def _subsystem_apply_reference(state, channel, mask):
if isket(state):
state = ket2dm(state)
if isoper(channel):
full_oper = tensor([channel if mask[j]
else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))])
return full_oper * state * full_oper.dag()
else:
# Go to Choi, then Kraus
# chan_mat = array(channel.data.todense())
choi_matrix = super_to_choi(channel)
vals, vecs = eig(choi_matrix.full())
vecs = list(map(array, zip(*vecs)))
kraus_list = [sqrt(vals[j]) * vec2mat(vecs[j])
for j in range(len(vals))]
# Kraus operators to be padded with identities:
k_qubit_kraus_list = product(kraus_list, repeat=sum(mask))
rho_out = Qobj(inpt=zeros(state.shape), dims=state.dims)
for operator_iter in k_qubit_kraus_list:
operator_iter = iter(operator_iter)
op_iter_list = [next(operator_iter).conj().T if mask[j]
else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))]
full_oper = tensor(list(map(Qobj, op_iter_list)))
rho_out = rho_out + full_oper * state * full_oper.dag()
return Qobj(rho_out)
def _jc_liouvillian(N):
from qutip.tensor import tensor
from qutip.operators import destroy, qeye
from qutip.superoperator import liouvillian
wc = 1.0 * 2 * np.pi # cavity frequency
wa = 1.0 * 2 * np.pi # atom frequency
g = 0.05 * 2 * np.pi # coupling strength
kappa = 0.005 # cavity dissipation rate
gamma = 0.05 # atom dissipation rate
n_th_a = 1 # temperature in frequency units
use_rwa = 0
# operators
a = tensor(destroy(N), qeye(2))
sm = tensor(qeye(N), destroy(2))
# Hamiltonian
if use_rwa:
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())
else:
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() + a) * (sm + sm.dag())
c_op_list = []
rate = kappa * (1 + n_th_a)
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * a)
rate = kappa * n_th_a
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * a.dag())
rate = gamma
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm)
return liouvillian(H, c_op_list)
def test_sp_bandwidth():
"Sparse: Bandwidth"
# Bandwidth test 1
A = create(25)+destroy(25)+qeye(25)
band = sp_bandwidth(A.data)
assert_equal(band[0], 3)
assert_equal(band[1] == band[2] == 1, 1)
# Bandwidth test 2
A = np.array([[1, 0, 0, 0, 1, 0, 0, 0],
[0, 1, 1, 0, 0, 1, 0, 1],
[0, 1, 1, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 1, 0],
[1, 0, 1, 0, 1, 0, 0, 0],
[0, 1, 0, 0, 0, 1, 0, 1],
[0, 0, 0, 1, 0, 0, 1, 0],
[0, 1, 0, 0, 0, 1, 0, 1]], dtype=np.int32)
A = sp.csr_matrix(A)
out1 = sp_bandwidth(A)
assert_equal(out1[0], 13)
assert_equal(out1[1] == out1[2] == 6, 1)
# Bandwidth test 3
perm = reverse_cuthill_mckee(A)
B = sp_permute(A, perm, perm)
out2 = sp_bandwidth(B)
assert_equal(out2[0], 5)
assert_equal(out2[1] == out2[2] == 2, 1)
# Asymmetric bandwidth test
A = destroy(25)+qeye(25)
out1 = sp_bandwidth(A.data)
assert_equal(out1[0], 2)
assert_equal(out1[1], 0)
assert_equal(out1[2], 1)
def test_unitarity_known():
"""
Metrics: Unitarity for known cases.
"""
def case(q_oper, known_unitarity):
assert_almost_equal(unitarity(q_oper), known_unitarity)
yield case, to_super(sigmax()), 1.0
yield case, sum(map(
to_super, [qeye(2), sigmax(), sigmay(), sigmaz()]
)) / 4, 0.0
yield case, sum(map(
to_super, [qeye(2), sigmax()]
)) / 2, 1 / 3.0
def test_rand_unitary_haar_unitarity():
"""
Random Qobjs: Tests that unitaries are actually unitary.
