def state_log_likelihood(state, results, qubits) -> float:
"""
The log Likelihood function used in the diluted MLE tomography routine.
Equation 2 of [DIMLE1]
:param state: The state (given as a density matrix) that we think we have.
:param results: Measured results from a state tomography experiment
:param qubits: Qubits that were tomographized.
:return: The log likelihood that our state is the one we believe it is.
"""
ll = 0
for res in results:
n = res.total_counts
op_matrix = pauli2matrix(res.setting.out_operator, qubits)
meas_exp = res.expectation
pred_exp = np.real(np.trace(op_matrix @ state))
for sign in [1, -1]:
f_j = n * (1 + sign * meas_exp) / 2
pr_j = (1 + sign * pred_exp) / 2
if pr_j <= 0:
continue
ll += f_j * np.log10(pr_j)
return ll
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 linear_inv_process_estimate(results: List[ExperimentResult], qubits: List[int]) -> np.ndarray:
"""
Estimate a quantum process using linear inversion.
This is the simplest process tomography post processing. To use this function,
collect process tomography data with :py:func:`generate_process_tomography_experiment`
and :py:func:`~forest.benchmarking.observable_estimation.estimate_observables`.
For more details on this post-processing technique,
see https://en.wikipedia.org/wiki/Quantum_tomography#Linear_inversion or
see section 3.5 of [WOOD]_
: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; the first qubit in the list is the left-most
tensor factor.
:return: A point estimate of the quantum process represented by a Choi matrix
"""
# state2matrix and pauli2matrix use pyquil tensor factor ordering where the least significant
# qubit, e.g. qubit 0, is the right-most tensor factor. We stick with the standard convention
# here that the first qubit in the list is the left-most tensor factor, so we have to reverse
# the qubits before passing to state2matrix and pauli2matrix
qs = qubits[::-1]
measurement_matrix = np.vstack([
vec(np.kron(state2matrix(result.setting.in_state, qs).conj(),
pauli2matrix(result.setting.observable, qs))).conj().T
for result in results
])
expectations = np.array([result.expectation for result in results])
rho = pinv(measurement_matrix) @ expectations
# add in identity term
dim = 2 ** len(qubits)
return unvec(rho) + np.eye(dim**2) / dim
def state_log_likelihood(state: np.ndarray, results: Iterator[ExperimentResult],
qubits: Sequence[int]) -> float:
"""
The log Likelihood function used in the diluted MLE tomography routine.
Equation 2 of [DIMLE1]_
:param state: The state (given as a density matrix) that we think we have.
:param results: Measured results from a state tomography experiment
:param qubits: All qubits that were tomographized. This specifies the order in
which qubits will be kron'ed together; the first qubit in the list is the left-most
tensor factor. This should agree with the provided state.
:return: The log likelihood that our state is the one we believe it is.
"""
# state2matrix and pauli2matrix use pyquil tensor factor ordering where the least significant
# qubit, e.g. qubit 0, is the right-most tensor factor. We stick with the standard convention
# here that the first qubit in the list is the left-most tensor factor, so we have to reverse
# the qubits before passing to state2matrix and pauli2matrix
qs = qubits[::-1]
ll = 0
for res in results:
n = res.total_counts
op_matrix = pauli2matrix(res.setting.observable, qs)
meas_exp = res.expectation
pred_exp = np.real(np.trace(op_matrix @ state))
for sign in [1, -1]:
f_j = n * (1 + sign * meas_exp) / 2
pr_j = (1 + sign * pred_exp) / 2
if pr_j <= 0:
continue
ll += f_j * np.log10(pr_j)
return ll
def _R(state, results, qubits):
r"""
This implements Eqn 4 in [DIMLE1]
As stated in [DIMLE1] eqn 4 reads
R(rho) = (1/N) \sum_j (f_j/Pr_j) Pi_j
N = total number of measurements
f_j = number of times j'th outcome was observed
Pi_j = measurement operator or projector, with \sum_j Pi_j = Id and Pi_j \geq 0
Pr_j = Tr[Pi_j \rho] (up to some normalization of the Pi_j)
We are working with results whose out_operators are elements of the un-normalized Pauli
basis. Each Pauli P_j can be split into projectors onto the plus and minus eigenspaces
P_k = Pi_k^+ - Pi_k^- ; Pi_k^+ = (I + P_k) / 2 ; Pi_k^- = (I - P_k) / 2
where each Pi \geq 0 as required above. Hence for each P_k we associate two Pi_k,
and subsequently two f_k. We can express these in terms of Exp[P_k] := exp_k
plus: f_k^+ / N = (1 + exp_k) / 2
minus: f_k^- / N = (1 - exp_k) / 2
We use these f_k and Pi_k to arrive at the code below.
Finally, since our Pauli's are not normalized, i.e. Pi_k^+ + Pi_k^- = Id, in order to enforce
the condition \sum_j Pi_j = Id stated above we need to divide our final answer by the number
of Paulis.
:param state: The state (given as a density matrix) that we think we have.
:param results: Measured results from a state tomography experiment. (assumes Pauli basis)
:param qubits: Qubits that were tomographized.
