters
----------
dims : list
A list of the dimensions of each subsystem of a composite quantum
system.
excitations : integer
The maximum number of excitations that are to be included in the
state space.
n : integer
The average number of exciations in the thermal state. `n` can be
a float (which then applies to each mode), or a list/array of the same
length as dims, in which each element corresponds specifies the
temperature of the corresponding mode.
Returns
-------
dm : Qobj
Thermal state density matrix.
"""
nstates, state2idx, idx2state = enr_state_dictionaries(dims, excitations)
if not isinstance(n, (list, np.ndarray)):
n = np.ones(len(dims)) * n
else:
n = np.asarray(n)
diags = [np.prod((n / (n + 1))**np.array(state)) for state in idx2state]
diags /= np.sum(diags)
data = sp.spdiags(diags, 0, nstates, nstates, format='csr')
return Qobj(data, dims=[dims, dims])
def toffoli(N=None, controls=[0, 1], target=2):
"""Quantum object representing the Toffoli gate.
Returns
-------
toff_gate : qobj
Quantum object representation of Toffoli gate.
Examples
--------
>>> toffoli()
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], \
shape = [8, 8], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j]]
"""
if [controls[0], controls[1], target] != [0, 1, 2] and N is None:
N = 3
if N is not None:
return gate_expand_3toN(toffoli(), N, controls, target)
else:
return Qobj([[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 0, 1, 0]],
dims=[[2, 2, 2], [2, 2, 2]])
def snot(N=None, target=0):
"""Quantum object representing the SNOT (Hadamard) gate.
Returns
-------
snot_gate : qobj
Quantum object representation of SNOT gate.
Examples
--------
>>> snot() # doctest: +SKIP
Quantum object: dims = [[2], [2]], \
shape = [2, 2], type = oper, isHerm = True
Qobj data =
[[ 0.70710678+0.j 0.70710678+0.j]
[ 0.70710678+0.j -0.70710678+0.j]]
"""
if N is not None:
return gate_expand_1toN(snot(), N, target)
return 1 / np.sqrt(2.0) * Qobj([[1, 1], [1, -1]])
def destroy_x(states):
"""
Destruction (lowering) operator for atomic excitations.
Parameters
----------
states : array
List of states of the Hilbert space.
Returns
-------
oper : qobj
Qobj for destruction operator for atomic excitations.
"""
nstates = len(states)
sigma = np.eye(nstates)
for i in range(nstates):
for j in range(nstates):
sigma[i, j] = _destroy_excitons(states[i, 0], states[i, 1],
states[j, 0], states[j, 1])
return Qobj(sp.csr_matrix(sigma, dtype=complex), isherm=False)
def _super_tofrom_choi(q_oper):
"""
We exploit that the basis transformation between Choi and supermatrix
representations squares to the identity, so that if we munge Qobj.type,
we can use the same function.
Since this function doesn't respect :attr:`Qobj.type`, we mark it as
private; only those functions which wrap this in a way so as to preserve
type should be called externally.
"""
data = q_oper.data.toarray()
dims = q_oper.dims
new_dims = [[dims[1][1], dims[0][1]], [dims[1][0], dims[0][0]]]
d0 = np.prod(np.ravel(new_dims[0]))
d1 = np.prod(np.ravel(new_dims[1]))
s0 = np.prod(dims[0][0])
s1 = np.prod(dims[1][1])
return Qobj(dims=new_dims,
inpt=data.reshape([s0, s1, s0, s1]).transpose(3, 1, 2,
0).reshape(
(d0, d1)))
def _steadystate_direct_dense(L):
"""
Direct solver that use numpy dense matrices. Suitable for
small system, with a few states.
"""
if settings.debug:
print('Starting direct dense solver...')
dims = L.dims[0]
n = prod(L.dims[0][0])
b = np.zeros(n**2)
b[0] = 1.0
L = L.data.todense()
L[0, :] = np.diag(np.ones(n)).reshape((1, n**2))
v = np.linalg.solve(L, b)
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def maximally_mixed_dm(N):
"""
Returns the maximally mixed density matrix for a Hilbert space of
dimension N.
Parameters
----------
N : int
Number of basis states in Hilbert space.
Returns
-------
dm : qobj
Thermal state density matrix.
"""
if (not isinstance(N, (int, np.int64))) or N <= 0:
raise ValueError("N must be integer N > 0")
dm = sp.spdiags(np.ones(N, dtype=complex)/float(N), 0, N, N, format='csr')
return Qobj(dm, isherm=True)
def _steadystate_power(L, maxiter=10, tol=1e-6, itertol=1e-5):
"""
Inverse power method for steady state solving.
