Source code for pennylane.templates.subroutines.hilbert_schmidt
# Copyright 2022 Xanadu Quantum Technologies Inc.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""
This submodule contains the templates for the Hilbert-Schmidt tests.
"""
# pylint: disable-msg=too-many-arguments
import pennylane as qml
from pennylane.operation import AnyWires, Operation
[docs]class HilbertSchmidt(Operation):
r"""Create a Hilbert-Schmidt template that can be used to compute the Hilbert-Schmidt Test (HST).
The HST is a useful quantity to compile a target unitary `U` with an approximate unitary `V`. The HST
is used as a distance between `U` and `V`. The result of executing the HST is 0 if and only if `V` is equal to
`U` (up to a global phase). As suggested in [1], we can define a cost function using the Hilbert-Schmidt inner product
between the unitaries `U` and `V` as follows:
.. math::
C_{HST} = 1 - \frac{1}{d^2} \left|Tr(V^{\dagger}U)\right|^2,
where `d` is the dimension of the space in which the unitaries `U` and `V` act.
The quantity :math:`\frac{1}{d^2} \left|Tr(V^{\dagger}U)\right|^2` is obtained by executing the Hilbert-Schmidt Test.
It is equivalent to taking the outcome probability of the state :math:`|0 ... 0\rangle`
for the following circuit:
.. figure:: ../../_static/templates/subroutines/hst.png
:align: center
:width: 80%
:target: javascript:void(0);
It defines our decomposition for the Hilbert-Schmidt Test template.
Args:
*params (array): Parameters for the quantum function `V`.
v_function (callable): Quantum function that represents the approximate compiled unitary `V`.
v_wires (int or Iterable[Number, str]]): The wire(s) on which the approximate compiled unitary acts.
u_tape (.QuantumTape): `U`, the unitary to be compiled as a ``qml.tape.QuantumTape``.
Raises:
QuantumFunctionError: The argument ``u_tape`` must be a ``QuantumTape``.
QuantumFunctionError: ``v_function`` is not a valid quantum function.
QuantumFunctionError: ``U`` and ``V`` do not have the same number of wires.
QuantumFunctionError: The wires ``v_wires`` are a subset of ``V`` wires.
QuantumFunctionError: ``u_tape`` and ``v_tape`` must act on distinct wires.
**Reference**
[1] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T. Sornborger and Patrick J. Coles
Quantum-assisted Quantum Compiling.
`arxiv/1807.00800 <https://arxiv.org/pdf/1807.00800.pdf>`_
.. seealso:: :class:`~.LocalHilbertSchmidt`
.. details::
:title: Usage Details
Consider that we want to evaluate the Hilbert-Schmidt Test cost between the unitary ``U`` and an approximate
unitary ``V``. We need to define some functions where it is possible to use the :class:`~.HilbertSchmidt`
template. Here the considered unitary is ``Hadamard`` and we try to compute the cost for the approximate
unitary ``RZ``. For an angle that is equal to ``0`` (``Identity``), we have the maximal cost which is ``1``.
.. code-block:: python
with qml.QueuingManager.stop_recording():
u_tape = qml.tape.QuantumTape([qml.Hadamard(0)])
def v_function(params):
qml.RZ(params[0], wires=1)
dev = qml.device("default.qubit", wires=2)
@qml.qnode(dev)
def hilbert_test(v_params, v_function, v_wires, u_tape):
qml.HilbertSchmidt(v_params, v_function=v_function, v_wires=v_wires, u_tape=u_tape)
return qml.probs(u_tape.wires + v_wires)
def cost_hst(parameters, v_function, v_wires, u_tape):
return (1 - hilbert_test(v_params=parameters, v_function=v_function, v_wires=v_wires, u_tape=u_tape)[0])
Now that the cost function has been defined it can be called for specific parameters:
>>> cost_hst([0], v_function = v_function, v_wires = [1], u_tape = u_tape)
tensor(1., requires_grad=True)
"""
num_wires = AnyWires
grad_method = None
def _flatten(self):
metadata = (
("v_function", self.hyperparameters["v_function"]),
("v_wires", self.hyperparameters["v_wires"]),
("u_tape", self.hyperparameters["u_tape"]),
)
return self.data, metadata
@classmethod
def _primitive_bind_call(cls, *params, v_function, v_wires, u_tape, id=None):
# pylint: disable=arguments-differ
kwargs = {"v_function": v_function, "v_wires": v_wires, "u_tape": u_tape, "id": id}
return cls._primitive.bind(*params, **kwargs)
@classmethod
def _unflatten(cls, data, metadata):
return cls(*data, **dict(metadata))
def __init__(self, *params, v_function, v_wires, u_tape, id=None):
self._num_params = len(params)
if not isinstance(u_tape, qml.tape.QuantumScript):
raise qml.QuantumFunctionError("The argument u_tape must be a QuantumTape.")
