def test_dual_channel(sub_dimensions, n_trials=50):
"""
Qobj: dual_chan() preserves inner products with arbitrary density ops.
"""
S = rand_super_bcsz(np.prod(sub_dimensions))
S.dims = [[sub_dimensions, sub_dimensions],
[sub_dimensions, sub_dimensions]]
S = to_super(S)
left_dims, right_dims = S.dims
# Assume for the purposes of the test that S maps square operators to
# square operators.
in_dim = np.prod(right_dims[0])
out_dim = np.prod(left_dims[0])
S_dual = to_super(S.dual_chan())
primals = []
duals = []
for _ in [None] * n_trials:
X = rand_dm_ginibre(out_dim)
X.dims = left_dims
X = operator_to_vector(X)
Y = rand_dm_ginibre(in_dim)
Y.dims = right_dims
Y = operator_to_vector(Y)
primals.append((X.dag() * S * Y)[0, 0])
duals.append((X.dag() * S_dual.dag() * Y)[0, 0])
np.testing.assert_array_almost_equal(primals, duals)
def case(S, n_trials=50):
S = to_super(S)
left_dims, right_dims = S.dims
# Assume for the purposes of the test that S maps square operators to square operators.
in_dim = np.prod(right_dims[0])
out_dim = np.prod(left_dims[0])
S_dual = to_super(S.dual_chan())
primals = []
duals = []
for idx_trial in range(n_trials):
X = rand_dm_ginibre(out_dim)
X.dims = left_dims
X = operator_to_vector(X)
Y = rand_dm_ginibre(in_dim)
Y.dims = right_dims
Y = operator_to_vector(Y)
primals.append((X.dag() * S * Y)[0, 0])
duals.append((X.dag() * S_dual.dag() * Y)[0, 0])
np.testing.assert_array_almost_equal(primals, duals)
def case(S, n_trials=50):
S = to_super(S)
left_dims, right_dims = S.dims
# Assume for the purposes of the test that S maps square operators to square operators.
in_dim = np.prod(right_dims[0])
out_dim = np.prod(left_dims[0])
S_dual = to_super(S.dual_chan())
primals = []
duals = []
for idx_trial in range(n_trials):
X = rand_dm_ginibre(out_dim)
X.dims = left_dims
X = operator_to_vector(X)
Y = rand_dm_ginibre(in_dim)
Y.dims = right_dims
Y = operator_to_vector(Y)
primals.append((X.dag() * S * Y)[0, 0])
duals.append((X.dag() * S_dual.dag() * Y)[0, 0])
np.testing.assert_array_almost_equal(primals, duals)
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_QobjPermute():
"Qobj permute"
A = basis(3, 0)
B = basis(5, 4)
C = basis(4, 2)
psi = tensor(A, B, C)
psi2 = psi.permute([2, 0, 1])
assert psi2 == tensor(C, A, B)
psi_bra = psi.dag()
psi2_bra = psi_bra.permute([2, 0, 1])
assert psi2_bra == tensor(C, A, B).dag()
A = fock_dm(3, 0)
B = fock_dm(5, 4)
C = fock_dm(4, 2)
rho = tensor(A, B, C)
rho2 = rho.permute([2, 0, 1])
assert rho2 == tensor(C, A, B)
for _ in range(3):
A = rand_ket(3)
B = rand_ket(4)
C = rand_ket(5)
psi = tensor(A, B, C)
psi2 = psi.permute([1, 0, 2])
assert psi2 == tensor(B, A, C)
psi_bra = psi.dag()
psi2_bra = psi_bra.permute([1, 0, 2])
assert psi2_bra == tensor(B, A, C).dag()
for _ in range(3):
A = rand_dm(3)
B = rand_dm(4)
C = rand_dm(5)
rho = tensor(A, B, C)
rho2 = rho.permute([1, 0, 2])
assert rho2 == tensor(B, A, C)
rho_vec = operator_to_vector(rho)
rho2_vec = rho_vec.permute([[1, 0, 2], [4, 3, 5]])
assert rho2_vec == operator_to_vector(tensor(B, A, C))
rho_vec_bra = operator_to_vector(rho).dag()
rho2_vec_bra = rho_vec_bra.permute([[1, 0, 2], [4, 3, 5]])
assert rho2_vec_bra == operator_to_vector(tensor(B, A, C)).dag()
for _ in range(3):
super_dims = [3, 5, 4]
U = rand_unitary(np.prod(super_dims),
density=0.02,
dims=[super_dims, super_dims])
Unew = U.permute([2, 1, 0])
S_tens = to_super(U)
S_tens_new = to_super(Unew)
assert S_tens_new == S_tens.permute([[2, 1, 0], [5, 4, 3]])
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 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 test_QobjPermute():
"Qobj permute"
A = basis(3, 0)
B = basis(5, 4)
C = basis(4, 2)
psi = tensor(A, B, C)
psi2 = psi.permute([2, 0, 1])
assert_(psi2 == tensor(C, A, B))
psi_bra = psi.dag()
psi2_bra = psi_bra.permute([2, 0, 1])
assert_(psi2_bra == tensor(C, A, B).dag())
A = fock_dm(3, 0)
B = fock_dm(5, 4)
C = fock_dm(4, 2)
rho = tensor(A, B, C)
rho2 = rho.permute([2, 0, 1])
assert_(rho2 == tensor(C, A, B))
for ii in range(3):
A = rand_ket(3)
B = rand_ket(4)
C = rand_ket(5)
psi = tensor(A, B, C)
psi2 = psi.permute([1, 0, 2])
assert_(psi2 == tensor(B, A, C))
psi_bra = psi.dag()
psi2_bra = psi_bra.permute([1, 0, 2])
assert_(psi2_bra == tensor(B, A, C).dag())
for ii in range(3):
A = rand_dm(3)
B = rand_dm(4)
C = rand_dm(5)
rho = tensor(A, B, C)
rho2 = rho.permute([1, 0, 2])
assert_(rho2 == tensor(B, A, C))
rho_vec = operator_to_vector(rho)
rho2_vec = rho_vec.permute([[1, 0, 2],[4,3,5]])
assert_(rho2_vec == operator_to_vector(tensor(B, A, C)))
rho_vec_bra = operator_to_vector(rho).dag()
rho2_vec_bra = rho_vec_bra.permute([[1, 0, 2],[4,3,5]])
assert_(rho2_vec_bra == operator_to_vector(tensor(B, A, C)).dag())
for ii in range(3):
super_dims = [3, 5, 4]
U = rand_unitary(np.prod(super_dims), density=0.02, dims=[super_dims, super_dims])
Unew = U.permute([2,1,0])
S_tens = to_super(U)
S_tens_new = to_super(Unew)
assert_(S_tens_new == S_tens.permute([[2,1,0],[5,4,3]]))
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_super_tensor_property():
"""
Tensor: Super_tensor correctly tensors on underlying spaces.
