def linear_inv_state_estimate(results: List[ExperimentResult],
qubits: List[int]) -> np.ndarray:
"""
Estimate a quantum state using linear inversion.
This is the simplest state tomography post processing. To use this function,
collect state tomography data with :py:func:`generate_state_tomography_experiment`
and :py:func:`~pyquil.operator_estimation.measure_observables`.
For more details on this post-processing technique,
see https://en.wikipedia.org/wiki/Quantum_tomography#Linear_inversion or
see section 3.4 of
[WOOD] Initialization and characterization of open quantum systems
C. Wood,
PhD thesis from University of Waterloo, (2015).
http://hdl.handle.net/10012/9557
:param results: A tomographically complete list of results.
:param qubits: All qubits that were tomographized. This specifies the order in
which qubits will be kron'ed together.
:return: A point estimate of the quantum state rho.
"""
measurement_matrix = np.vstack([
vec(pauli2matrix(result.setting.out_operator, qubits=qubits)).T.conj()
for result in results
])
expectations = np.array([result.expectation for result in results])
rho = pinv(measurement_matrix) @ expectations
return unvec(rho)
def _grad_cost(A, n, estimate, eps=1e-6):
"""
Computes the gradient of the cost, leveraging the vectorized equation 6 of [PGD] given in the
appendix.
:param A: a matrix constructed from the input states and POVM elements (eq. A1) that aids
in calculating the model probabilities p.
:param n: vectorized form of the observed counts n_ij
:param estimate: the current model Choi representation of an estimated process for which we
compute the gradient.
:return: Gradient of the cost of the estimate given the data, n
"""
p = A @ vec(estimate)
# see appendix on "stalling"
p = np.clip(p, a_min=eps, a_max=None)
eta = n / p
return unvec(-A.conj().T @ eta)
