# Copyright 2018-2021 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
import warnings
from typing import Sequence, Callable

import numpy as np
from scipy.sparse.linalg import expm
import pennylane as qml

from pennylane import transform
from pennylane.tape import QuantumTape
from pennylane.queuing import QueuingManager

@transform
def append_time_evolution(
tape: QuantumTape, riemannian_gradient, t, n, exact=False
) -> (Sequence[QuantumTape], Callable):
r"""Append an approximate time evolution, corresponding to a Riemannian
gradient on the Lie group, to an existing circuit.

We want to implement the time evolution generated by an operator of the form

.. math::

\text{grad} f(U) = sum_i c_i O_i,

where :math:O_i are Pauli words and :math:c_t \in \mathbb{R}.
If exact is False, we Trotterize this operator and apply the unitary

.. math::

U' = \prod_{n=1}^{N_{Trot.}} \left(\prod_i \exp{-it / N_{Trot.} O_i}\right),

which is then appended to the current circuit.

If exact is True, we calculate the exact time evolution for the Riemannian gradient
by way of the matrix exponential.

.. math:

and append this unitary.

Args:
tape (QuantumTape or QNode or Callable): circuit to transform.
t (float): time evolution parameter.
n (int): number of Trotter steps.

Returns:
qnode (QNode) or quantum function (Callable) or tuple[List[QuantumTape], function]: The transformed circuit as described in :func:qml.transform <pennylane.transform>.

"""
new_operations = tape.operations
if exact:
with QueuingManager.stop_recording():
new_operations.append(
qml.QubitUnitary(
)
)
else:
with QueuingManager.stop_recording():

new_tape = type(tape)(new_operations, tape.measurements, shots=tape.shots)

def null_postprocessing(results):
"""A postprocesing function returned by a transform that only converts the batch of results
into a result for a single QuantumTape.
"""
return results[0]  # pragma: no cover

return [new_tape], null_postprocessing

def algebra_commutator(tape, observables, lie_algebra_basis_names, nqubits):
"""Calculate the Riemannian gradient in the Lie algebra with the parameter shift rule
(see :meth:RiemannianGradientOptimizer.get_omegas).

Args:
tape (.QuantumTape or .QNode): input circuit
observables (list[.Observable]): list of observables to be measured. Can be grouped.
lie_algebra_basis_names (list[str]): List of strings corresponding to valid Pauli words.
nqubits (int): the number of qubits.

Returns:
function or tuple[list[QuantumTape], function]:

- If the input is a QNode, an object representing the Riemannian gradient function
of the QNode that can be executed with the same arguments as the QNode to obtain
the Lie algebra commutator.

- If the input is a tape, a tuple containing a
list of generated tapes, together with a post-processing
function to be applied to the results of the evaluated tapes
in order to obtain the Lie algebra commutator.
"""
tapes_plus_total = []
tapes_min_total = []
for obs in observables:
for o in obs:
# create a list of tapes for the plus and minus shifted circuits
queues_plus = [qml.queuing.AnnotatedQueue() for _ in lie_algebra_basis_names]
queues_min = [qml.queuing.AnnotatedQueue() for _ in lie_algebra_basis_names]

# loop through all operations on the input tape
for op in tape.operations:
for t in queues_plus + queues_min:
with t:
qml.apply(op)
for i, t in enumerate(queues_plus):
with t:
qml.PauliRot(
np.pi / 2,
lie_algebra_basis_names[i],
wires=list(range(nqubits)),
)
qml.expval(o)
for i, t in enumerate(queues_min):
with t:
qml.PauliRot(
-np.pi / 2,
lie_algebra_basis_names[i],
wires=list(range(nqubits)),
)
qml.expval(o)
tapes_plus_total.extend(
[
qml.tape.QuantumScript(*qml.queuing.process_queue(q))
for q, p in zip(queues_plus, lie_algebra_basis_names)
]
)
tapes_min_total.extend(
[
qml.tape.QuantumScript(*qml.queuing.process_queue(q))
for q, p in zip(queues_min, lie_algebra_basis_names)
]
)
return tapes_plus_total + tapes_min_total

Riemannian gradient descent algorithms can be used to optimize a function directly on a Lie group
as opposed to on an Euclidean parameter space. Consider the function
:math:f(U) = \text{Tr}(U \rho_0 U^\dagger H)
for a given Hamiltonian :math:H, unitary :math:U\in \text{SU}(2^N) and initial state
:math:\rho_0. One can show that this function is minimized by the flow equation

.. math::

where :math:\text{grad} is the Riemannian gradient operator on :math:\text{SU}(2^N).
By discretizing the flow above, we see that a step of this optimizer
iterates the Riemannian gradient flow on :math:\text{SU}(2^N) as

.. math::

where :math:\epsilon is a user-defined hyperparameter corresponding to the step size.