"""
U1 = rand_unitary(3)
U2 = rand_unitary(5)
U = tensor(U1, U2)
S_tens = to_super(U)
S_supertens = super_tensor(to_super(U1), to_super(U2))
assert_(S_tens == S_supertens)
assert_equal(S_supertens.superrep, 'super')
def test_super_tensor_property():
"""
Tensor: Super_tensor correctly tensors on underlying spaces.
"""
U1 = rand_unitary(3)
U2 = rand_unitary(5)
U = tensor(U1, U2)
S_tens = to_super(U)
S_supertens = super_tensor(to_super(U1), to_super(U2))
assert_(S_tens == S_supertens)
assert_equal(S_supertens.superrep, 'super')
def test_dag_preserves_superrep():
"""
Checks that dag() preserves superrep.
"""
def case(qobj):
orig_superrep = qobj.superrep
assert_equal(qobj.dag().superrep, orig_superrep)
for dim in (2, 4, 8):
qobj = rand_super_bcsz(dim)
yield case, to_super(qobj)
# These two shouldn't even do anything, since qobj
# is Hermicity-preserving.
yield case, to_choi(qobj)
yield case, to_chi(qobj)
def test_SuperPreservesSelf(self):
"""
Superoperator: to_super(q) returns q if q is already a
supermatrix.
"""
superop = rand_super()
assert_(superop is to_super(superop))
def test_SuperPreservesSelf(self):
"""
Superoperator: to_super(q) returns q if q is already a
supermatrix.
"""
superop = rand_super()
assert_(superop is to_super(superop))
def test_composite_oper():
"""
Composite: Tests compositing unitaries and superoperators.
"""
U1 = rand_unitary(3)
U2 = rand_unitary(5)
S1 = to_super(U1)
S2 = to_super(U2)
S3 = rand_super(4)
S4 = rand_super(7)
assert_(composite(U1, U2) == tensor(U1, U2))
assert_(composite(S3, S4) == super_tensor(S3, S4))
assert_(composite(U1, S4) == super_tensor(S1, S4))
assert_(composite(S3, U2) == super_tensor(S3, S2))
def test_SuperPreservesSelf(self, superoperator):
"""
Superoperator: to_super(q) returns q if q is already a
supermatrix.
"""
assert superoperator is to_super(superoperator)
def test_composite_oper():
"""
Composite: Tests compositing unitaries and superoperators.
"""
U1 = rand_unitary(3)
U2 = rand_unitary(5)
S1 = to_super(U1)
S2 = to_super(U2)
S3 = rand_super(4)
S4 = rand_super(7)
assert_(composite(U1, U2) == tensor(U1, U2))
assert_(composite(S3, S4) == super_tensor(S3, S4))
assert_(composite(U1, S4) == super_tensor(S1, S4))
assert_(composite(S3, U2) == super_tensor(S3, S2))
def test_CheckMulType():
"Qobj multiplication type"
# ket-bra and bra-ket multiplication
psi = basis(5)
dm = psi * psi.dag()
assert_(dm.isoper)
assert_(dm.isherm)
nrm = psi.dag() * psi
assert_equal(np.prod(nrm.shape), 1)
assert_((abs(nrm) == 1)[0, 0])
# operator-operator multiplication
H1 = rand_herm(3)
H2 = rand_herm(3)
out = H1 * H2
assert_(out.isoper)
out = H1 * H1
assert_(out.isoper)
assert_(out.isherm)
out = H2 * H2
assert_(out.isoper)
assert_(out.isherm)
U = rand_unitary(5)
out = U.dag() * U
assert_(out.isoper)
assert_(out.isherm)
N = num(5)
out = N * N
assert_(out.isoper)
assert_(out.isherm)
# operator-ket and bra-operator multiplication
op = sigmax()
ket1 = basis(2)
ket2 = op * ket1
assert_(ket2.isket)
bra1 = basis(2).dag()
bra2 = bra1 * op
assert_(bra2.isbra)
assert_(bra2.dag() == ket2)
# superoperator-operket and operbra-superoperator multiplication
sop = to_super(sigmax())
opket1 = operator_to_vector(fock_dm(2))
opket2 = sop * opket1
assert(opket2.isoperket)
opbra1 = operator_to_vector(fock_dm(2)).dag()
opbra2 = opbra1 * sop
assert(opbra2.isoperbra)
assert_(opbra2.dag() == opket2)
def test_CheckMulType():
"Qobj multiplication type"
# ket-bra and bra-ket multiplication
psi = basis(5)
dm = psi * psi.dag()
assert_(dm.isoper)
assert_(dm.isherm)
nrm = psi.dag() * psi
assert_equal(np.prod(nrm.shape), 1)
assert_((abs(nrm) == 1)[0, 0])
# operator-operator multiplication
H1 = rand_herm(3)
H2 = rand_herm(3)
out = H1 * H2
assert_(out.isoper)
out = H1 * H1
assert_(out.isoper)
assert_(out.isherm)
out = H2 * H2
assert_(out.isoper)
assert_(out.isherm)
U = rand_unitary(5)
out = U.dag() * U
assert_(out.isoper)
assert_(out.isherm)
N = num(5)
out = N * N
assert_(out.isoper)
assert_(out.isherm)
# operator-ket and bra-operator multiplication
op = sigmax()
ket1 = basis(2)
ket2 = op * ket1
assert_(ket2.isket)
bra1 = basis(2).dag()
bra2 = bra1 * op
assert_(bra2.isbra)
assert_(bra2.dag() == ket2)
# superoperator-operket and operbra-superoperator multiplication
sop = to_super(sigmax())
opket1 = operator_to_vector(fock_dm(2))
opket2 = sop * opket1
assert(opket2.isoperket)
opbra1 = operator_to_vector(fock_dm(2)).dag()
opbra2 = opbra1 * sop
assert(opbra2.isoperbra)
assert_(opbra2.dag() == opket2)
def test_average_gate_fidelity_target():
"""
Metrics: Tests that for random unitaries U, AGF(U, U) = 1.