"""
U = rand_unitary_haar(5)
I = qeye(5)
assert_(U * U.dag() == I)
def test_stinespring_dims(self, dimension):
"""
Stinespring: Check that dims of channels are preserved.
"""
chan = super_tensor(to_super(sigmax()), to_super(qeye(dimension)))
A, B = to_stinespring(chan)
assert A.dims == [[2, dimension, 1], [2, dimension]]
assert B.dims == [[2, dimension, 1], [2, dimension]]
def test_stinespring_dims(self):
"""
Stinespring: Check that dims of channels are preserved.
"""
# FIXME: not the most general test, since this assumes a map
# from square matrices to square matrices on the same space.
chan = super_tensor(to_super(sigmax()), to_super(qeye(3)))
A, B = to_stinespring(chan)
assert_equal(A.dims, [[2, 3, 1], [2, 3]])
assert_equal(B.dims, [[2, 3, 1], [2, 3]])
def _prop_identity(self, U, tol=1e-6):
"""
Returns True if and only if U is proportional to the
identity.
"""
if U[0, 0] != 0:
norm_U = U / U[0, 0]
return (qeye(U.dims[0]) - norm_U).norm() <= tol
else:
return False
def _powers(op, N):
"""
Generator that yields powers of an operator `op`,
through to `N`.
"""
acc = qeye(op.dims[0])
yield acc
for _ in range(N - 1):
acc *= op
yield acc
def _opto_liouvillian(N):
from qutip.tensor import tensor
from qutip.operators import destroy, qeye
from qutip.superoperator import liouvillian
Nc = 5 # Number of cavity states
Nm = N # Number of mech states
kappa = 0.3 # Cavity damping rate
E = 0.1 # Driving Amplitude
g0 = 2.4*kappa # Coupling strength
Qm = 1e4 # Mech quality factor
gamma = 1/Qm # Mech damping rate
n_th = 1 # Mech bath temperature
delta = -0.43 # Detuning
a = tensor(destroy(Nc), qeye(Nm))
b = tensor(qeye(Nc), destroy(Nm))
num_b = b.dag()*b
num_a = a.dag()*a
H = -delta*(num_a)+num_b+g0*(b.dag()+b)*num_a+E*(a.dag()+a)
cc = np.sqrt(kappa)*a
cm = np.sqrt(gamma*(1.0 + n_th))*b
cp = np.sqrt(gamma*n_th)*b.dag()
c_ops = [cc,cm,cp]
return liouvillian(H, c_ops)
def _spin_hamiltonian(N):
from qutip.tensor import tensor
from qutip.operators import qeye, sigmax, sigmay, sigmaz
# array of spin energy splittings and coupling strengths. here we use
# uniform parameters, but in general we don't have too
h = 1.0 * 2 * np.pi * np.ones(N)
Jz = 0.1 * 2 * np.pi * np.ones(N)
Jx = 0.1 * 2 * np.pi * np.ones(N)
Jy = 0.1 * 2 * np.pi * np.ones(N)
# dephasing rate
gamma = 0.01 * np.ones(N)
si = qeye(2)
sx = sigmax()
sy = sigmay()
sz = sigmaz()
sx_list = []
sy_list = []
sz_list = []
for n in range(N):
op_list = []
for m in range(N):
op_list.append(si)
op_list[n] = sx
sx_list.append(tensor(op_list))
op_list[n] = sy
sy_list.append(tensor(op_list))
op_list[n] = sz
sz_list.append(tensor(op_list))
# construct the hamiltonian
H = 0
# energy splitting terms
for n in range(N):
H += - 0.5 * h[n] * sz_list[n]
# interaction terms
for n in range(N-1):
H += - 0.5 * Jx[n] * sx_list[n] * sx_list[n+1]
H += - 0.5 * Jy[n] * sy_list[n] * sy_list[n+1]
H += - 0.5 * Jz[n] * sz_list[n] * sz_list[n+1]
return H
def test_QobjUnitaryOper():
"Qobj unitarity"
# Check some standard operators
Sx = sigmax()
Sy = sigmay()
assert_unitarity(qeye(4), True, "qeye(4) should be unitary.")
assert_unitarity(Sx, True, "sigmax() should be unitary.")
assert_unitarity(Sy, True, "sigmax() should be unitary.")
assert_unitarity(sigmam(), False, "sigmam() should NOT be unitary.")
assert_unitarity(destroy(10), False, "destroy(10) should NOT be unitary.")