The Riemannian gradient in the Lie algebra is given by

.. math::

\text{grad}f(U^{(t)}) = -\left[U \rho U^\dagger, H\right] .

Hence we see that subsequent steps of this optimizer will append the unitary generated by the Riemannian

The exact Riemannian gradient flow on :math:\text{SU}(2^N) has desirable optimization properties
that can guarantee convergence to global minima under mild assumptions. However, this comes
at a cost. Since :math:\text{dim}(\text{SU}(2^N)) = 4^N-1, we need an exponential number
of parameters to calculate the gradient. This will not be problematic for small systems (:math:N<5),
but will quickly get out of control as the number of qubits increases.

To resolve this issue, we can restrict the Riemannian gradient to a subspace of the Lie algebra and calculate an
approximate Riemannian gradient flow. The choice of restriction will affect the optimization behaviour
and quality of the final solution.

T. Schulte-Herbrueggen et. al. (2008) <https://arxiv.org/abs/0802.4195>_
and the application to quantum circuits
Wiersema and Killoran (2022) <https://arxiv.org/abs/2202.06976>_.

Args:
circuit (.QNode): a user defined circuit that does not take any arguments and returns
the expectation value of a qml.Hamiltonian.
stepsize (float): the user-defined hyperparameter :math:\epsilon.
restriction (.Hamiltonian): Restrict the Lie algebra to a corresponding subspace of
the full Lie algebra. This restriction should be passed in the form of a
qml.Hamiltonian that consists only of Pauli words.
exact (bool): Flag that indicates wether we approximate the Riemannian gradient with a
Trotterization or calculate the exact evolution via a matrix exponential. The latter is
not hardware friendly and can only be done in simulation.

**Examples**

Define a Hamiltonian cost function to minimize:

>>> coeffs = [-1., -1., -1.]
>>> observables = [qml.X(0), qml.Z(1), qml.Y(0) @ qml.X(1)]
>>> hamiltonian = qml.Hamiltonian(coeffs, observables)

Create an initial state and return the expectation value of the Hamiltonian:

>>> @qml.qnode(qml.device("default.qubit", wires=2))
... def quant_fun():
...     qml.RX(0.1, wires=[0])
...     qml.RY(0.5, wires=[1])
...     qml.CNOT(wires=[0,1])
...     qml.RY(0.6, wires=[0])
...     return qml.expval(hamiltonian)

Instantiate the optimizer with the initial circuit and the cost function and set the stepsize
accordingly:

Applying 5 steps gets us close the ground state of :math:E\approx-2.23:

>>> for step in range(6):
...    circuit, cost = opt.step_and_cost()
...    print(f"Step {step} - cost {cost}")
Step 0 - cost -1.3351865007304005
Step 1 - cost -1.9937887238935206
Step 2 - cost -2.1524234485729834
Step 3 - cost -2.1955105378898487
Step 4 - cost -2.2137628169764256
Step 5 - cost -2.2234364822091575

The optimized circuit is returned at each step, and can be used as any other QNode:

>>> circuit()
-2.2283086057521713

"""

# pylint: disable=too-many-arguments
# pylint: disable=too-many-instance-attributes
def __init__(self, circuit, stepsize=0.01, restriction=None, exact=False, trottersteps=1):
if not isinstance(circuit, qml.QNode):
raise TypeError(f"circuit must be a QNode, received {type(circuit)}")

self.circuit = circuit
self.circuit.construct([], {})
self.hamiltonian = circuit.func().obs
if not isinstance(self.hamiltonian, (qml.ops.Hamiltonian, qml.ops.LinearCombination)):
raise TypeError(
f"circuit must return the expectation value of a Hamiltonian,"
)
self.nqubits = len(circuit.device.wires)

if self.nqubits > 4:
warnings.warn(
"The exact Riemannian gradient is exponentially expensive in the number of qubits, "
f"optimizing a {self.nqubits} qubit circuit may be slow.",
UserWarning,
)
if restriction is not None and not isinstance(
restriction, (qml.ops.Hamiltonian, qml.ops.LinearCombination)
):
raise TypeError(f"restriction must be a Hamiltonian, received {type(restriction)}")
(
self.lie_algebra_basis_ops,
self.lie_algebra_basis_names,
) = self.get_su_n_operators(restriction)
self.exact = exact
self.trottersteps = trottersteps
self.coeffs, self.observables = self.hamiltonian.terms()
self.stepsize = stepsize

[docs]    def step(self):
r"""Update the circuit with one step of the optimizer.