"""
for _ in range(10):
U = rand_unitary_haar(13)
SU = to_super(U)
assert_almost_equal(average_gate_fidelity(SU, target=U), 1)
def test_average_gate_fidelity_target():
"""
Metrics: Tests that for random unitaries U, AGF(U, U) = 1.
"""
for _ in range(10):
U = rand_unitary_haar(13)
SU = to_super(U)
assert_almost_equal(average_gate_fidelity(SU, target=U), 1)
def test_CheckMulType():
"Qobj multiplication type"
# ket-bra and bra-ket multiplication
psi = basis(5)
dm = psi * psi.dag()
assert dm.isoper
assert dm.isherm
nrm = psi.dag() * psi
assert np.prod(nrm.shape) == 1
assert abs(nrm)[0, 0] == 1
# operator-operator multiplication
H1 = rand_herm(3)
H2 = rand_herm(3)
out = H1 * H2
assert out.isoper
out = H1 * H1
assert out.isoper
assert out.isherm
out = H2 * H2
assert out.isoper
assert out.isherm
U = rand_unitary(5)
out = U.dag() * U
assert out.isoper
assert out.isherm
N = num(5)
out = N * N
assert out.isoper
assert out.isherm
# operator-ket and bra-operator multiplication
op = sigmax()
ket1 = basis(2)
ket2 = op * ket1
assert ket2.isket
bra1 = basis(2).dag()
bra2 = bra1 * op
assert bra2.isbra
assert bra2.dag() == ket2
# superoperator-operket and operbra-superoperator multiplication
sop = to_super(sigmax())
opket1 = operator_to_vector(fock_dm(2))
opket2 = sop * opket1
assert opket2.isoperket
opbra1 = operator_to_vector(fock_dm(2)).dag()
opbra2 = opbra1 * sop
assert opbra2.isoperbra
assert opbra2.dag() == opket2
def test_tensor_swap_other():
dims = (2, 3, 4, 5, 7)
for dim in dims:
S = to_super(rand_super_bcsz(dim))
# Swapping the inner indices on a superoperator should give a Choi matrix.
J = to_choi(S)
case_tensor_swap(S, [(1, 2)], [[[dim], [dim]], [[dim], [dim]]], J)
def test_tensor_swap_other():
dims = (2, 3, 4, 5, 7)
for dim in dims:
S = to_super(rand_super_bcsz(dim))
# Swapping the inner indices on a superoperator should give a Choi matrix.
J = to_choi(S)
case_tensor_swap(S, [(1, 2)], [[[dim], [dim]], [[dim], [dim]]], J)
def case(map, state):
S = to_super(map)
A, B = to_stinespring(map)
q1 = vector_to_operator(S * operator_to_vector(state))
# FIXME: problem if Kraus index is implicitly
# ptraced!
q2 = (A * state * B.dag()).ptrace((0, ))
assert_((q1 - q2).norm('tr') <= thresh)
def test_tensor_contract_ident():
qobj = identity([2, 3, 4])
ans = 3 * identity([2, 4])
assert_(ans == tensor_contract(qobj, (1, 4)))
# Now try for superoperators.
# For now, we just ensure the dims are correct.
sqobj = to_super(qobj)
correct_dims = [[[2, 4], [2, 4]], [[2, 4], [2, 4]]]
assert_equal(correct_dims, tensor_contract(sqobj, (1, 4), (7, 10)).dims)
def case(map, state):
S = to_super(map)
A, B = to_stinespring(map)
q1 = vector_to_operator(
S * operator_to_vector(state)
)
# FIXME: problem if Kraus index is implicitly
# ptraced!
q2 = (A * state * B.dag()).ptrace((0,))
assert_((q1 - q2).norm('tr') <= thresh)
def test_SuperChoiSuper(self, superoperator):
"""
Superoperator: Converting superoperator to Choi matrix and back.
"""
choi_matrix = to_choi(superoperator)
test_supe = to_super(choi_matrix)
# Assert both that the result is close to expected, and has the right
# type.
assert (test_supe - superoperator).norm() < tol
assert choi_matrix.type == "super" and choi_matrix.superrep == "choi"
assert test_supe.type == "super" and test_supe.superrep == "super"
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_SuperChoiSuper(self):
"""
Superoperator: Converting superoperator to Choi matrix and back.
"""
superoperator = rand_super()
choi_matrix = to_choi(superoperator)
test_supe = to_super(choi_matrix)
# Assert both that the result is close to expected, and has the right
# type.
assert_((test_supe - superoperator).norm() < tol)
assert_(choi_matrix.type == "super" and choi_matrix.superrep == "choi")
assert_(test_supe.type == "super" and test_supe.superrep == "super")
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_dag_preserves_superrep():
"""
Checks that dag() preserves superrep.