"""
if settings.debug:
print('Starting iterative power method Solver...')
use_solver(assumeSortedIndices=True)
rhoss = Qobj()
sflag = issuper(L)
if sflag:
rhoss.dims = L.dims[0]
else:
rhoss.dims = [L.dims[0], 1]
n = prod(rhoss.shape)
L = L.data.tocsc() - (tol**2) * sp.eye(n, n, format='csc')
L.sort_indices()
v = mat2vec(rand_dm(rhoss.shape[0], 0.5 / rhoss.shape[0] + 0.5).full())
it = 0
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
v = spsolve(L, v)
v = v / la.norm(v, np.inf)
it += 1
if it >= maxiter:
raise Exception('Failed to find steady state after ' + str(maxiter) +
' iterations')
# normalise according to type of problem
if sflag:
trow = sp.eye(rhoss.shape[0], rhoss.shape[0], format='coo')
trow = sp_reshape(trow, (1, n))
data = v / sum(trow.dot(v))
else:
data = data / la.norm(v)
data = sp.csr_matrix(vec2mat(data))
rhoss.data = 0.5 * (data + data.conj().T)
rhoss.isherm = True
return rhoss.tidyup() if settings.auto_tidyup else rhoss
def test_NonSquareKrausSuperChoi(self):
"""
Superoperator: Convert non-square Kraus operator to Super + Choi matrix
and back.
"""
zero = asarray([[1], [0]], dtype=complex)
one = asarray([[0], [1]], dtype=complex)
zero_log = kron(kron(zero, zero), zero)
one_log = kron(kron(one, one), one)
# non-square Kraus operator (isometry)
kraus = Qobj(zero_log @ zero.T + one_log @ one.T)
super = sprepost(kraus, kraus.dag())
choi = to_choi(super)
op1 = to_kraus(super)
op2 = to_kraus(choi)
op3 = to_super(choi)
assert choi.type == "super" and choi.superrep == "choi"
assert super.type == "super" and super.superrep == "super"
assert (op1[0] - kraus).norm() < tol
assert (op2[0] - kraus).norm() < tol
assert (op3 - super).norm() < tol
def sprepost(A, B):
"""Superoperator formed from pre-multiplication by operator A and post-
multiplication of operator B.
Parameters
----------
A : Qobj
Quantum operator for pre-multiplication.
B : Qobj
Quantum operator for post-multiplication.
Returns
--------
super : Qobj
Superoperator formed from input quantum objects.
"""
dims = [[_drop_projected_dims(A.dims[0]), _drop_projected_dims(B.dims[1])],
[_drop_projected_dims(A.dims[1]), _drop_projected_dims(B.dims[0])]]
data = sp.kron(B.data.T, A.data, format='csr')
return Qobj(data, dims=dims, superrep='super')
def chi_to_choi(q_oper):
"""
Converts a Chi matrix to a Choi matrix.
NOTE: this is only supported for qubits right now. Need to extend to
Heisenberg-Weyl for other subsystem dimensions.
"""
nq = _nq(q_oper.dims)
B = _pauli_basis(nq)
# Force the basis change to match the dimensions of
# the input.
B.dims = q_oper.dims
# We normally should not multiply objects of different
# superreps, so Qobj warns about that. Here, however, we're actively
# converting between, so the superrep of B is irrelevant.
# To suppress warnings, we pretend that B is also a chi.
B.superrep = 'chi'
# The Chi matrix has tr(chi) == d², so we need to divide out
# by that to get back to the Choi form.
return Qobj((B * q_oper * B.dag()) / q_oper.shape[0], superrep='choi')
def _add_common_result_attribs(self, result, st_time, end_time):
"""
Update the result object attributes which are common to all
optimisers and outcomes
"""
dyn = self.dynamics
result.num_iter = self.num_iter
result.num_fid_func_calls = self.num_fid_func_calls
result.wall_time = end_time - st_time
result.fid_err = dyn.fid_computer.get_fid_err()
result.grad_norm_final = dyn.fid_computer.grad_norm
result.final_amps = dyn.ctrl_amps
final_evo = dyn.full_evo
if isinstance(final_evo, Qobj):
result.evo_full_final = final_evo
else:
result.evo_full_final = Qobj(final_evo, dims=dyn.sys_dims)
# *** update stats ***
if self.stats is not None:
self.stats.wall_time_optim_end = end_time
self.stats.calculate()
result.stats = copy.copy(self.stats)
def extract(self, idx_or_label):
"""
Extract a Qobj representing specified ADO from a full representation of
the ADO states.
Parameters
----------
idx : int or label
The index of the ADO to extract. If an ADO label, e.g.
``(0, 1, 0, ...)`` is supplied instead, then the ADO
is extracted by label instead.
Returns
-------
Qobj
A :obj:`Qobj` representing the state of the specified ADO.