# Check multiplcation of unitary is unitary
assert_unitarity(Sx*Sy, True, "sigmax()*sigmay() should be unitary.")
# Check some other operations clear unitarity
assert_unitarity(Sx+Sy, False, "sigmax()+sigmay() should NOT be unitary.")
assert_unitarity(4*Sx, False, "4*sigmax() should NOT be unitary.")
assert_unitarity(Sx*4, False, "sigmax()*4 should NOT be unitary.")
assert_unitarity(4+Sx, False, "4+sigmax() should NOT be unitary.")
assert_unitarity(Sx+4, False, "sigmax()+4 should NOT be unitary.")
def test_fidelity_known_cases():
"""
Metrics: Checks fidelity against known cases.
"""
ket0 = basis(2, 0)
ket1 = basis(2, 1)
ketp = (ket0 + ket1).unit()
# A state that almost overlaps with |+> should serve as
# a nice test case, especially since we can analytically
# calculate its overlap with |+>.
ketpy = (ket0 + np.exp(1j * np.pi / 4) * ket1).unit()
mms = qeye(2).unit()
assert_almost_equal(fidelity(ket0, ketp), 1 / np.sqrt(2))
assert_almost_equal(fidelity(ket0, ket1), 0)
assert_almost_equal(fidelity(ket0, mms), 1 / np.sqrt(2))
assert_almost_equal(fidelity(ketp, ketpy),
np.sqrt(
(1 / 8) + (1 / 2 + 1 / (2 * np.sqrt(2))) ** 2
)
)
def _spectrum_pi(H, wlist, c_ops, a_op, b_op, use_pinv=False):
"""
Internal function for calculating the spectrum of the correlation function
:math:`\left<A(\\tau)B(0)\\right>`.
"""
L = H if issuper(H) else liouvillian(H, c_ops)
tr_mat = tensor([qeye(n) for n in L.dims[0][0]])
N = np.prod(L.dims[0][0])
A = L.full()
b = spre(b_op).full()
a = spre(a_op).full()
tr_vec = np.transpose(mat2vec(tr_mat.full()))
rho_ss = steadystate(L)
rho = np.transpose(mat2vec(rho_ss.full()))
I = np.identity(N * N)
P = np.kron(np.transpose(rho), tr_vec)
Q = I - P
spectrum = np.zeros(len(wlist))
for idx, w in enumerate(wlist):
if use_pinv:
MMR = np.linalg.pinv(-1.0j * w * I + A)
else:
MMR = np.dot(Q, np.linalg.solve(-1.0j * w * I + A, Q))
s = np.dot(tr_vec,
np.dot(a, np.dot(MMR, np.dot(b, np.transpose(rho)))))
spectrum[idx] = -2 * np.real(s[0, 0])
return spectrum
def test_chi_known(self):
"""
Superoperator: Chi-matrix for known cases is correct.
"""
def case(S, chi_expected, silent=True):
chi_actual = to_chi(S)
chiq = Qobj(chi_expected, dims=[[[2], [2]], [[2], [2]]], superrep='chi')
if not silent:
print(chi_actual)
print(chi_expected)
assert_almost_equal((chi_actual - chiq).norm('tr'), 0)
yield case, sigmax(), [
[0, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]
]
yield case, to_super(sigmax()), [
[0, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]
]
yield case, qeye(2), [
[4, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]
]
yield case, (-1j * sigmax() * pi / 4).expm(), [
[2, 2j, 0, 0],
[-2j, 2, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]
]
def test_dnorm_qubit_known_cases():
"""
Metrics: check agreement for known qubit channels.