"""
def case(qobj):
orig_superrep = qobj.superrep
assert_equal(qobj.dag().superrep, orig_superrep)
for dim in (2, 4, 8):
qobj = rand_super_bcsz(dim)
yield case, to_super(qobj)
# These two shouldn't even do anything, since qobj
# is Hermicity-preserving.
yield case, to_choi(qobj)
yield case, to_chi(qobj)
def test_SuperChoiChiSuper(self):
"""
Superoperator: Converting two-qubit superoperator through
Choi and chi representations goes back to right superoperator.
"""
superoperator = super_tensor(rand_super(2), rand_super(2))
choi_matrix = to_choi(superoperator)
chi_matrix = to_chi(choi_matrix)
test_supe = to_super(chi_matrix)
# Assert both that the result is close to expected, and has the right
# type.
assert_((test_supe - superoperator).norm() < tol)
assert_(choi_matrix.type == "super" and choi_matrix.superrep == "choi")
assert_(chi_matrix.type == "super" and chi_matrix.superrep == "chi")
assert_(test_supe.type == "super" and test_supe.superrep == "super")
def test_stinespring_agrees(self, dimension):
"""
Stinespring: Partial Tr over pair agrees w/ supermatrix.
"""
map = rand_super_bcsz(dimension)
state = rand_dm_ginibre(dimension)
S = to_super(map)
A, B = to_stinespring(map)
q1 = vector_to_operator(S * operator_to_vector(state))
# FIXME: problem if Kraus index is implicitly
# ptraced!
q2 = (A * state * B.dag()).ptrace((0, ))
assert (q1 - q2).norm('tr') <= tol
def test_SuperChoiChiSuper(self):
"""
Superoperator: Converting two-qubit superoperator through
Choi and chi representations goes back to right superoperator.
"""
superoperator = super_tensor(rand_super(2), rand_super(2))
choi_matrix = to_choi(superoperator)
chi_matrix = to_chi(choi_matrix)
test_supe = to_super(chi_matrix)
# Assert both that the result is close to expected, and has the right
# type.
assert_((test_supe - superoperator).norm() < tol)
assert_(choi_matrix.type == "super" and choi_matrix.superrep == "choi")
assert_(chi_matrix.type == "super" and chi_matrix.superrep == "chi")
assert_(test_supe.type == "super" and test_supe.superrep == "super")
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() < 1e-8)
assert_((op2[0] - kraus).norm() < 1e-8)
assert_((op3 - super).norm() < 1e-8)
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 hinton(rho, xlabels=None, ylabels=None, title=None, ax=None, cmap=None,
label_top=True):
"""Draws a Hinton diagram for visualizing a density matrix or superoperator.
Parameters
----------
rho : qobj
Input density matrix or superoperator.
xlabels : list of strings or False
list of x labels
ylabels : list of strings or False
list of y labels
title : string
title of the plot (optional)
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
cmap : a matplotlib colormap instance
Color map to use when plotting.
label_top : bool
If True, x-axis labels will be placed on top, otherwise
they will appear below the plot.
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
Raises
------
ValueError
Input argument is not a quantum object.
"""
# Apply default colormaps.
# TODO: abstract this away into something that makes default
# colormaps.
cmap = (
(cm.Greys_r if settings.colorblind_safe else cm.RdBu)
if cmap is None else cmap
)
# Extract plotting data W from the input.
if isinstance(rho, Qobj):
if rho.isoper:
W = rho.full()
# Create default labels if none are given.
if xlabels is None or ylabels is None:
labels = _cb_labels(rho.dims[0])
xlabels = xlabels if xlabels is not None else list(labels[0])
ylabels = ylabels if ylabels is not None else list(labels[1])
elif rho.isoperket:
W = vector_to_operator(rho).full()
elif rho.isoperbra:
W = vector_to_operator(rho.dag()).full()
elif rho.issuper:
if not _isqubitdims(rho.dims):
raise ValueError("Hinton plots of superoperators are "
"currently only supported for qubits.")
# Convert to a superoperator in the Pauli basis,
# so that all the elements are real.
sqobj = to_super(rho)
nq = int(np.log2(sqobj.shape[0]) / 2)
B = _pauli_basis(nq) / np.sqrt(2**nq)
# To do this, we have to hack a bit and force the dims to match,
# since the _pauli_basis function makes different assumptions
# about indices than we need here.
B.dims = sqobj.dims
sqobj = B.dag() * sqobj * B
W = sqobj.full().T
# Create default labels, too.
if (xlabels is None) or (ylabels is None):
labels = list(map("".join, it.product("IXYZ", repeat=nq)))
xlabels = xlabels if xlabels is not None else labels
ylabels = ylabels if ylabels is not None else labels
else:
raise ValueError(
"Input quantum object must be an operator or superoperator."
)
else:
W = rho
if ax is None:
fig, ax = plt.subplots(1, 1, figsize=(8, 6))
else:
fig = None
if not (xlabels or ylabels):
ax.axis('off')
ax.axis('equal')
ax.set_frame_on(False)
height, width = W.shape
w_max = 1.25 * max(abs(np.diag(np.matrix(W))))
if w_max <= 0.0:
w_max = 1.0
ax.fill(array([0, width, width, 0]), array([0, 0, height, height]),
color=cmap(128))
for x in range(width):
for y in range(height):
_x = x + 1
_y = y + 1
if np.real(W[x, y]) > 0.0:
_blob(_x - 0.5, height - _y + 0.5, abs(W[x,
y]), w_max, min(1, abs(W[x, y]) / w_max), cmap=cmap)
else:
_blob(_x - 0.5, height - _y + 0.5, -abs(W[
x, y]), w_max, min(1, abs(W[x, y]) / w_max), cmap=cmap)
# color axis
norm = mpl.colors.Normalize(-abs(W).max(), abs(W).max())
cax, kw = mpl.colorbar.make_axes(ax, shrink=0.75, pad=.1)
mpl.colorbar.ColorbarBase(cax, norm=norm, cmap=cmap)
# x axis
ax.xaxis.set_major_locator(plt.IndexLocator(1, 0.5))
if xlabels:
ax.set_xticklabels(xlabels)
if label_top:
ax.xaxis.tick_top()
ax.tick_params(axis='x', labelsize=14)
# y axis
ax.yaxis.set_major_locator(plt.IndexLocator(1, 0.5))
if ylabels:
ax.set_yticklabels(list(reversed(ylabels)))
ax.tick_params(axis='y', labelsize=14)
return fig, ax
def rand_super_bcsz(N=2, enforce_tp=True, rank=None, dims=None):
"""
Returns a random superoperator drawn from the Bruzda
et al ensemble for CPTP maps [BCSZ08]_. Note that due to
finite numerical precision, for ranks less than full-rank,
zero eigenvalues may become slightly negative, such that the
returned operator is not actually completely positive.
