# Source code for pennylane.resource.measurement

# Copyright 2018-2022 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
"""
This module contains the functions needed for estimating the number of measurements and the error
for computing expectation values.
"""
from pennylane import numpy as np

[docs]def estimate_shots(coeffs, variances=None, error=0.0016):
r"""Estimate the number of measurements required to compute an expectation value with a target
error.

See also :func:estimate_error.

Args:
coeffs (list[tensor_like]): list of coefficient groups
variances (list[float]): variances of the Pauli word groups
error (float): target error in computing the expectation value

Returns:
int: the number of measurements

**Example**

>>> coeffs = [np.array([-0.32707061, 0.7896887]), np.array([0.18121046])]
>>> qml.resource.estimate_shots(coeffs)
419218

.. details::
:title: Theory

An estimation for the number of measurements :math:M required to predict the expectation
value of an observable :math:H = \sum_i A_i, with :math:A = \sum_j c_j O_j representing
a linear combination of Pauli words, can be obtained following Eq. (34) of
[PRX Quantum 2, 040320 (2021) <https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.2.040320>_]
as

.. math::

M = \frac{\left ( \sum_i \sqrt{\text{Var}(A_i)} \right )^2}{\epsilon^2},

with :math:\epsilon and :math:\text{Var}(A) denoting the target error in computing
:math:\left \langle H \right \rangle and the variance in computing
:math:\left \langle A \right \rangle, respectively. It has been shown in Eq. (10) of
[arXiv:2201.01471v3 <https://arxiv.org/abs/2201.01471v3>_] that
the variances can be computed from the covariances between the Pauli words as

.. math::

\text{Var}(A_i) = \sum_{jk} c_j c_k \text{Cov}(O_j, O_k),

where

.. math::

\text{Cov}(O_j, O_k) = \left \langle O_j O_k \right \rangle - \left \langle O_j \right \rangle \left \langle O_k \right \rangle.

The values of :math:\text{Cov}(O_j, O_k) are not known a priori and should be either
computed from affordable classical methods, such as the configuration interaction with
singles and doubles (CISD), or approximated with other methods. If the variances are not
provided to the function as input, they will be approximated following Eqs. (6-7) of
[Phys. Rev. Research 4, 033154, 2022 <https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.033154>_]
by assuming
:math:\text{Cov}(O_j, O_k) =0 for :math:j \neq k and using :math:\text{Var}(O_i) \leq 1
from

.. math::

\text{Var}(O_i) = \left \langle O_i^2 \right \rangle - \left \langle O_i \right \rangle^2 = 1 - \left \langle O_i \right \rangle^2.

This approximation gives

.. math::

M \approx \frac{\left ( \sum_i \sqrt{\sum_j c_{ij}^2} \right )^2}{\epsilon^2},

where :math:i and :math:j run over the observable groups and the Pauli words inside the
group, respectively.
"""
if variances:
return int(np.ceil(np.sum(np.sqrt(variances)) ** 2 / error**2))

group_sum = [np.sum(coeff**2) for coeff in coeffs]
return int(np.ceil(np.sum(np.sqrt(group_sum)) ** 2 / error**2))

[docs]def estimate_error(coeffs, variances=None, shots=1000):
r"""Estimate the error in computing an expectation value with a given number of measurements.

See also :func:estimate_shots.

Args:
coeffs (list[tensor_like]): list of coefficient groups
variances (list[float]): variances of the Pauli word groups
shots (int): the number of measurements

Returns:
float: target error in computing the expectation value

**Example**

>>> coeffs = [np.array([-0.32707061, 0.7896887]), np.array([0.18121046])]
>>> qml.resource.estimate_error(coeffs, shots=100000)
0.00327597

.. details::
:title: Theory

An estimation for the error :math:\epsilon in predicting the expectation
value of an observable :math:H = \sum_i A_i with :math:A = \sum_j c_j O_j representing a
linear combination of Pauli words can be obtained following Eq. (34) of
[PRX Quantum 2, 040320 (2021) <https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.2.040320>_]
as

.. math::

\epsilon = \frac{\sum_i \sqrt{\text{Var}(A_i)}}{\sqrt{M}},

with :math:M and :math:\text{Var}(A) denoting the number of measurements and the
variance in computing :math:\left \langle A \right \rangle, respectively. It has been
shown in Eq. (10) of
[arXiv:2201.01471v3 <https://arxiv.org/abs/2201.01471v3>_] that the variances can be
computed from the covariances between the Pauli words as

.. math::

\text{Var}(A_i) = \sum_{jk} c_j c_k \text{Cov}(O_j, O_k),

where

.. math::

\text{Cov}(O_j, O_k) = \left \langle O_j O_k \right \rangle - \left \langle O_j \right \rangle \left \langle O_k \right \rangle.

The values of :math:\text{Cov}(O_j, O_k) are not known a priori and should be either
computed from affordable classical methods, such as the configuration interaction with
singles and doubles (CISD), or approximated with other methods. If the variances are not
provided to the function as input, they will be approximated following Eqs. (6-7) of
[Phys. Rev. Research 4, 033154, 2022 <https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.033154>_]
by assuming
:math:\text{Cov}(O_j, O_k) =0 for :math:j \neq k and using :math:\text{Var}(O_i) \leq 1
from

.. math::

\text{Var}(O_i) = \left \langle O_i^2 \right \rangle - \left \langle O_i \right \rangle^2 = 1 - \left \langle O_i \right \rangle^2.

This approximation gives

.. math::

\epsilon \approx \frac{\sum_i \sqrt{\sum_j c_{ij}^2}}{\sqrt{M}},

where :math:i and :math:j run over the observable groups and the Pauli words inside the
group, respectively.
"""
if variances:
return np.sum(np.sqrt(variances)) / np.sqrt(shots)

group_sum = [np.sum(coeff**2) for coeff in coeffs]
return np.sum(np.sqrt(group_sum)) / np.sqrt(shots)


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