def test_local_pbasis_default_densitymatrix(self):
"""Test default states kwarg"""
default_states = [qi.random_density_matrix(2, seed=30 + i) for i in range(3)]
basis = LocalPreparationBasis("fitter_basis", default_states=default_states)
for i, state in enumerate(default_states):
basis_state = qi.DensityMatrix(basis.matrix([i], [0]))
fid = qi.state_fidelity(state, basis_state)
self.assertTrue(isclose(fid, 1))
def test_local_pbasis_qubit_states_no_default(self):
"""Test matrix method raises for invalid qubit with no default states"""
size = 2
qubits = [0, 2]
qubit_states = {
qubits[0]: [qi.random_density_matrix(2, seed=30 + i) for i in range(size)],
qubits[1]: [qi.random_statevector(2, seed=40 + i) for i in range(size)],
}
basis = LocalPreparationBasis("fitter_basis", qubit_states=qubit_states)
# No default states so should raise an exception
with self.assertRaises(ValueError):
basis.matrix([0, 0], [0, 1])
def test_local_pbasis_default_and_qubit_states(self):
"""Test qubit states kwarg"""
size = 3
qubits = [2, 0]
default_states = [qi.random_density_matrix(2, seed=20 + i) for i in range(size)]
qubit_states = {2: [qi.random_statevector(2, seed=40 + i) for i in range(size)]}
basis = LocalPreparationBasis(
"fitter_basis", default_states=default_states, qubit_states=qubit_states
)
# Check states
indices = it.product(range(size), repeat=2)
states0 = qubit_states[qubits[0]] if qubits[0] in qubit_states else default_states
states1 = qubit_states[qubits[1]] if qubits[1] in qubit_states else default_states
for index in indices:
basis_state = qi.DensityMatrix(basis.matrix(index, qubits))
target = qi.DensityMatrix(states0[index[0]]).expand(states1[index[1]])
fid = qi.state_fidelity(basis_state, target)
self.assertTrue(isclose(fid, 1))
def test_local_pbasis_qubit_states(self):
"""Test qubit states kwarg"""
size = 3
qubits = [0, 2]
qubit_states = {
qubits[0]: [qi.random_density_matrix(2, seed=30 + i) for i in range(size)],
qubits[1]: [qi.random_statevector(2, seed=40 + i) for i in range(size)],
}
basis = LocalPreparationBasis("fitter_basis", qubit_states=qubit_states)
# Check states
indices = it.product(range(size), repeat=2)
for index in indices:
basis_state = qi.DensityMatrix(basis.matrix(index, qubits))
target0 = qi.DensityMatrix(qubit_states[qubits[0]][index[0]])
target1 = qi.DensityMatrix(qubit_states[qubits[1]][index[1]])
target = target0.expand(target1)
fid = qi.state_fidelity(basis_state, target)
self.assertTrue(isclose(fid, 1))
def test_local_pbasis_no_inst(self):
"""Test circuits method raises if no instructions"""
default_states = [qi.random_statevector(2, seed=30 + i) for i in range(2)]
basis = LocalPreparationBasis("fitter_basis", default_states=default_states)
with self.assertRaises(NotImplementedError):
basis.circuit([0], [0])
def test_local_pbasis_unitary(self):
"""Test custom local measurement basis"""
size = 5
unitaries = [qi.random_unitary(2, seed=10 + i) for i in range(size)]
basis = LocalPreparationBasis("unitary_basis", unitaries)
self._test_ideal_basis(basis, [0, 1])
def test_local_pbasis_inst(self):
"""Test custom local measurement basis"""
basis = LocalPreparationBasis("custom_basis", [XGate(), YGate(), HGate()])
self._test_ideal_basis(basis, [0, 1])
def linear_inversion(
outcome_data: np.ndarray,
shot_data: np.ndarray,
measurement_data: np.ndarray,
preparation_data: np.ndarray,
measurement_basis: Optional[MeasurementBasis] = None,
preparation_basis: Optional[PreparationBasis] = None,
measurement_qubits: Optional[Tuple[int, ...]] = None,
preparation_qubits: Optional[Tuple[int, ...]] = None,
) -> Tuple[np.ndarray, Dict]:
r"""Linear inversion tomography fitter.