u_wires = u_tape.wires
self.hyperparameters["u_tape"] = u_tape
if not callable(v_function):
raise qml.QuantumFunctionError(
"The argument v_function must be a callable quantum function."
)
self.hyperparameters["v_function"] = v_function
v_tape = qml.tape.make_qscript(v_function)(*params)
self.hyperparameters["v_tape"] = v_tape
self.hyperparameters["v_wires"] = qml.wires.Wires(v_wires)
if len(u_wires) != len(v_wires):
raise qml.QuantumFunctionError("U and V must have the same number of wires.")
if not qml.wires.Wires(v_wires).contains_wires(v_tape.wires):
raise qml.QuantumFunctionError("All wires in v_tape must be in v_wires.")
# Intersection of wires
if len(qml.wires.Wires.shared_wires([u_tape.wires, v_tape.wires])) != 0:
raise qml.QuantumFunctionError("u_tape and v_tape must act on distinct wires.")
wires = qml.wires.Wires(u_wires + v_wires)
super().__init__(*params, wires=wires, id=id)
[docs] def map_wires(self, wire_map: dict):
raise NotImplementedError("Mapping the wires of HilbertSchmidt is not implemented.")
@property
def num_params(self):
return self._num_params
[docs] @staticmethod
def compute_decomposition(
params, wires, u_tape, v_tape, v_function=None, v_wires=None
): # pylint: disable=arguments-differ,unused-argument,too-many-positional-arguments
r"""Representation of the operator as a product of other operators."""
n_wires = len(u_tape.wires + v_tape.wires)
first_range = range(n_wires // 2)
second_range = range(n_wires // 2, n_wires)
# Hadamard first layer
decomp_ops = [qml.Hadamard(wires[i]) for i in first_range]
# CNOT first layer
decomp_ops.extend(
qml.CNOT(wires=[wires[i], wires[j]]) for i, j in zip(first_range, second_range)
)
# Unitary U
for op_u in u_tape.operations:
# The operation has been defined outside of this function, to queue it we call qml.apply.
if qml.QueuingManager.recording():
qml.apply(op_u)
decomp_ops.append(op_u)
# Unitary V conjugate
# Since we don't currently have an easy way to apply the complex conjugate of a tape, we manually
# apply the complex conjugate of each operation in the V tape and append it to the decomposition
# using the QubitUnitary operation.
decomp_ops.extend(
qml.QubitUnitary(op_v.matrix().conjugate(), wires=op_v.wires)
for op_v in v_tape.operations
)
# CNOT second layer
decomp_ops.extend(
qml.CNOT(wires=[wires[i], wires[j]])
for i, j in zip(reversed(first_range), reversed(second_range))
)
# Hadamard second layer
decomp_ops.extend(qml.Hadamard(wires[i]) for i in first_range)
return decomp_ops
[docs]class LocalHilbertSchmidt(HilbertSchmidt):
r"""Create a Local Hilbert-Schmidt template that can be used to compute the Local Hilbert-Schmidt Test (LHST).
The result of the LHST is a useful quantity for compiling a unitary `U` with an approximate unitary `V`. The
LHST is used as a distance between `U` and `V`. It is similar to the Hilbert-Schmidt test, but the measurement is
made only on one qubit at the end of the circuit. The LHST cost is always smaller than the HST cost and is useful
for large unitaries.
