# Source code for pennylane.fourier.reconstruct

```
# Copyright 2018-2021 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.
"""Contains a function that computes the fourier series of
a quantum expectation value."""
from functools import wraps
from inspect import signature
import warnings
import numpy as np
from autoray import numpy as anp
import pennylane as qml
def _reconstruct_equ(fun, num_frequency, x0=None, f0=None, interface=None):
r"""Reconstruct a univariate Fourier series with consecutive integer
frequencies, using trigonometric interpolation and equidistant shifts.
This technique is based on
`Dirichlet kernels <https://en.wikipedia.org/wiki/Dirichlet_kernel>`_, see
`Vidal and Theis (2018) <https://arxiv.org/abs/1812.06323>`_ or
`Wierichs et al. (2022) <https://doi.org/10.22331/q-2022-03-30-677>`_.
Args:
fun (callable): Univariate finite Fourier series to reconstruct.
It must have signature ``float -> float`` .
num_frequency (int): Number of integer frequencies in ``fun``.
All integer frequencies below ``num_frequency`` are assumed
to be present in ``fun`` as well; if they are not, the output
is correct put the reconstruction could have been performed
with fewer evaluations of ``fun`` .
x0 (float): Center to which to shift the reconstruction.
The points at which ``fun`` is evaluated are *not* affected
by ``x0`` .
f0 (float): Value of ``fun`` at zero; Providing ``f0`` saves one
evaluation of ``fun``.
interface (str): Which auto-differentiation framework to use as
interface. This determines in which interface the output
reconstructed function is intended to be used.
Returns:
callable: Reconstructed Fourier series with ``num_frequency`` frequencies.
This function is a purely classical function. Furthermore, it is fully
differentiable.
"""
if not abs(int(num_frequency)) == num_frequency:
raise ValueError(f"num_frequency must be a non-negative integer, got {num_frequency}")
a = (num_frequency + 0.5) / np.pi
b = 0.5 / np.pi
shifts_pos = qml.math.arange(1, num_frequency + 1) / a
shifts_neg = -shifts_pos[::-1]
shifts = qml.math.concatenate([shifts_neg, [0.0], shifts_pos])
shifts = anp.asarray(shifts, like=interface)
f0 = fun(0.0) if f0 is None else f0
evals = (
list(map(fun, shifts[:num_frequency])) + [f0] + list(map(fun, shifts[num_frequency + 1 :]))
)
evals = anp.asarray(evals, like=interface)
x0 = anp.asarray(np.float64(0.0), like=interface) if x0 is None else x0
def _reconstruction(x):
"""Univariate reconstruction based on equidistant shifts and Dirichlet kernels.
The derivative at of ``sinc`` are not well-implemented in TensorFlow and Autograd,
use the Fourier transform reconstruction if this derivative is needed.
"""
_x = x - x0 - shifts
return qml.math.tensordot(
qml.math.sinc(a * _x) / qml.math.sinc(b * _x),
evals,
axes=[[0], [0]],
)
return _reconstruction
_warn_text_f0_ignored = (
"The provided value of the function at zero will be ignored due to the "
"provided shift values. This may lead to additional evaluations of the "
"function to be reconstructed."
)
def _reconstruct_gen(fun, spectrum, shifts=None, x0=None, f0=None, interface=None):
r"""Reconstruct a univariate (real-valued) Fourier series with given spectrum.
Args:
fun (callable): Univariate finite Fourier series to reconstruct.
It must have signature ``float -> float`` .
spectrum (Collection): Frequency spectrum of the Fourier series;
non-positive frequencies are ignored.
shifts (Sequence): Shift angles at which to evaluate ``fun`` for the reconstruction.
Chosen equidistantly within the interval :math:`[0, 2\pi/f_\text{max}]`
if ``shifts=None`` , where :math:`f_\text{max}` is the biggest
frequency in ``spectrum``.
x0 (float): Center to which to shift the reconstruction.