"""
def case(chan1, chan2, expected, significant=4):
# We again take a generous tolerance so that we don't kill off
# SCS solvers.
assert_approx_equal(
dnorm(chan1, chan2), expected,
significant=significant
)
id_chan = to_choi(qeye(2))
S_eye = to_super(id_chan)
X_chan = to_choi(sigmax())
depol = to_choi(Qobj(
diag(ones((4,))),
dims=[[[2], [2]], [[2], [2]]], superrep='chi'
))
S_H = to_super(hadamard_transform())
W = swap()
# We need to restrict the number of iterations for things on the boundary,
# such as perfectly distinguishable channels.
yield case, id_chan, X_chan, 2
yield case, id_chan, depol, 1.5
# Next, we'll generate some test cases based on comparisons to pre-existing
# dnorm() implementations. In particular, the targets for the following
# test cases were generated using QuantumUtils for MATLAB (https://goo.gl/oWXhO9).
def overrotation(x):
return to_super((1j * np.pi * x * sigmax() / 2).expm())
for x, target in {
1.000000e-03: 3.141591e-03,
3.100000e-03: 9.738899e-03,
1.000000e-02: 3.141463e-02,
3.100000e-02: 9.735089e-02,
1.000000e-01: 3.128689e-01,
3.100000e-01: 9.358596e-01
}.items():
yield case, overrotation(x), id_chan, target
def had_mixture(x):
return (1 - x) * S_eye + x * S_H
for x, target in {
1.000000e-03: 2.000000e-03,
3.100000e-03: 6.200000e-03,
1.000000e-02: 2.000000e-02,
3.100000e-02: 6.200000e-02,
1.000000e-01: 2.000000e-01,
3.100000e-01: 6.200000e-01
}.items():
yield case, had_mixture(x), id_chan, target
def swap_map(x):
S = (1j * x * W).expm()
S._type = None
S.dims = [[[2], [2]], [[2], [2]]]
S.superrep = 'super'
return S
for x, target in {
1.000000e-03: 2.000000e-03,
3.100000e-03: 6.199997e-03,
1.000000e-02: 1.999992e-02,
3.100000e-02: 6.199752e-02,
1.000000e-01: 1.999162e-01,
3.100000e-01: 6.173918e-01
}.items():
yield case, swap_map(x), id_chan, target
# Finally, we add a known case from Johnston's QETLAB documentation,
# || Phi - I ||,_♢ where Phi(X) = UXU⁺ and U = [[1, 1], [-1, 1]] / sqrt(2).
yield case, Qobj([[1, 1], [-1, 1]]) / np.sqrt(2), qeye(2), np.sqrt(2)
def propagator(H, t, c_op_list=[], args={}, options=None,
unitary_mode='batch', parallel=False,
progress_bar=None, **kwargs):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
unitary_mode = str ('batch', 'single')
Solve all basis vectors simulaneously ('batch') or individually
('single').
parallel : bool {False, True}
Run the propagator in parallel mode. This will override the
unitary_mode settings if set to True.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
td_type = _td_format_check(H, c_op_list, solver='me')
if isinstance(H, (types.FunctionType, types.BuiltinFunctionType,
functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
if parallel:
unitary_mode = 'single'
u = np.zeros([N, N, len(tlist)], dtype=complex)
output = parallel_map(_parallel_sesolve,range(N),
task_args=(N,H, tlist,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
if unitary_mode == 'single':
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
elif unitary_mode =='batch':
u = np.zeros(len(tlist), dtype=object)
_rows = np.array([(N+1)*m for m in range(N)])
_cols = np.zeros_like(_rows)
_data = np.ones_like(_rows,dtype=complex)
psi0 = Qobj(sp.coo_matrix((_data,(_rows,_cols))).tocsr())
if td_type[1] > 0 or td_type[2] > 0:
H2 = []
for k in range(len(H)):
if isinstance(H[k], list):
H2.append([tensor(qeye(N), H[k][0]), H[k][1]])
else:
H2.append(tensor(qeye(N), H[k]))
else:
H2 = tensor(qeye(N), H)
output = sesolve(H2, psi0, tlist, [] , args = args, _safe_mode=False,
options=Options(normalize_output=False))
for k, t in enumerate(tlist):
u[k] = sp_reshape(output.states[k].data, (N, N))
unit_row_norm(u[k].data, u[k].indptr, u[k].shape[0])
u[k] = u[k].T.tocsr()
else:
raise Exception('Invalid unitary mode.')