.. figure:: ../../_static/templates/subroutines/lhst.png
:align: center
:width: 80%
:target: javascript:void(0);
Args:
params (array): Parameters for the quantum function `V`.
v_function (Callable): Quantum function that represents the approximate compiled unitary `V`.
v_wires (int or Iterable[Number, str]]): the wire(s) on which the approximate compiled unitary acts.
u_tape (.QuantumTape): `U`, the unitary to be compiled as a ``qml.tape.QuantumTape``.
Raises:
QuantumFunctionError: The argument u_tape must be a QuantumTape
QuantumFunctionError: ``v_function`` is not a valid Quantum function.
QuantumFunctionError: `U` and `V` do not have the same number of wires.
QuantumFunctionError: The wires ``v_wires`` are a subset of `V` wires.
QuantumFunctionError: u_tape and v_tape must act on distinct wires.
**Reference**
[1] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T. Sornborger and Patrick J. Coles
Quantum-assisted Quantum Compiling.
`arxiv/1807.00800 <https://arxiv.org/pdf/1807.00800.pdf>`_
.. seealso:: :class:`~.HilbertSchmidt`
.. details::
:title: Usage Details
Consider that we want to evaluate the Local Hilbert-Schmidt Test cost between the unitary ``U`` and an
approximate unitary ``V``. We need to define some functions where it is possible to use the
:class:`~.LocalHilbertSchmidt` template. Here the considered unitary is ``CZ`` and we try to compute the
cost for the approximate unitary.
.. code-block:: python
import numpy as np
with qml.QueuingManager.stop_recording():
u_tape = qml.tape.QuantumTape([qml.CZ(wires=(0,1))])
def v_function(params):
qml.RZ(params[0], wires=2)
qml.RZ(params[1], wires=3)
qml.CNOT(wires=[2, 3])
qml.RZ(params[2], wires=3)
qml.CNOT(wires=[2, 3])
dev = qml.device("default.qubit", wires=4)
@qml.qnode(dev)
def local_hilbert_test(v_params, v_function, v_wires, u_tape):
qml.LocalHilbertSchmidt(v_params, v_function=v_function, v_wires=v_wires, u_tape=u_tape)
return qml.probs(u_tape.wires + v_wires)
def cost_lhst(parameters, v_function, v_wires, u_tape):
return (1 - local_hilbert_test(v_params=parameters, v_function=v_function, v_wires=v_wires, u_tape=u_tape)[0])
Now that the cost function has been defined it can be called for specific parameters:
>>> cost_lhst([3*np.pi/2, 3*np.pi/2, np.pi/2], v_function = v_function, v_wires = [2,3], u_tape = u_tape)
tensor(0.5, requires_grad=True)
"""
[docs] @staticmethod
def compute_decomposition(
params, wires, u_tape, v_tape, v_function=None, v_wires=None
): # pylint: disable=arguments-differ,unused-argument,too-many-positional-arguments
r"""Representation of the operator as a product of other operators (static method)."""
n_wires = len(u_tape.wires + v_tape.wires)
first_range = range(n_wires // 2)
second_range = range(n_wires // 2, n_wires)
# Hadamard first layer
decomp_ops = [qml.Hadamard(wires[i]) for i in first_range]
# CNOT first layer
decomp_ops.extend(
qml.CNOT(wires=[wires[i], wires[j]]) for i, j in zip(first_range, second_range)
)
# Unitary U
if qml.QueuingManager.recording():
decomp_ops.extend(qml.apply(op_u) for op_u in u_tape.operations)
else:
decomp_ops.extend(u_tape.operations)
# Unitary V conjugate
# Since we don't currently have an easy way to apply the complex conjugate of a tape, we manually
# apply the complex conjugate of each operation in the V tape and append it to the decomposition
# using the QubitUnitary operation.
decomp_ops.extend(
qml.QubitUnitary(op_v.matrix().conjugate(), wires=op_v.wires)
for op_v in v_tape.operations
)
# Single qubit measurement
decomp_ops.extend((qml.CNOT(wires=[wires[0], wires[n_wires // 2]]), qml.Hadamard(wires[0])))
return decomp_ops
_modules/pennylane/templates/subroutines/hilbert_schmidt
Download Python script
Download Notebook
View on GitHub