Overview
This fitter uses linear inversion to reconstructs the maximum-likelihood
estimate of the least-squares log-likelihood function
.. math::
\hat{\rho}
&= -\mbox{argmin }\log\mathcal{L}{\rho} \\
&= \mbox{argmin }\sum_i (\mbox{Tr}[E_j\rho] - \hat{p}_i)^2 \\
&= \mbox{argmin }\|Ax - y \|_2^2
where
* :math:`A = \sum_j |j \rangle\!\langle\!\langle E_j|` is the matrix of measured
basis elements.
* :math:`y = \sum_j \hat{p}_j |j\rangle` is the vector of estimated measurement
outcome probabilites for each basis element.
* :math:`x = |\rho\rangle\!\rangle` is the vectorized density matrix.
Additional Details
The linear inversion solution is given by
.. math::
\hat{\rho} = \sum_i \hat{p}_i D_i
where measurement probabilities :math:`\hat{p}_i = f_i / n_i` are estimated
from the observed count frequencies :math:`f_i` in :math:`n_i` shots for each
basis element :math:`i`, and :math:`D_i` is the *dual basis* element constructed
from basis :math:`\{E_i\}` via:
.. math:
|D_i\rangle\!\rangle = M^{-1}|E_i \rangle\!\rangle \\
M = \sum_j |E_j\rangle\!\rangle\!\langle\!\langle E_j|
.. note::
The Linear inversion fitter treats the input measurement and preparation
bases as local bases and constructs separate 1-qubit dual basis for each
individual qubit.
Linear inversion is only possible if the input bases are local and a spanning
set for the vector space of the reconstructed matrix
(*tomographically complete*). If the basis is not tomographically complete
the :func:`~qiskit_experiments.library.tomography.fitters.scipy_linear_lstsq`
or :func:`~qiskit_experiments.library.tomography.fitters.cvxpy_linear_lstsq`
function can be used to solve the same objective function via
least-squares optimization.
Args:
outcome_data: basis outcome frequency data.
shot_data: basis outcome total shot data.
measurement_data: measurement basis indice data.
preparation_data: preparation basis indice data.
measurement_basis: the tomography measurement basis.
preparation_basis: the tomography preparation basis.
measurement_qubits: Optional, the physical qubits that were measured.
If None they are assumed to be [0, ..., M-1] for
M measured qubits.
preparation_qubits: Optional, the physical qubits that were prepared.
If None they are assumed to be [0, ..., N-1] for
N preparated qubits.
Raises:
AnalysisError: If the fitted vector is not a square matrix
Returns:
The fitted matrix rho.
"""
# Construct dual bases
meas_dual_basis = None
if measurement_basis:
if not measurement_qubits:
measurement_qubits = tuple(range(measurement_data.shape[1]))
meas_duals = {
i: _dual_povms(measurement_basis, i)
for i in measurement_qubits
}
meas_dual_basis = LocalMeasurementBasis(
f"Dual_{measurement_basis.name}", qubit_povms=meas_duals)
prep_dual_basis = None
if preparation_basis:
if not preparation_qubits:
preparation_qubits = tuple(range(preparation_data.shape[1]))
prep_duals = {
i: _dual_states(preparation_basis, i)
for i in preparation_qubits
}
prep_dual_basis = LocalPreparationBasis(
f"Dual_{preparation_basis.name}", qubit_states=prep_duals)
if shot_data is None:
shot_data = np.ones(len(outcome_data))
# Construct linear inversion matrix
rho_fit = 0.0
for i, outcomes in enumerate(outcome_data):
shots = shot_data[i]
midx = measurement_data[i]
pidx = preparation_data[i]
# Get prep basis component
if prep_dual_basis:
# TODO: Add prep qubits
p_mat = np.transpose(
prep_dual_basis.matrix(pidx, preparation_qubits))
else:
p_mat = None
# Get probabilities and optional measurement basis component
for outcome, freq in enumerate(outcomes):
if freq == 0:
# Skip component with zero probability
continue
if meas_dual_basis:
# TODO: Add meas qubits
dual_op = meas_dual_basis.matrix(midx, outcome,
measurement_qubits)
if prep_dual_basis:
dual_op = np.kron(p_mat, dual_op)
else:
dual_op = p_mat
# Add component to linear inversion reconstruction
prob = freq / shots
rho_fit = rho_fit + prob * dual_op
return rho_fit, {}