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(sqrt_N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(sqrt_N,sqrt_N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(sqrt_N,sqrt_N), dtype=complex))
output = mesolve(H, rho0, tlist, [], [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(N,N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(N,N), dtype=complex))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
if unitary_mode == 'batch':
return Qobj(u[-1], dims=dims)
else:
return Qobj(u[:, :, 1], dims=dims)
else:
if unitary_mode == 'batch':
return np.array([Qobj(u[k], dims=dims) for k in range(len(tlist))], dtype=object)
else:
return np.array([Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))], dtype=object)
def test_QobjExpmZeroOper():
"Qobj expm zero_oper (#493)"
A = Qobj(np.zeros((5,5), dtype=complex))
B = A.expm()
assert_(B == qeye(5))
def spectrum_pi(H, wlist, c_ops, a_op, b_op, use_pinv=False):
"""
Calculate the spectrum corresponding to a correlation function
:math:`\left<A(\\tau)B(0)\\right>`, i.e., the Fourier transform of the
correlation function:
.. math::
S(\omega) = \int_{-\infty}^{\infty} \left<A(\\tau)B(0)\\right>
e^{-i\omega\\tau} d\\tau.
Parameters
----------
H : :class:`qutip.qobj`
system Hamiltonian.
wlist : *list* / *array*
list of frequencies for :math:`\\omega`.
c_ops : list of :class:`qutip.qobj`
list of collapse operators.
a_op : :class:`qutip.qobj`
operator A.
b_op : :class:`qutip.qobj`
operator B.
Returns
-------
s_vec: *array*
An *array* with spectrum :math:`S(\omega)` for the frequencies
specified in `wlist`.
"""
L = H if issuper(H) else liouvillian_fast(H, c_ops)
tr_mat = tensor([qeye(n) for n in L.dims[0][0]])
N = prod(L.dims[0][0])
A = L.full()
b = spre(b_op).full()
a = spre(a_op).full()
tr_vec = transpose(mat2vec(tr_mat.full()))
rho_ss = steadystate(L)
rho = transpose(mat2vec(rho_ss.full()))
I = np.identity(N * N)
P = np.kron(transpose(rho), tr_vec)
Q = I - P
s_vec = np.zeros(len(wlist))
for idx, w in enumerate(wlist):
if use_pinv:
MMR = numpy.linalg.pinv(-1.0j * w * I + A)
else:
MMR = np.dot(Q, np.linalg.solve(-1.0j * w * I + A, Q))
s = np.dot(tr_vec, np.dot(a, np.dot(MMR, np.dot(b, transpose(rho)))))
s_vec[idx] = -2 * np.real(s[0, 0])
return s_vec
def dnorm(A, B=None, solver="CVXOPT", verbose=False, force_solve=False):
"""
Calculates the diamond norm of the quantum map q_oper, using
the simplified semidefinite program of [Wat12]_.
The diamond norm SDP is solved by using CVXPY_.
Parameters
----------
A : Qobj
Quantum map to take the diamond norm of.
B : Qobj or None
If provided, the diamond norm of :math:`A - B` is
taken instead.
solver : str
Solver to use with CVXPY. One of "CVXOPT" (default)
or "SCS". The latter tends to be significantly faster,
but somewhat less accurate.
verbose : bool
If True, prints additional information about the
solution.
force_solve : bool
If True, forces dnorm to solve the associated SDP, even if a special
case is known for the argument.