"""
if isinstance(idx_or_label, int):
idx = idx_or_label
else:
idx = self._ados.idx(idx_or_label)
return Qobj(self._ado_state[idx, :].T, dims=self.rho.dims)
def _get_aug_mat(self, k, j):
"""
Generate the matrix [[A, E], [0, A]] where
A is the overall dynamics generator
E is the control dynamics generator
for a given timeslot and control
returns this augmented matrix
"""
dyn = self.parent
dg = dyn._get_phased_dyn_gen(k)
if dyn.oper_dtype == Qobj:
A = dg.data * dyn.tau[k]
E = dyn._get_phased_ctrl_dyn_gen(k, j).data * dyn.tau[k]
Z = sp.csr_matrix(dg.data.shape)
aug = Qobj(sp.vstack([sp.hstack([A, E]), sp.hstack([Z, A])]))
else:
A = dg * dyn.tau[k]
E = dyn._get_phased_ctrl_dyn_gen(k, j) * dyn.tau[k]
Z = np.zeros(dg.shape)
aug = np.vstack([np.hstack([A, E]), np.hstack([Z, A])])
return aug
def get_states(self, nstep, ntraj):
if debug:
print(inspect.stack()[0][3])
from scipy.sparse import csr_matrix
if (odeconfig.options.average_states):
states = np.array([Qobj()] * nstep)
if (self.sparse_dms):
# averaged sparse density matrices
for i in range(nstep):
qtf90.qutraj_run.get_rho_sparse(i + 1)
val = qtf90.qutraj_run.csr_val
col = qtf90.qutraj_run.csr_col - 1
ptr = qtf90.qutraj_run.csr_ptr - 1
m = qtf90.qutraj_run.csr_nrows
k = qtf90.qutraj_run.csr_ncols
states[i] = Qobj(csr_matrix((val, col, ptr),
(m, k)).toarray(),
dims=self.dm_dims, shape=self.dm_shape)
else:
# averaged dense density matrices
for i in range(nstep):
states[i] = Qobj(qtf90.qutraj_run.sol[0, i, :, :],
dims=self.dm_dims, shape=self.dm_shape)
else:
# all trajectories as kets
if (ntraj == 1):
states = np.array([Qobj()] * nstep)
for i in range(nstep):
states[i] = Qobj(numpy.matrix(
qtf90.qutraj_run.sol[0, 0, i, :]).transpose(),
dims=self.psi0_dims, shape=self.psi0_shape)
else:
states = np.array([np.array([Qobj()] * nstep)] * ntraj)
for traj in range(ntraj):
for i in range(nstep):
states[traj][i] = Qobj(numpy.matrix(
qtf90.qutraj_run.sol[0, traj, i, :]).transpose(),
dims=self.psi0_dims, shape=self.psi0_shape)
return states
def qzero(dimensions):
"""
Zero operator.
Parameters
----------
dimensions : (int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the
dimension is the product over this list, but the ``dims`` property of
the new Qobj are set to this list. This can produce either `oper` or
`super` depending on the passed `dimensions`.
Returns
-------
qzero : qobj
Zero operator Qobj.
"""
size, dimensions = _implicit_tensor_dimensions(dimensions)
# A sparse matrix with no data is equal to a zero matrix.
return Qobj(fast_csr_matrix(shape=(size, size), dtype=complex),
dims=dimensions,
isherm=True)
def _steadystate_direct_dense(L):
"""
Direct solver that use numpy dense matrices. Suitable for
small system, with a few states.
"""
if settings.debug:
print('Starting direct dense solver...')
dims = L.dims[0]
n = prod(L.dims[0][0])
b = np.zeros(n**2)
b[0] = ss_args['weight']
L = L.data.todense()
L[0, :] = np.diag(ss_args['weight'] * np.ones(n)).reshape((1, n**2))
_dense_start = time.time()
v = np.linalg.solve(L, b)
_dense_end = time.time()
ss_args['info']['solution_time'] = _dense_end - _dense_start
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def spre(A):
"""Superoperator formed from pre-multiplication by operator A.
Parameters
----------
A : qobj
Quantum operator for pre-multiplication.
Returns
--------
super :qobj
Superoperator formed from input quantum object.
"""
if not isinstance(A, Qobj):
raise TypeError('Input is not a quantum object')
if not A.isoper:
raise TypeError('Input is not a quantum operator')
S = Qobj(isherm=A.isherm, superrep='super')
S.dims = [[A.dims[0], A.dims[1]], [A.dims[0], A.dims[1]]]
S.data = sp.kron(sp.identity(np.prod(A.shape[1])), A.data, format='csr')
return S
def _partial_transpose_reference(rho, mask):
"""
This is a reference implementation that explicitly loops over
all states and performs the transpose. It's slow but easy to
understand and useful for testing.