:return: the operator of equation 4 in [DIMLE1] which fixes rho by left and right multiplication
"""
# this small number ~ 10^-304 is added so that we don't get divide by zero errors
machine_eps = np.finfo(float).tiny
update = np.zeros_like(state, dtype=complex)
IdH = np.eye(update.shape[0])
for res in results:
op_matrix = pauli2matrix(res.setting.out_operator, qubits)
meas_exp = res.expectation
pred_exp = np.trace(op_matrix @ state)
for sign in [1, -1]:
f_j_over_n = (1 + sign * meas_exp) / 2
pr_j = (1 + sign * pred_exp) / 2
pi_j = (IdH + sign * op_matrix) / 2
update += f_j_over_n / (pr_j + machine_eps) * pi_j
return update / len(results)
def _extract_from_results(results: List[ExperimentResult], qubits: List[int]):
"""
Construct the matrix A such that the probabilities p_ij of outcomes n_ij given an estimate E
can be cast in a vectorized form.
Specifically::
p = vec(p_ij) = A x vec(E)
This yields convenient vectorized calculations of the cost and its gradient, in terms of A, n,
and E.
"""
A = []
n = []
grand_total_shots = 0
for result in results:
# 'lift' the result's ExperimentSetting input TensorProductState to the corresponding
# matrix. This is simply the density matrix of the state that was prepared.
in_state_matrix = state2matrix(result.setting.in_state, qubits=qubits)
# 'lift' the result's ExperimentSetting output PauliTerm to the corresponding matrix.
operator = pauli2matrix(result.setting.out_operator, qubits=qubits)
proj_plus = (np.eye(2 ** len(qubits)) + operator) / 2
proj_minus = (np.eye(2 ** len(qubits)) - operator) / 2
# Constructing A per eq. (A1) of [PGD]
# TODO: figure out if we can avoid re-splitting into Pi+ and Pi- counts
A += [
# vec() turns into a column vector; transpose to a row vector; index into the
# 1 row to avoid an extra tensor dimension when we call np.asarray(A).
vec(np.kron(in_state_matrix, proj_plus.T)).T[0],
vec(np.kron(in_state_matrix, proj_minus.T)).T[0],
]
expected_plus_ones = (1 + result.expectation) / 2
n += [
result.total_counts * expected_plus_ones,
result.total_counts * (1 - expected_plus_ones)
]
grand_total_shots += result.total_counts
n_qubits = len(qubits)
dimension = 2 ** n_qubits
A = np.asarray(A) / dimension ** 2
n = np.asarray(n)[:, np.newaxis] / grand_total_shots
return A, n
def lasso_state_estimate(results: List[ExperimentResult],
qubits: List[int]) -> np.ndarray:
"""
Estimate a quantum state using compressed sensing
[FLAMMIA] Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
Steven T. Flammia et. al.
(2012).
https://arxiv.org/pdf/1205.2300.pdf
: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.
"""
pauli_num = len(results)
qubit_num = len(qubits)
d = 2 ** qubit_num
pauli_list = []
expectation_list = []
y = np.zeros((m,1))
mu = 4 * pauli_num / np.sqrt(1000 * pauli_num)
for i in range(m):
#Convert the Pauli term into a tensor
r = results[i]
p_tensor = pauli2matrix(r.setting.out_operator, qubits)
e = r.expectation
y[i] = e
pauli_list.append(p_tensor)
expectation_list.append(e)
x = cp.Variable((d,d),complex = True)
A = cp.vstack([cp.trace(cp.matmul(pauli_list[i], x)) * np.sqrt(d / m) for i in range(m)])
#Minimize trace norm
obj = cp.Minimize(0.5 * cp.norm((A - y), 2) + mu * cp.norm(x, 'nuc'))
constraints = [cp.trace(x) == 1]
# Form and solve problem.
prob = cp.Problem(obj, constraints)
prob.solve()
rho = x.value
return rho
def compressed_sensing_state_estimate(results: List[ExperimentResult],
qubits: List[int]) -> np.ndarray:
"""
Estimate a quantum state using compressed sensing
[FLAMMIA] Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
Steven T. Flammia et. al.
(2012).
https://arxiv.org/pdf/1205.2300.pdf
: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.
"""
pauli_num = len(results)
qubit_num = len(qubits)
d = 2 ** qubit_num
pauli_list = []
expectation_list = []
y = np.zeros((m,1))
for i in range(pauli_num):
#Convert the Pauli term into a tensor
r = results[i]
p_tensor = pauli2matrix(r.setting.out_operator, qubits)
e = r.expectation
#A[i] = e * scale_factor
pauli_list.append(p_tensor)
expectation_list.append(e)
s = cp.Variable((d,d),complex = True)
obj = cp.Minimize(cp.norm(s, 'nuc'))
constraints = [cp.trace(s) == 1]
for i in range(len(pauli_list)):
trace_bool = cp.trace(cp.matmul(pauli_list[i], s)) - expectation_list[i] == 0
constraints.append(trace_bool)
# Form and solve problem.
prob = cp.Problem(obj, constraints)
prob.solve()
rho = s.value
return rho
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:`~observable_estimation.estimate_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]_
.. [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; the first qubit in the list is the left-most
tensor factor.
:return: A point estimate of the quantum state rho.
"""
# state2matrix and pauli2matrix use pyquil tensor factor ordering where the least significant
# qubit, e.g. qubit 0, is the right-most tensor factor. We stick with the standard convention
# here that the first qubit in the list is the left-most tensor factor, so we have to reverse
# the qubits before passing to state2matrix and pauli2matrix
qs = qubits[::-1]
measurement_matrix = np.vstack([
vec(pauli2matrix(result.setting.observable, qubits=qs)).T.conj() for result in results])
expectations = np.array([result.expectation for result in results])
rho = pinv(measurement_matrix) @ expectations
# add in the traceful identity term
dim = 2**len(qubits)
return unvec(rho) + np.eye(dim) / dim