The points at which ``fun`` is evaluated are *not* affected
by ``x0`` .
f0 (float): Value of ``fun`` at zero; If :math:`0` is among the ``shifts``
and ``f0`` is provided, one evaluation of ``fun`` is saved.
interface (str): Which auto-differentiation framework to use as
interface. This determines in which interface the output
reconstructed function is intended to be used.
Returns:
callable: Reconstructed Fourier series with :math:`R` frequencies in ``spectrum`` .
This function is a purely classical function. Furthermore, it is fully differentiable.
"""
# pylint: disable=unused-argument, too-many-arguments
have_f0 = f0 is not None
have_shifts = shifts is not None
spectrum = anp.asarray(spectrum, like=interface)
spectrum = spectrum[spectrum > 0]
f_max = qml.math.max(spectrum)
# If no shifts are provided, choose equidistant ones
if not have_shifts:
R = qml.math.shape(spectrum)[0]
shifts = qml.math.arange(-R, R + 1) * 2 * np.pi / (f_max * (2 * R + 1)) * R
zero_idx = R
need_f0 = True
elif have_f0:
zero_idx = qml.math.where(qml.math.isclose(shifts, qml.math.zeros_like(shifts[0])))
zero_idx = zero_idx[0][0] if (len(zero_idx) > 0 and len(zero_idx[0]) > 0) else None
need_f0 = zero_idx is not None
# Take care of shifts close to zero if f0 was provided
if have_f0 and need_f0:
# Only one shift may be zero at a time
shifts = qml.math.concatenate(
[shifts[zero_idx : zero_idx + 1], shifts[:zero_idx], shifts[zero_idx + 1 :]]
)
shifts = anp.asarray(shifts, like=interface)
evals = anp.asarray([f0] + list(map(fun, shifts[1:])), like=interface)
else:
shifts = anp.asarray(shifts, like=interface)
if have_f0 and not need_f0:
warnings.warn(_warn_text_f0_ignored)
evals = anp.asarray(list(map(fun, shifts)), like=interface)
L = len(shifts)
# Construct the coefficient matrix case by case
C1 = qml.math.ones((L, 1))
C2 = qml.math.cos(qml.math.tensordot(shifts, spectrum, axes=0))
C3 = qml.math.sin(qml.math.tensordot(shifts, spectrum, axes=0))
C = qml.math.hstack([C1, C2, C3])
# Solve the system of linear equations
cond = qml.math.linalg.cond(C)
if cond > 1e8:
warnings.warn(
f"The condition number of the Fourier transform matrix is very large: {cond}.",
UserWarning,
)
W = qml.math.linalg.solve(C, evals)
# Extract the Fourier coefficients
R = (L - 1) // 2
a0 = W[0]
a = anp.asarray(W[1 : R + 1], like=interface)
b = anp.asarray(W[R + 1 :], like=interface)
x0 = anp.asarray(np.float64(0.0), like=interface) if x0 is None else x0
# Construct the Fourier series
def _reconstruction(x):
"""Univariate reconstruction based on arbitrary shifts."""
x = x - x0
return (
a0
+ qml.math.tensordot(qml.math.cos(spectrum * x), a, axes=[[0], [0]])
+ qml.math.tensordot(qml.math.sin(spectrum * x), b, axes=[[0], [0]])
)
return _reconstruction
def _parse_ids(ids, info_dict):
"""Parse different formats of ``ids`` into the right dictionary format,
potentially using the information in ``info_dict`` to complete it.
"""
if ids is None:
# Infer all id information from info_dict
return {outer_key: inner_dict.keys() for outer_key, inner_dict in info_dict.items()}
if isinstance(ids, str):
# ids only provides a single argument name but no parameter indices
return {ids: info_dict[ids].keys()}
if not isinstance(ids, dict):
# ids only provides argument names but no parameter indices
return {_id: info_dict[_id].keys() for _id in ids}
return ids
def _parse_shifts(shifts, R, arg_name, par_idx, atol, need_f0):
"""Processes shifts for a single reconstruction and determines
wheter the function at the reconstruction point, ``f0`` will be
needed.