Returns
-------
dn : float
Diamond norm of q_oper.
Raises
------
ImportError
If CVXPY cannot be imported.
.. _cvxpy: http://www.cvxpy.org/en/latest/
"""
if cvxpy is None: # pragma: no cover
raise ImportError("dnorm() requires CVXPY to be installed.")
# We follow the strategy of using Watrous' simpler semidefinite
# program in its primal form. This is the same strategy used,
# for instance, by both pyGSTi and SchattenNorms.jl. (By contrast,
# QETLAB uses the dual problem.)
# Check if A and B are both unitaries. If so, then we can without
# loss of generality choose B to be the identity by using the
# unitary invariance of the diamond norm,
# || A - B ||_♢ = || A B⁺ - I ||_♢.
# Then, using the technique mentioned by each of Johnston and
# da Silva,
# || A B⁺ - I ||_♢ = max_{i, j} | \lambda_i(A B⁺) - \lambda_j(A B⁺) |,
# where \lambda_i(U) is the ith eigenvalue of U.
if (
# There's a lot of conditions to check for this path.
not force_solve and B is not None and
# Only check if they aren't superoperators.
A.type == "oper" and B.type == "oper" and
# The difference of unitaries optimization is currently
# only implemented for d == 2. Much of the code below is more general,
# though, in anticipation of generalizing the optimization.
A.shape[0] == 2
):
# Make an identity the same size as A and B to
# compare against.
I = qeye(A.dims[0])
# Compare to B first, so that an error is raised
# as soon as possible.
Bd = B.dag()
if (
(B * Bd - I).norm() < 1e-6 and
(A * A.dag() - I).norm() < 1e-6
):
# Now we are on the fast path, so let's compute the
# eigenvalues, then find the diameter of the smallest circle
# containing all of them.
#
# For now, this is only implemented for dim = 2, such that
# generalizing here will allow for generalizing the optimization.
# A reasonable approach would probably be to use Welzl's algorithm
# (https://en.wikipedia.org/wiki/Smallest-circle_problem).
U = A * B.dag()
eigs = U.eigenenergies()
eig_distances = np.abs(eigs[:, None] - eigs[None, :])
return np.max(eig_distances)
# Force the input superoperator to be a Choi matrix.
J = to_choi(A)
if B is not None:
J -= to_choi(B)
# Watrous 2012 also points out that the diamond norm of Lambda
# is the same as the completely-bounded operator-norm (∞-norm)
# of the dual map of Lambda. We can evaluate that norm much more
# easily if Lambda is completely positive, since then the largest
# eigenvalue is the same as the largest singular value.
if not force_solve and J.iscp:
S_dual = to_super(J.dual_chan())
vec_eye = operator_to_vector(qeye(S_dual.dims[1][1]))
op = vector_to_operator(S_dual * vec_eye)
# The 2-norm was not implemented for sparse matrices as of the time
# of this writing. Thus, we must yet again go dense.
return la.norm(op.data.todense(), 2)
# If we're still here, we need to actually solve the problem.
# Assume square...
dim = np.prod(J.dims[0][0])
# The constraints only depend on the dimension, so
# we can cache them efficiently.
problem, Jr, Ji, X, rho0, rho1 = dnorm_problem(dim)
# Load the parameters with the Choi matrix passed in.
J_dat = J.data
Jr.value = sp.csr_matrix((J_dat.data.real, J_dat.indices, J_dat.indptr),
shape=J_dat.shape)
Ji.value = sp.csr_matrix((J_dat.data.imag, J_dat.indices, J_dat.indptr),
shape=J_dat.shape)
# Finally, set up and run the problem.
problem.solve(solver=solver, verbose=verbose)
return problem.value
def qudit_swap(dim):
# We should likely generalize this and include it in qip.gates.
W = qeye([dim, dim])
return tensor_swap(W, (0, 1))