Parameters
----------
N : int
Square root of the dimension of the superoperator to be returned.
enforce_tp : bool
If True, the trace-preserving condition of [BCSZ08]_ is enforced;
otherwise only complete positivity is enforced.
rank : int or None
Rank of the sampled superoperator. If None, a full-rank
superoperator is generated.
dims : list
Dimensions of quantum object. Used for specifying
tensor structure. Default is dims=[[[N],[N]], [[N],[N]]].
Returns
-------
rho : Qobj
A superoperator acting on vectorized dim × dim density operators,
sampled from the BCSZ distribution.
"""
if dims is not None:
# TODO: check!
pass
else:
dims = [[[N],[N]], [[N],[N]]]
if rank is None:
rank = N**2
if rank > N**2:
raise ValueError("Rank cannot exceed superoperator dimension.")
# We use mainly dense matrices here for speed in low
# dimensions. In the future, it would likely be better to switch off
# between sparse and dense matrices as the dimension grows.
# We start with a Ginibre uniform matrix X of the appropriate rank,
# and use it to construct a positive semidefinite matrix X X⁺.
X = randnz((N**2, rank), norm='ginibre')
# Precompute X X⁺, as we'll need it in two different places.
XXdag = np.dot(X, X.T.conj())
if enforce_tp:
# We do the partial trace over the first index by using dense reshape
# operations, so that we can avoid bouncing to a sparse representation
# and back.
Y = np.einsum('ijik->jk', XXdag.reshape((N, N, N, N)))
# Now we have the matrix 𝟙 ⊗ Y^{-1/2}, which we can find by doing
# the square root and the inverse separately. As a possible improvement,
# iterative methods exist to find inverse square root matrices directly,
# as this is important in statistics.
Z = np.kron(
np.eye(N),
sqrtm(la.inv(Y))
)
# Finally, we dot everything together and pack it into a Qobj,
# marking the dimensions as that of a type=super (that is,
# with left and right compound indices, each representing
# left and right indices on the underlying Hilbert space).
D = Qobj(np.dot(Z, np.dot(XXdag, Z)))
else:
D = N * Qobj(XXdag / np.trace(XXdag))
D.dims = [
# Left dims
[[N], [N]],
# Right dims
[[N], [N]]
]
# Since [BCSZ08] gives a row-stacking Choi matrix, but QuTiP
# expects a column-stacking Choi matrix, we must permute the indices.
D = D.permute([[1], [0]])
D.dims = dims
# Mark that we've made a Choi matrix.
D.superrep = 'choi'
return sr.to_super(D)
def test_known_iscptp(self):
"""
Superoperator: ishp, iscp, istp and iscptp known cases.
"""
def case(qobj, shouldhp, shouldcp, shouldtp):
hp = qobj.ishp
cp = qobj.iscp
tp = qobj.istp
cptp = qobj.iscptp
shouldcptp = shouldcp and shouldtp
if (hp == shouldhp and cp == shouldcp and tp == shouldtp
and cptp == shouldcptp):
return
fails = []
if hp != shouldhp:
fails.append(("ishp", shouldhp, hp))
if tp != shouldtp:
fails.append(("istp", shouldtp, tp))
if cp != shouldcp:
fails.append(("iscp", shouldcp, cp))
if cptp != shouldcptp:
fails.append(("iscptp", shouldcptp, cptp))
raise AssertionError("Expected {}.".format(" and ".join([
"{} == {} (got {})".format(fail, expected, got)
for fail, expected, got in fails
])))
# Conjugation by a creation operator should
# have be CP (and hence HP), but not TP.
a = create(2).dag()
S = sprepost(a, a.dag())
case(S, True, True, False)
# A single off-diagonal element should not be CP,
# nor even HP.
S = sprepost(a, a)
case(S, False, False, False)
# Check that unitaries are CPTP and HP.
case(identity(2), True, True, True)
case(sigmax(), True, True, True)
# Check that unitaries on bipartite systems are CPTP and HP.
case(tensor(sigmax(), identity(2)), True, True, True)
# Check that a linear combination of bipartitie unitaries is CPTP and HP.
S = (to_super(tensor(sigmax(), identity(2))) +
to_super(tensor(identity(2), sigmay()))) / 2
case(S, True, True, True)
# The partial transpose map, whose Choi matrix is SWAP, is TP
# and HP but not CP (one negative eigenvalue).
W = Qobj(swap(), type='super', superrep='choi')
case(W, True, False, True)
# Subnormalized maps (representing erasure channels, for instance)
# can be CP but not TP.
subnorm_map = Qobj(identity(4) * 0.9, type='super', superrep='super')
case(subnorm_map, True, True, False)
# Check that things which aren't even operators aren't identified as
# CPTP.
case(basis(2), False, False, False)
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 had_mixture(x):
id_chan = to_choi(qeye(2))
S_eye = to_super(id_chan)
S_H = to_super(hadamard_transform())
return (1 - x) * S_eye + x * S_H
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 hinton(rho, xlabels=None, ylabels=None, title=None, ax=None, cmap=None,
label_top=True):
"""Draws a Hinton diagram for visualizing a density matrix or superoperator.