"""
# pylint: disable=too-many-arguments
_shifts = shifts.get(arg_name)
if _shifts is not None:
_shifts = _shifts.get(par_idx)
if _shifts is not None:
# Check whether the _shifts have the correct size
if len(_shifts) != 2 * R + 1:
raise ValueError(
f"The number of provided shifts ({len(_shifts)}) does not fit to the "
f"number of frequencies (2R+1={2*R+1}) for parameter {par_idx} in "
f"argument {arg_name}."
)
if any(qml.math.isclose(_shifts, qml.math.zeros_like(_shifts), rtol=0, atol=atol)):
# If 0 is among the shifts, f0 is needed
return _shifts, True
# If 0 is not among the shifts, f0 is not needed
return _shifts, (False or need_f0)
# If no shifts are given, f0 is needed always
return _shifts, True
def _prepare_jobs(ids, nums_frequency, spectra, shifts, atol):
r"""For inputs to reconstruct, determine how the given information yields
function reconstruction tasks and collect them into a dictionary ``jobs``.
Also determine whether the function at zero is needed.
Args:
ids (dict or Sequence or str): Indices for the QNode parameters with respect to which
the QNode should be reconstructed as a univariate function, per QNode argument.
Each key of the dict, entry of the list, or the single ``str`` has to be the name
of an argument of ``qnode`` .
If a ``dict`` , the values of ``ids`` have to contain the parameter indices
for the respective array-valued QNode argument represented by the key.
These indices always are tuples, i.e. ``()`` for scalar and ``(i,)`` for
one-dimensional arguments.
If a ``list`` , the parameter indices are inferred from ``nums_frequency`` if
given or ``spectra`` else.
If ``None``, all keys present in ``nums_frequency`` / ``spectra`` are considered.
nums_frequency (dict[dict]): Numbers of integer frequencies -- and biggest
frequency -- per QNode parameter. The keys have to be argument names of ``qnode``
and the inner dictionaries have to be mappings from parameter indices to the
respective integer number of frequencies. If the QNode frequencies are not contiguous
integers, the argument ``spectra`` should be used to save evaluations of ``qnode`` .
Takes precedence over ``spectra`` and leads to usage of equidistant shifts.
spectra (dict[dict]): Frequency spectra per QNode parameter.
The keys have to be argument names of ``qnode`` and the inner dictionaries have to
be mappings from parameter indices to the respective frequency spectrum for that
parameter. Ignored if ``nums_frequency!=None``.
shifts (dict[dict]): Shift angles for the reconstruction per QNode parameter.
The keys have to be argument names of ``qnode`` and the inner dictionaries have to
be mappings from parameter indices to the respective shift angles to be used for that
parameter. For :math:`R` non-zero frequencies, there must be :math:`2R+1` shifts
given. Ignored if ``nums_frequency!=None``.
atol (float): Absolute tolerance used to analyze shifts lying close to 0.
Returns:
dict[dict]: Indices for the QNode parameters with respect to which the QNode
will be reconstructed. Cast to the dictionary structure explained above.
If the input ``ids`` was a dictionary, it is returned unmodified.
callable: The reconstruction method to use, one out of two internal methods.
dict[dict[dict]]: Keyword arguments for the reconstruction method specifying
how to carry out the reconstruction. The outer-most keys are QNode argument
names, the middle keys are parameter indices like the inner keys of
``nums_frequency`` or ``spectra`` and the inner-most dictionary contains the
keyword arguments, i.e. the keys are keyword argument names for the
reconstruction method
bool: Whether any of the reconstruction jobs will require the evaluation
of the function at the position of reconstruction itself.
