# Source code for pennylane.optimize.qng

# Copyright 2018 Xanadu Quantum Technologies Inc.

# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
#pylint: disable=too-many-branches
from scipy import linalg

from pennylane.utils import _flatten, unflatten

r"""Optimizer with adaptive learning rate, via calculation
of the diagonal or block-diagonal approximation to the Fubini-Study metric tensor.
A quantum generalization of natural gradient descent.

The QNG optimizer uses a step- and parameter-dependent learning rate,
with the learning rate dependent on the pseudo-inverse
of the Fubini-Study metric tensor :math:g:

.. math::
x^{(t+1)} = x^{(t)} - \eta g(f(x^{(t)}))^{-1} \nabla f(x^{(t)}),

where :math:f(x^{(t)}) = \langle 0 | U(x^{(t)})^\dagger \hat{B} U(x^{(t)}) | 0 \rangle
is an expectation value of some observable measured on the variational
quantum circuit :math:U(x^{(t)}).

Consider a quantum node represented by the variational quantum circuit

.. math::

U(\mathbf{\theta}) = W(\theta_{i+1}, \dots, \theta_{N})X(\theta_{i})
V(\theta_1, \dots, \theta_{i-1}),

where all parametrized gates can be written of the form :math:X(\theta_{i}) = e^{i\theta_i K_i}.
That is, the gate :math:K_i is the *generator* of the parametrized operation :math:X(\theta_i)
corresponding to the :math:i-th parameter.

For each parametric layer :math:\ell in the variational quantum circuit
containing :math:n parameters, the :math:n\times n block-diagonal submatrix
of the Fubini-Study tensor :math:g_{ij}^{(\ell)} is calculated directly on the
quantum device in a single evaluation:

.. math::

g_{ij}^{(\ell)} = \langle \psi_\ell | K_i K_j | \psi_\ell \rangle
- \langle \psi_\ell | K_i | \psi_\ell\rangle
\langle \psi_\ell |K_j | \psi_\ell\rangle

where :math:|\psi_\ell\rangle =  V(\theta_1, \dots, \theta_{i-1})|0\rangle
(that is, :math:|\psi_\ell\rangle is the quantum state prior to the application
of parameterized layer :math:\ell).

Combining the quantum natural gradient optimizer with the analytic parameter-shift
rule to optimize a variational circuit with :math:d parameters and :math:L layers,
a total of :math:2d+L quantum evaluations are required per optimization step.

For more details, see:

James Stokes, Josh Izaac, Nathan Killoran, Giuseppe Carleo.
"Quantum Natural Gradient." arXiv:1909.02108 <https://arxiv.org/abs/1909.02108>_, 2019.

.. note::

The QNG optimizer **only supports single QNodes** as objective functions.

In particular:

* For hybrid classical-quantum models, the "mixed geometry" of the model
makes it unclear which metric should be used for which parameter.
For example, parameters of quantum nodes are better suited to
one metric (such as the QNG), whereas others (e.g., parameters of classical nodes)
are likely better suited to another metric.

* For multi-QNode models, we don't know what geometry is appropriate
if a parameter is shared amongst several QNodes.

.. seealso::

See the :ref:quantum natural gradient example <quantum_natural_gradient>
for more details on Fubini-Study metric tensor and this optimization class.

Args:
stepsize (float): the user-defined hyperparameter :math:\eta
diag_approx (bool): If True, forces a diagonal approximation
where the calculated metric tensor only contains diagonal
elements :math:G_{ii}. In some cases, this may reduce the
time taken per optimization step.
tol (float): tolerance used when finding the inverse of the
"""
def __init__(self, stepsize=0.01, diag_approx=False):
super().__init__(stepsize)
self.diag_approx = diag_approx
self.metric_tensor_inv = None

[docs]    def step(self, qnode, x, recompute_tensor=True):
"""Update x with one step of the optimizer.

Args:
qnode (QNode): the QNode for optimization
x (array): NumPy array containing the current values of the variables to be updated
recompute_tensor (bool): Whether or not the metric tensor should
be recomputed. If not, the metric tensor from the previous
optimization step is used.

Returns:
array: the new variable values :math:x^{(t+1)}
"""
# pylint: disable=arguments-differ
if not hasattr(qnode, "metric_tensor"):
raise ValueError("Objective function must be encoded as a single QNode")

if recompute_tensor or self.metric_tensor is None:
# pseudo-inverse metric tensor
metric_tensor = qnode.metric_tensor(x, diag_approx=self.diag_approx)
self.metric_tensor_inv = linalg.pinvh(metric_tensor)

return x_out

r"""Update the variables x to take a single optimization step. Flattens and unflattens
the inputs to maintain nested iterables as the parameters of the optimization.

Args:
function at point :math:x^{(t)}: :math:\nabla f(x^{(t)})
x (array): the current value of the variables :math:x^{(t)}
array: the new values :math:x^{(t+1)}