Parameters
----------
rho : qobj
Input density matrix or superoperator.
xlabels : list of strings or False
list of x labels
ylabels : list of strings or False
list of y labels
title : string
title of the plot (optional)
ax : a matplotlib axes instance
The axes context in which the plot will be drawn.
cmap : a matplotlib colormap instance
Color map to use when plotting.
label_top : bool
If True, x-axis labels will be placed on top, otherwise
they will appear below the plot.
Returns
-------
fig, ax : tuple
A tuple of the matplotlib figure and axes instances used to produce
the figure.
Raises
------
ValueError
Input argument is not a quantum object.
"""
# Apply default colormaps.
# TODO: abstract this away into something that makes default
# colormaps.
cmap = (
(cm.Greys_r if settings.colorblind_safe else cm.RdBu)
if cmap is None else cmap
)
# Extract plotting data W from the input.
if isinstance(rho, Qobj):
if rho.isoper:
W = rho.full()
# Create default labels if none are given.
if xlabels is None or ylabels is None:
labels = _cb_labels(rho.dims[0])
xlabels = xlabels if xlabels is not None else list(labels[0])
ylabels = ylabels if ylabels is not None else list(labels[1])
elif rho.isoperket:
W = vector_to_operator(rho).full()
elif rho.isoperbra:
W = vector_to_operator(rho.dag()).full()
elif rho.issuper:
if not _isqubitdims(rho.dims):
raise ValueError("Hinton plots of superoperators are "
"currently only supported for qubits.")
# Convert to a superoperator in the Pauli basis,
# so that all the elements are real.
sqobj = to_super(rho)
nq = int(np.log2(sqobj.shape[0]) / 2)
B = _pauli_basis(nq) / np.sqrt(2**nq)
# To do this, we have to hack a bit and force the dims to match,
# since the _pauli_basis function makes different assumptions
# about indices than we need here.
B.dims = sqobj.dims
sqobj = B.dag() * sqobj * B
W = sqobj.full()
# Create default labels, too.
if (xlabels is None) or (ylabels is None):
labels = list(map("".join, it.product("IXYZ", repeat=nq)))
xlabels = xlabels if xlabels is not None else labels
ylabels = ylabels if ylabels is not None else labels
else:
raise ValueError(
"Input quantum object must be an operator or superoperator."
)
else:
W = rho
if ax is None:
fig, ax = plt.subplots(1, 1, figsize=(8, 6))
else:
fig = None
if not (xlabels or ylabels):
ax.axis('off')
ax.axis('equal')
ax.set_frame_on(False)
height, width = W.shape
w_max = 1.25 * max(abs(np.diag(np.matrix(W))))
if w_max <= 0.0:
w_max = 1.0
ax.fill(array([0, width, width, 0]), array([0, 0, height, height]),
color=cmap(128))
for x in range(width):
for y in range(height):
_x = x + 1
_y = y + 1
if np.real(W[x, y]) > 0.0:
_blob(_x - 0.5, height - _y + 0.5, abs(W[x,
y]), w_max, min(1, abs(W[x, y]) / w_max), cmap=cmap)
else:
_blob(_x - 0.5, height - _y + 0.5, -abs(W[
x, y]), w_max, min(1, abs(W[x, y]) / w_max), cmap=cmap)
# color axis
norm = mpl.colors.Normalize(-abs(W).max(), abs(W).max())
cax, kw = mpl.colorbar.make_axes(ax, shrink=0.75, pad=.1)
mpl.colorbar.ColorbarBase(cax, norm=norm, cmap=cmap)
# x axis
ax.xaxis.set_major_locator(plt.IndexLocator(1, 0.5))
if xlabels:
ax.set_xticklabels(xlabels)
if label_top:
ax.xaxis.tick_top()
ax.tick_params(axis='x', labelsize=14)
# y axis
ax.yaxis.set_major_locator(plt.IndexLocator(1, 0.5))
if ylabels:
ax.set_yticklabels(list(reversed(ylabels)))
ax.tick_params(axis='y', labelsize=14)
return fig, ax
def rand_super_bcsz(N=2, enforce_tp=True, rank=None, dims=None):
"""
Returns a random superoperator drawn from the Bruzda
et al ensemble for CPTP maps [BCSZ08]_. Note that due to
finite numerical precision, for ranks less than full-rank,
zero eigenvalues may become slightly negative, such that the
returned operator is not actually completely positive.
Parameters
----------
N : int
Square root of the dimension of the superoperator to be returned.
enforce_tp : bool
If True, the trace-preserving condition of [BCSZ08]_ is enforced;
otherwise only complete positivity is enforced.
rank : int or None
Rank of the sampled superoperator. If None, a full-rank
superoperator is generated.
dims : list
Dimensions of quantum object. Used for specifying
tensor structure. Default is dims=[[[N],[N]], [[N],[N]]].
Returns
-------
rho : Qobj
A superoperator acting on vectorized dim × dim density operators,
sampled from the BCSZ distribution.
"""
if dims is not None:
# TODO: check!
pass
else:
dims = [[[N], [N]], [[N], [N]]]
if rank is None:
rank = N ** 2
if rank > N ** 2:
raise ValueError("Rank cannot exceed superoperator dimension.")