"""
if nums_frequency is None:
if spectra is None:
raise ValueError("Either nums_frequency or spectra must be given.")
ids = _parse_ids(ids, spectra)
if shifts is None:
shifts = {}
need_f0 = False
recon_fn = _reconstruct_gen
jobs = {}
# If no shifts are provided, compute them
for arg_name, inner_dict in ids.items():
_jobs = {}
for par_idx in inner_dict:
# Determine spectrum and number of frequencies, discounting for 0
_spectrum = spectra[arg_name][par_idx]
R = len(_spectrum) - 1
_shifts, need_f0 = _parse_shifts(shifts, R, arg_name, par_idx, atol, need_f0)
# Store job
if R > 0:
_jobs[par_idx] = {"shifts": _shifts, "spectrum": _spectrum}
else:
# R=0 belongs to a constant function
_jobs[par_idx] = None
jobs[arg_name] = _jobs
else:
jobs = {}
need_f0 = True
ids = _parse_ids(ids, nums_frequency)
recon_fn = _reconstruct_equ
for arg_name, inner_dict in ids.items():
_jobs = {}
for par_idx in inner_dict:
_num_frequency = nums_frequency[arg_name][par_idx]
_jobs[par_idx] = {"num_frequency": _num_frequency} if _num_frequency > 0 else None
jobs[arg_name] = _jobs
return ids, recon_fn, jobs, need_f0
[docs]def reconstruct(qnode, ids=None, nums_frequency=None, spectra=None, shifts=None):
r"""Reconstruct an expectation value QNode along a single parameter direction.
This means we restrict the QNode to vary only one parameter, a univariate restriction.
For common quantum gates, such restrictions are finite Fourier series with known
frequency spectra. Thus they may be reconstructed using Dirichlet kernels or
a non-uniform Fourier transform.
Args:
qnode (pennylane.QNode): Quantum node to be reconstructed, representing a
circuit that outputs an expectation value.
ids (dict or Sequence or str): Indices for the QNode parameters with respect to which
the QNode should be reconstructed as a univariate function, per QNode argument.
Each key of the dict, entry of the list, or the single ``str`` has to be the name
of an argument of ``qnode`` .
If a ``dict`` , the values of ``ids`` have to contain the parameter indices
for the respective array-valued QNode argument represented by the key.
These indices always are tuples, i.e., ``()`` for scalar and ``(i,)`` for
one-dimensional arguments.
If a ``list`` , the parameter indices are inferred from ``nums_frequency`` if
given or ``spectra`` else.
If ``None``, all keys present in ``nums_frequency`` / ``spectra`` are considered.
nums_frequency (dict[dict]): Numbers of integer frequencies -- and biggest
frequency -- per QNode parameter. The keys have to be argument names of ``qnode``
and the inner dictionaries have to be mappings from parameter indices to the
respective integer number of frequencies. If the QNode frequencies are not contiguous
integers, the argument ``spectra`` should be used to save evaluations of ``qnode`` .
Takes precedence over ``spectra`` and leads to usage of equidistant shifts.
spectra (dict[dict]): Frequency spectra per QNode parameter.
The keys have to be argument names of ``qnode`` and the inner dictionaries have to
be mappings from parameter indices to the respective frequency spectrum for that
parameter. Ignored if ``nums_frequency!=None``.
shifts (dict[dict]): Shift angles for the reconstruction per QNode parameter.
The keys have to be argument names of ``qnode`` and the inner dictionaries have to
be mappings from parameter indices to the respective shift angles to be used for that
parameter. For :math:`R` non-zero frequencies, there must be :math:`2R+1` shifts
given. Ignored if ``nums_frequency!=None``.
Returns:
function: Function which accepts the same arguments as the QNode and one additional
keyword argument ``f0`` to provide the QNode value at the given arguments.
When called, this function will return a dictionary of dictionaries,
formatted like ``nums_frequency`` or ``spectra`` ,
that contains the univariate reconstructions per QNode parameter.
For each provided ``id`` in ``ids``, the QNode is restricted to varying the single QNode
parameter corresponding to the ``id`` . This univariate function is then reconstructed
via a Fourier transform or Dirichlet kernels, depending on the provided input.
Either the frequency ``spectra`` of the QNode with respect to its input parameters or
the numbers of frequencies, ``nums_frequency`` , per parameter must be provided.