Returns:
float: the optimized circuit and the objective function output prior
to the step.
"""
return self.step_and_cost()[0]

[docs]    def step_and_cost(self):
r"""Update the circuit with one step of the optimizer and return the corresponding
objective function value prior to the step.

Returns:
tuple[.QNode, float]: the optimized circuit and the objective function output prior
to the step.
"""
# pylint: disable=not-callable

cost = self.circuit()
omegas = self.get_omegas()
non_zero_omegas = -omegas[omegas != 0]

nonzero_idx = np.nonzero(omegas)[0]
non_zero_lie_algebra_elements = [self.lie_algebra_basis_names[i] for i in nonzero_idx]

non_zero_omegas,
[qml.pauli.string_to_pauli_word(ps) for ps in non_zero_lie_algebra_elements],
)
new_circuit = append_time_evolution(
)

# we can set diff_method=None because the gradient of the QNode is computed
# directly in this optimizer
self.circuit = qml.QNode(new_circuit, self.circuit.device, diff_method=None)
return self.circuit, cost

[docs]    def get_su_n_operators(self, restriction):
r"""Get the SU(N) operators. The dimension of the group is :math:N^2-1.

Args:
restriction (.Hamiltonian): Restrict the Riemannian gradient to a subspace.

Returns:
tuple[list[array[complex]], list[str]]: list of :math:N^2 \times N^2 NumPy complex arrays and corresponding Pauli words.
"""

operators = []
names = []
# construct the corresponding pennylane observables
wire_map = dict(zip(range(self.nqubits), range(self.nqubits)))
if restriction is None:
for ps in qml.pauli.pauli_group(self.nqubits):
operators.append(ps)
names.append(qml.pauli.pauli_word_to_string(ps, wire_map=wire_map))
else:
for ps in set(restriction.ops):
operators.append(ps)
names.append(qml.pauli.pauli_word_to_string(ps, wire_map=wire_map))

return operators, names

[docs]    def get_omegas(self):
r"""Measure the coefficients of the Riemannian gradient with respect to a Pauli word basis.
We want to calculate the components of the Riemannian gradient in the Lie algebra
with respect to a Pauli word basis. For a Hamiltonian of the form :math:H = \sum_i c_i O_i,
where :math:c_i\in\mathbb{R}, this can be achieved by calculating

.. math::

\omega_{i,j} = \text{Tr}(c_i[\rho, O_i] P_j)

where :math:P_j is a Pauli word in the set of Pauli monomials on :math:N qubits.

Via the parameter shift rule, the commutator can be calculated as

.. math::

[\rho, O_i] = \frac{1}{2}(V(\pi/2) \rho V^\dagger(\pi/2) - V(-\pi/2) \rho V^\dagger(-\pi/2))

where :math:V is the unitary generated by the Pauli word :math:V(\theta) = \exp\{-i\theta P_j\}.

Returns:
array: array of omegas for each direction in the Lie algebra.
"""

obs_groupings, _ = qml.pauli.group_observables(self.observables, self.coeffs)
# get all circuits we need to calculate the coefficients
circuits = algebra_commutator(
self.circuit.qtape,
obs_groupings,
self.lie_algebra_basis_names,
self.nqubits,
)

if isinstance(self.circuit.device, qml.devices.Device):
program, config = self.circuit.device.preprocess()

circuits = qml.execute(
circuits,
self.circuit.device,
transform_program=program,
config=config,
)
else:
circuits = qml.execute(
)  # pragma: no cover

program, _ = self.circuit.device.preprocess()

circuits_plus = np.array(circuits[: len(circuits) // 2]).reshape(
len(self.coeffs), len(self.lie_algebra_basis_names)
)
circuits_min = np.array(circuits[len(circuits) // 2 :]).reshape(
len(self.coeffs), len(self.lie_algebra_basis_names)
)

# For each observable O_i in the Hamiltonian, we have to calculate all Lie coefficients
omegas = circuits_plus - circuits_min

return np.dot(self.coeffs, omegas)


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