# We use mainly dense matrices here for speed in low
# dimensions. In the future, it would likely be better to switch off
# between sparse and dense matrices as the dimension grows.
# We start with a Ginibre uniform matrix X of the appropriate rank,
# and use it to construct a positive semidefinite matrix X X⁺.
X = randnz((N ** 2, rank), norm="ginibre")
# Precompute X X⁺, as we'll need it in two different places.
XXdag = np.dot(X, X.T.conj())
if enforce_tp:
# We do the partial trace over the first index by using dense reshape
# operations, so that we can avoid bouncing to a sparse representation
# and back.
Y = np.einsum("ijik->jk", XXdag.reshape((N, N, N, N)))
# Now we have the matrix 𝟙 ⊗ Y^{-1/2}, which we can find by doing
# the square root and the inverse separately. As a possible improvement,
# iterative methods exist to find inverse square root matrices directly,
# as this is important in statistics.
Z = np.kron(np.eye(N), sqrtm(la.inv(Y)))
# Finally, we dot everything together and pack it into a Qobj,
# marking the dimensions as that of a type=super (that is,
# with left and right compound indices, each representing
# left and right indices on the underlying Hilbert space).
D = Qobj(np.dot(Z, np.dot(XXdag, Z)))
else:
D = N * Qobj(XXdag / np.trace(XXdag))
D.dims = [
# Left dims
[[N], [N]],
# Right dims
[[N], [N]],
]
# Since [BCSZ08] gives a row-stacking Choi matrix, but QuTiP
# expects a column-stacking Choi matrix, we must permute the indices.
D = D.permute([[1], [0]])
D.dims = dims
# Mark that we've made a Choi matrix.
D.superrep = "choi"
return sr.to_super(D)
def overrotation(x):
return to_super((1j * np.pi * x * sigmax() / 2).expm())
def overrotation(x):
return to_super((1j * np.pi * x * sigmax() / 2).expm())
class TestSuperopReps:
"""
A test class for the QuTiP function for applying superoperators to
subsystems.
"""
def test_SuperChoiSuper(self, superoperator):
"""
Superoperator: Converting superoperator to Choi matrix and back.
"""
choi_matrix = to_choi(superoperator)
test_supe = to_super(choi_matrix)
# Assert both that the result is close to expected, and has the right
# type.
assert (test_supe - superoperator).norm() < tol
assert choi_matrix.type == "super" and choi_matrix.superrep == "choi"
assert test_supe.type == "super" and test_supe.superrep == "super"
@pytest.mark.parametrize('dimension', [2, 4])
def test_SuperChoiChiSuper(self, dimension):
"""
Superoperator: Converting two-qubit superoperator through
Choi and chi representations goes back to right superoperator.
"""
superoperator = super_tensor(
rand_super(dimension),
rand_super(dimension),
)
choi_matrix = to_choi(superoperator)
chi_matrix = to_chi(choi_matrix)
test_supe = to_super(chi_matrix)
# Assert both that the result is close to expected, and has the right
# type.
assert (test_supe - superoperator).norm() < tol
assert choi_matrix.type == "super" and choi_matrix.superrep == "choi"
assert chi_matrix.type == "super" and chi_matrix.superrep == "chi"
assert test_supe.type == "super" and test_supe.superrep == "super"
def test_ChoiKrausChoi(self, superoperator):
"""
Superoperator: Convert superoperator to Choi matrix and back.
"""
choi_matrix = to_choi(superoperator)
kraus_ops = to_kraus(choi_matrix)
test_choi = kraus_to_choi(kraus_ops)
# Assert both that the result is close to expected, and has the right
# type.
assert (test_choi - choi_matrix).norm() < tol
assert choi_matrix.type == "super" and choi_matrix.superrep == "choi"
assert test_choi.type == "super" and test_choi.superrep == "choi"
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 test_NeglectSmallKraus(self):
"""
Superoperator: Convert Kraus to Choi matrix and back. Neglect tiny
Kraus operators.
"""
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())
# 1 non-zero Kraus operator the rest are zero
sixteen_kraus_ops = to_kraus(super, tol=0.0)
# default is tol=1e-9
one_kraus_op = to_kraus(super)
assert len(sixteen_kraus_ops) == 16 and len(one_kraus_op) == 1
assert (one_kraus_op[0] - kraus).norm() < tol
def test_SuperPreservesSelf(self, superoperator):
"""
Superoperator: to_super(q) returns q if q is already a
supermatrix.
"""
assert superoperator is to_super(superoperator)
def test_ChoiPreservesSelf(self, superoperator):
"""
Superoperator: to_choi(q) returns q if q is already Choi.
"""
choi = to_choi(superoperator)
assert choi is to_choi(choi)
def test_random_iscptp(self, superoperator):
"""
Superoperator: Randomly generated superoperators are
correctly reported as CPTP and HP.
"""
assert superoperator.iscptp
assert superoperator.ishp
# Conjugation by a creation operator
a = create(2).dag()
S = sprepost(a, a.dag())
# A single off-diagonal element
S_ = sprepost(a, a)
# Check that a linear combination of bipartite unitaries is CPTP and HP.
S_U = (to_super(tensor(sigmax(), identity(2))) +
to_super(tensor(identity(2), sigmay()))) / 2
# The partial transpose map, whose Choi matrix is SWAP
ptr_swap = Qobj(swap(), type='super', superrep='choi')
# Subnormalized maps (representing erasure channels, for instance)
subnorm_map = Qobj(identity(4) * 0.9, type='super', superrep='super')
@pytest.mark.parametrize(['qobj', 'shouldhp', 'shouldcp', 'shouldtp'], [
pytest.param(S, True, True, False, id="conjugatio by create op"),
pytest.param(S_, False, False, False, id="single off-diag"),
pytest.param(identity(2), True, True, True, id="Identity"),
pytest.param(sigmax(), True, True, True, id="Pauli X"),
pytest.param(
tensor(sigmax(), identity(2)),
True,
True,
True,
id="bipartite system",
),
pytest.param(
S_U,
True,
True,
True,
id="linear combination of bip. unitaries",
),
pytest.param(ptr_swap, True, False, True, id="partial transpose map"),
pytest.param(subnorm_map, True, True, False, id="subnorm map"),
pytest.param(basis(2), False, False, False, id="not an operator"),
])
def test_known_iscptp(self, qobj, shouldhp, shouldcp, shouldtp):
"""
Superoperator: ishp, iscp, istp and iscptp known cases.