For quantum-circuit specific details, we refer the reader to
`Vidal and Theis (2018) <https://arxiv.org/abs/1812.06323>`__ ,
`Vidal and Theis (2020) <https://www.frontiersin.org/articles/10.3389/fphy.2020.00297/full>`__ ,
`Schuld, Sweke and Meyer (2021) <https://journals.aps.org/pra/abstract/10.1103/PhysRevA.103.032430>`__ ,
and
`Wierichs, Izaac, Wang and Lin (2022) <https://doi.org/10.22331/q-2022-03-30-677>`__ .
An introduction to the concept of quantum circuits as Fourier series can also be found in
the
`Quantum models as Fourier series <https://pennylane.ai/qml/demos/tutorial_expressivity_fourier_series.html>`__
and
`General parameter-shift rules <https://pennylane.ai/qml/demos/tutorial_general_parshift.html>`__
demos as well as the
:mod:`qml.fourier <pennylane.fourier>` module docstring.
**Example**
Consider the following QNode:
.. code-block:: python
dev = qml.device("default.qubit", wires=2)
@qml.qnode(dev)
def circuit(x, Y):
qml.RX(x, wires=0)
qml.RY(Y[0], wires=0)
qml.RY(Y[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RY(5* Y[1], wires=1)
return qml.expval(qml.Z(0) @ qml.Z(1))
x = 0.4
Y = np.array([1.9, -0.5])
f = 2.3
circuit_value = circuit(x, Y)
It has three variational parameters ``x`` (a scalar) and two entries of ``Y``
(an array-like).
A reconstruction job could then be with respect to the two entries of ``Y``,
which enter the circuit with one and six integer frequencies, respectively
(see the additional examples below for details on how to obtain the frequency
spectrum if it is not known):
>>> nums_frequency = {"Y": {(0,): 1, (1,): 6}}
>>> with qml.Tracker(circuit.device) as tracker:
... rec = qml.fourier.reconstruct(circuit, {"Y": [(0,), (1,)]}, nums_frequency)(x, Y)
>>> rec.keys()
dict_keys(['Y'])
>>> print(*rec["Y"].items(), sep="\n")
((0,), <function _reconstruct_equ.<locals>._reconstruction at 0x7fbd685aee50>)
((1,), <function _reconstruct_equ.<locals>._reconstruction at 0x7fbd6866eee0>)
>>> recon_Y0 = rec["Y"][(0,)]
>>> recon_Y1 = rec["Y"][(1,)]
>>> np.isclose(recon_Y0(Y[0]), circuit_value)
True
>>> np.isclose(recon_Y1(Y[1]+1.3), circuit(x, Y+np.eye(2)[1]*1.3))
True
We successfully reconstructed the dependence on the two entries of ``Y`` ,
keeping ``x`` and the respective other entry in ``Y`` at their initial values.
Let us also see how many executions of the device were used to obtain the
reconstructions:
>>> tracker.totals
{'batches': 15, 'simulations': 15, 'executions': 15}
The example above used that we already knew the frequency spectra of the
QNode of interest. However, this is in general not the case and we may need
to compute the spectrum first. This can be done with
:func:`.fourier.qnode_spectrum` :
>>> spectra = qml.fourier.qnode_spectrum(circuit)(x, Y)
>>> spectra.keys()
dict_keys(['x', 'Y'])
>>> spectra["x"]
{(): [-1.0, 0.0, 1.0]}
>>> print(*spectra["Y"].items(), sep="\n")
((0,), [-1.0, 0.0, 1.0])
((1,), [-6.0, -5.0, -4.0, -1.0, 0.0, 1.0, 4.0, 5.0, 6.0])
For more detailed explanations, usage details and additional examples, see
the usage details section below.
.. details::
:title: Usage Details
**Input formatting**
As described briefly above, the essential inputs to ``reconstruct`` that provide information
about the QNode are given as dictionaries of dictionaries, where the outer keys reference
the argument names of ``qnode`` and the inner keys reference the parameter indices within
each array-valued QNode argument. These parameter indices always are tuples, so that
for scalar-valued QNode parameters, the parameter index is ``()`` by convention and the
``i`` -th parameter of a one-dimensional array can be accessed via ``(i,)`` .