"""
A_pt = np.zeros(rho.shape, dtype=complex)
for psi_A in state_number_enumerate(rho.dims[0]):
m = state_number_index(rho.dims[0], psi_A)
for psi_B in state_number_enumerate(rho.dims[1]):
n = state_number_index(rho.dims[1], psi_B)
m_pt = state_number_index(
rho.dims[1], np.choose(mask, [psi_A, psi_B]))
n_pt = state_number_index(
rho.dims[0], np.choose(mask, [psi_B, psi_A]))
A_pt[m_pt, n_pt] = rho.data[m, n]
return Qobj(A_pt, dims=rho.dims)
def destroy_a_1(states):
"""
Destruction (lowering) operator for photonic excitations.
Parameters
----------
states : list/array
List of states of the Hilbert space.
Returns
-------
oper : qobj
Qobj for destruction operator for photonic excitations.
"""
nstates = len(states)
a = np.eye(nstates)
for i in range(nstates):
for j in range(nstates):
a[i,
j] = _destroy_photons_1(states[i, 0], states[i, 1], states[i, 2],
states[i, 3], states[j, 0], states[j, 1],
states[j, 2], states[j, 3])
return Qobj(sp.csr_matrix(a, dtype=complex), isherm=False)
def spost(A):
"""Superoperator formed from post-multiplication by operator A
Parameters
----------
A : qobj
Quantum operator for post multiplication.
Returns
-------
super : qobj
Superoperator formed from input qauntum object.
"""
if not isinstance(A, Qobj):
raise TypeError('Input is not a quantum object')
if not A.isoper:
raise TypeError('Input is not a quantum operator')
S = Qobj(isherm=A.isherm, superrep='super')
S.dims = [[A.dims[0], A.dims[1]], [A.dims[0], A.dims[1]]]
S.data = zcsr_kron(A.data.T, fast_identity(np.prod(A.shape[0])))
return S
def snot():
"""Quantum object representing the SNOT (Hadamard) gate.
Returns
-------
snot_gate : qobj
Quantum object representation of SNOT (Hadamard) gate.
Examples
--------
>>> snot()
Quantum object: dims = [[2], [2]], \
shape = [2, 2], type = oper, isHerm = True
Qobj data =
[[ 0.70710678+0.j 0.70710678+0.j]
[ 0.70710678+0.j -0.70710678+0.j]]
"""
u = qstate('u')
d = qstate('d')
Q = 1.0 / sqrt(2.0) * (d * d.dag() + u * d.dag() + d * u.dag() -
u * u.dag())
return Qobj(Q)
def destroy(N, offset=0):
'''Destruction (lowering) operator.
Parameters
----------
N : int
Dimension of Hilbert space.
offset : int (default 0)
The lowest number state that is included in the finite number state
representation of the operator.
Returns
-------
oper : qobj
Qobj for lowering operator.
Examples
--------
>>> destroy(4)
Quantum object: dims = [[4], [4]], \
shape = [4, 4], type = oper, isHerm = False
Qobj data =
[[ 0.00000000+0.j 1.00000000+0.j 0.00000000+0.j 0.00000000+0.j]
[ 0.00000000+0.j 0.00000000+0.j 1.41421356+0.j 0.00000000+0.j]
[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 1.73205081+0.j]
[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j]]
'''
if not isinstance(N, (int, np.integer)): # raise error if N not integer
raise ValueError("Hilbert space dimension must be integer value")
return Qobj(sp.spdiags(np.sqrt(range(offset, N + offset)),
1,
N,
N,
format='csr'),
isherm=False)
def swapalpha(alpha, N=None, targets=[0, 1]):
"""
Quantum object representing the SWAPalpha gate.
Returns
-------
swapalpha_gate : qobj
Quantum object representation of SWAPalpha gate
Examples
--------
>>> swapalpha(alpha)
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.5*(1 + exp(j*pi*alpha) 0.5*(1 - exp(j*pi*alpha) 0.+0.j]
[ 0.+0.j 0.5*(1 - exp(j*pi*alpha) 0.5*(1 + exp(j*pi*alpha) 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]
"""
if (targets[0] == 1 and targets[1] == 0) and N is None:
N = 2
if N is not None:
return gate_expand_2toN(cnot(), N, targets=targets)
else:
return Qobj([[1, 0, 0, 0],
[
0, 0.5 * (1 + np.exp(1.0j * np.pi * alpha)), 0.5 *
(1 - np.exp(1.0j * np.pi * alpha)), 0
],
[
0, 0.5 * (1 - np.exp(1.0j * np.pi * alpha)), 0.5 *
(1 + np.exp(1.0j * np.pi * alpha)), 0
], [0, 0, 0, 1]],
dims=[[2, 2], [2, 2]])