"""
assert qobj.ishp == shouldhp
assert qobj.iscp == shouldcp
assert qobj.istp == shouldtp
assert qobj.iscptp == (shouldcp and shouldtp)
def test_choi_tr(self, dimension):
"""
Superoperator: Trace returned by to_choi matches docstring.
"""
assert abs(to_choi(identity(dimension)).tr() - dimension) <= tol
def test_stinespring_cp(self, dimension):
"""
Stinespring: A and B match for CP maps.
"""
superop = rand_super_bcsz(dimension)
A, B = to_stinespring(superop)
assert norm(A - B) < tol
@pytest.mark.repeat(3)
def test_stinespring_agrees(self, dimension):
"""
Stinespring: Partial Tr over pair agrees w/ supermatrix.
"""
map = rand_super_bcsz(dimension)
state = rand_dm_ginibre(dimension)
S = to_super(map)
A, B = to_stinespring(map)
q1 = vector_to_operator(S * operator_to_vector(state))
# FIXME: problem if Kraus index is implicitly
# ptraced!
q2 = (A * state * B.dag()).ptrace((0, ))
assert (q1 - q2).norm('tr') <= tol
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]]
@pytest.mark.parametrize('dimension', [2, 4, 8])
def test_chi_choi_roundtrip(self, dimension):
superop = rand_super_bcsz(dimension)
superop = to_chi(superop)
rt_superop = to_chi(to_choi(superop))
dif = norm(rt_superop - superop)
assert dif == pytest.approx(0, abs=1e-7)
assert rt_superop.type == superop.type
assert rt_superop.dims == superop.dims
chi_sigmax = [[0, 0, 0, 0], [0, 4, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
chi_diag2 = [[4, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
rotX_pi_4 = (-1j * sigmax() * pi / 4).expm()
chi_rotX_pi_4 = [[2, 2j, 0, 0], [-2j, 2, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
@pytest.mark.parametrize(['superop', 'chi_expected'], [
pytest.param(sigmax(), chi_sigmax),
pytest.param(to_super(sigmax()), chi_sigmax),
pytest.param(qeye(2), chi_diag2),
pytest.param(rotX_pi_4, chi_rotX_pi_4)
])
def test_chi_known(self, superop, chi_expected):
"""
Superoperator: Chi-matrix for known cases is correct.
"""
chi_actual = to_chi(superop)
chiq = Qobj(
chi_expected,
dims=[[[2], [2]], [[2], [2]]],
superrep='chi',
)
assert (chi_actual - chiq).norm() < tol
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 dnorm(A, B=None, solver="CVXOPT", verbose=False, force_solve=False,
sparse=True):
"""
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.
sparse : bool
Whether to use sparse matrices in the convex optimisation problem.
Default True.
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])
J_dat = J.data
if not sparse:
# The parameters and constraints only depend on the dimension, so
# we can cache them efficiently.
problem, Jr, Ji = dnorm_problem(dim)
# Load the parameters with the Choi matrix passed in.
Jr.value = sp.csr_matrix((J_dat.data.real, J_dat.indices,
J_dat.indptr),
shape=J_dat.shape).toarray()
Ji.value = sp.csr_matrix((J_dat.data.imag, J_dat.indices,
J_dat.indptr),
shape=J_dat.shape).toarray()
else:
# The parameters do not depend solely on the dimension,
# so we can not cache them efficiently.
problem = dnorm_sparse_problem(dim, J_dat)
problem.solve(solver=solver, verbose=verbose)
return problem.value
def test_known_iscptp(self):
"""
Superoperator: ishp, iscp, istp and iscptp known cases.
"""
def case(qobj, shouldhp, shouldcp, shouldtp):
hp = qobj.ishp
cp = qobj.iscp
tp = qobj.istp
cptp = qobj.iscptp
shouldcptp = shouldcp and shouldtp
if (
hp == shouldhp and
cp == shouldcp and
tp == shouldtp and
cptp == shouldcptp
):
return
fails = []
if hp != shouldhp:
fails.append(("ishp", shouldhp, hp))
if tp != shouldtp:
fails.append(("istp", shouldtp, tp))
if cp != shouldcp:
fails.append(("iscp", shouldcp, cp))
if cptp != shouldcptp:
fails.append(("iscptp", shouldcptp, cptp))
raise AssertionError("Expected {}.".format(" and ".join([
"{} == {} (got {})".format(fail, expected, got)
for fail, expected, got in fails
])))
# Conjugation by a creation operator should
# have be CP (and hence HP), but not TP.
a = create(2).dag()
S = sprepost(a, a.dag())
case(S, True, True, False)
# A single off-diagonal element should not be CP,
# nor even HP.
S = sprepost(a, a)
case(S, False, False, False)
# Check that unitaries are CPTP and HP.
case(identity(2), True, True, True)
case(sigmax(), True, True, True)
# Check that unitaries on bipartite systems are CPTP and HP.
case(tensor(sigmax(), identity(2)), True, True, True)
# Check that a linear combination of bipartitie unitaries is CPTP and HP.
S = (
to_super(tensor(sigmax(), identity(2))) + to_super(tensor(identity(2), sigmay()))
) / 2
case(S, True, True, True)
# The partial transpose map, whose Choi matrix is SWAP, is TP
# and HP but not CP (one negative eigenvalue).
W = Qobj(swap(), type='super', superrep='choi')
case(W, True, False, True)
# Subnormalized maps (representing erasure channels, for instance)
# can be CP but not TP.
subnorm_map = Qobj(identity(4) * 0.9, type='super', superrep='super')
case(subnorm_map, True, True, False)
# Check that things which aren't even operators aren't identified as
# CPTP.
case(basis(2), False, False, False)