For example, providing ``nums_frequency``
- for a scalar argument: ``nums_frequency = {"x": {(): 4}}``
- for a one-dimensional argument: ``nums_frequency = {"Y": {(0,): 2, (1,): 9, (4,): 1}}``
- for a three-dimensional argument: ``nums_frequency = {"Z": {(0, 2, 5): 2, (1, 1, 4): 1}}``
This applies to ``nums_frequency`` , ``spectra`` , and ``shifts`` .
Note that the information provided in ``nums_frequency`` / ``spectra`` is essential for
the correctness of the reconstruction.
On the other hand, the input format for ``ids`` is flexible and allows a collection of
parameter indices for each QNode argument name (as a ``dict`` ), a collection of argument
names (as a ``list``, ``set``, ``tuple`` or similar), or a single argument name
(as a ``str`` ) to be defined. For ``ids=None`` , all argument names contained in
``nums_frequency`` -- or ``spectra`` if ``nums_frequency`` is not used -- are considered.
For inputs that do not specify parameter indices per QNode argument name (all formats but
``dict`` ), these parameter indices are inferred from ``nums_frequency`` / ``spectra`` .
**Reconstruction cost**
The reconstruction cost -- in terms of calls to ``qnode`` -- depend on the number of
frequencies given via ``nums_frequency`` or ``spectra`` . A univariate reconstruction
for :math:`R` frequencies takes :math:`2R+1` evaluations. If multiple univariate
reconstructions are performed at the same point with various numbers of frequencies
:math:`R_k` , the cost are :math:`1+2\sum_k R_k` if the shift :math:`0` is used in all
of them. This is in particular the case if ``nums_frequency`` or ``spectra`` with
``shifts=None`` is used.
If the number of frequencies is too large or the given frequency spectrum contains
more than the spectrum of ``qnode`` , the reconstruction is performed suboptimally
but remains correct.
For integer-valued spectra with gaps, the equidistant reconstruction is thus suboptimal
and the non-equidistant version method be used (also see the examples below).
**Numerical stability**
In general, the reconstruction with equidistant shifts for equidistant frequencies
(used if ``nums_frequency`` is provided) is more stable numerically than the more
general Fourier reconstruction (used if ``nums_frequency=None`` ).
If the system of equations to be solved in the Fourier transform is
ill-conditioned, a warning is raised as the output might become unstable.
Examples for this are shift values or frequencies that lie very close to each other.
**Differentiability**
The returned scalar functions are differentiable in all interfaces with respect
to their scalar input variable. They expect these inputs to be in the same
interface as the one used by the QNode. More advanced differentiability, for example
of the reconstructions with respect to QNode properties, is not supported
reliably yet.
.. warning::
When using ``TensorFlow`` or ``Autograd`` *and* ``nums_frequency`` ,
the reconstructed functions are not differentiable at the point of
reconstruction. One workaround for this is to use ``spectra`` as
input instead and to thereby use the Fourier transform instead of
Dirichlet kernels. Alternatively, the original QNode evaluation can
be used.
**More examples**
Consider the QNode from the example above, now with an additional, tunable frequency
``f`` for the Pauli-X rotation that is controlled by ``x`` :
.. code-block:: python
@qml.qnode(dev)
def circuit(x, Y, f=1.0):
qml.RX(f * x, wires=0)
qml.RY(Y[0], wires=0)
qml.RY(Y[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RY(5* Y[1], wires=1)
return qml.expval(qml.Z(0) @ qml.Z(1))
f = 2.3
circuit_value = circuit(x, Y)
We repeat the reconstruction job for the dependence on ``Y[1]`` .
Note that even though information about ``Y[0]`` is contained in ``nums_frequency`` ,
``ids`` determines which reconstructions are performed.
>>> with qml.Tracker(circuit.device) as tracker:
... rec = qml.fourier.reconstruct(circuit, {"Y": [(1,)]}, nums_frequency)(x, Y)
>>> tracker.totals
{'executions': 13}
As expected, we required :math:`2R+1=2\cdot 6+1=13` circuit executions. However, not
all frequencies below :math:`f_\text{max}=6` are present in the circuit, so that
a reconstruction using knowledge of the full frequency spectrum will be cheaper:
>>> spectra = {"Y": {(1,): [0., 1., 4., 5., 6.]}}
>>> with tracker:
... rec = qml.fourier.reconstruct(circuit, {"Y": [(1,)]}, None, spectra)(x, Y)
>>> tracker.totals
{'executions': 9}
We again obtain the full univariate dependence on ``Y[1]`` but with considerably
fewer executions on the quantum device.
Once we obtained the classical function that describes the dependence, no
additional circuit evaluations are performed:
>>> with tracker:
... for Y1 in np.arange(-np.pi, np.pi, 20):
... rec["Y"][(1,)](-2.1)
>>> tracker.totals
{}
If we want to reconstruct the dependence of ``circuit`` on ``x`` , we cannot use
``nums_frequency`` if ``f`` is not an integer. One could rescale ``x`` to obtain
the frequency :math:`1` again, or directly use ``spectra`` . We will combine the
latter with another reconstruction with respect to ``Y[0]`` :
>>> spectra = {"x": {(): [0., f]}, "Y": {(0,): [0., 1.]}}
>>> with tracker:
... rec = qml.fourier.reconstruct(circuit, None, None, spectra)(x, Y, f=f)
>>> tracker.totals
{'executions': 5}
>>> recon_x = rec["x"][()]
>>> np.isclose(recon_x(x+0.5), circuit(x+0.5, Y, f=f)
True
Note that by convention, the parameter index for a scalar variable is ``()`` and
that the frequency :math:`0` always needs to be included in the spectra.
Furthermore, we here skipped the input ``ids`` so that the reconstruction
was performed for all keys in ``spectra`` .
The reconstruction with a single non-zero frequency
costs three evaluations of ``circuit`` for each, ``x`` and ``Y[0]`` . Performing
both reconstructions at the same position allowed us to save one of the
evaluations and reduce the number of calls to :math:`5`.
"""
# pylint: disable=cell-var-from-loop, unused-argument
atol = 1e-8
ids, recon_fn, jobs, need_f0 = _prepare_jobs(ids, nums_frequency, spectra, shifts, atol)
sign_fn = qnode.func if isinstance(qnode, qml.QNode) else qnode
arg_names = list(signature(sign_fn).parameters.keys())
arg_idx_from_names = {arg_name: i for i, arg_name in enumerate(arg_names)}
@wraps(qnode)
def wrapper(*args, f0=None, **kwargs):
if f0 is None and need_f0:
f0 = qnode(*args, **kwargs)
interface = qml.math.get_interface(args[0])
def constant_fn(x):
"""Univariate reconstruction of a constant Fourier series."""
return f0
# Carry out the reconstruction jobs
reconstructions = {}
for arg_name, inner_dict in jobs.items():
_reconstructions = {}
arg_idx = arg_idx_from_names[arg_name]
for par_idx, job in inner_dict.items():
if job is None:
_reconstructions[par_idx] = constant_fn
else:
if len(qml.math.shape(args[arg_idx])) == 0:
shift_vec = qml.math.ones_like(args[arg_idx])
x0 = args[arg_idx]
else:
shift_vec = qml.math.zeros_like(args[arg_idx])
shift_vec = qml.math.scatter_element_add(shift_vec, par_idx, 1.0)
x0 = args[arg_idx][par_idx]
def _univariate_fn(x):
new_arg = args[arg_idx] + shift_vec * x
new_args = args[:arg_idx] + (new_arg,) + args[arg_idx + 1 :]
return qnode(*new_args, **kwargs)
_reconstructions[par_idx] = recon_fn(
_univariate_fn, **job, x0=x0, f0=f0, interface=interface
)
reconstructions[arg_name] = _reconstructions
return reconstructions
return wrapper
```

_modules/pennylane/fourier/reconstruct

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