# Quellcode für qiskit.aqua.operators.expectations.cvar_expectation

```
# This code is part of Qiskit.
#
# (C) Copyright IBM 2020.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""The CVaR (Conditional Value at Risk) expectation class."""
import logging
from typing import Union, Optional
from ..operator_base import OperatorBase
from ..list_ops import ListOp, ComposedOp
from ..state_fns import CVaRMeasurement, OperatorStateFn
from .expectation_base import ExpectationBase
from .pauli_expectation import PauliExpectation
from .aer_pauli_expectation import AerPauliExpectation
logger = logging.getLogger(__name__)
[Doku]class CVaRExpectation(ExpectationBase):
r"""Compute the Conditional Value at Risk (CVaR) expectation value.
The standard approach to calculating the expectation value of a Hamiltonian w.r.t. a
state is to take the sample mean of the measurement outcomes. This corresponds to an estimator
of the energy. However in several problem settings with a diagonal Hamiltonian, e.g.
in combinatorial optimization where the Hamiltonian encodes a cost function, we are not
interested in calculating the energy but in the lowest possible value we can find.
To this end, we might consider using the best observed sample as a cost function during
variational optimization. The issue here, is that this can result in a non-smooth optimization
surface. To resolve this issue, we can smooth the optimization surface by using not just the
best observed sample, but instead average over some fraction of best observed samples.
This is exactly what the CVaR estimator accomplishes [1].
It is empirically shown, that this can lead to faster convergence for combinatorial
optimization problems.
Let :math:`\alpha` be a real number in :math:`[0,1]` which specifies the fraction of best
observed samples which are used to compute the objective function. Observe that if
:math:`\alpha = 1`, CVaR is equivalent to a standard expectation value. Similarly,
if :math:`\alpha = 0`, then CVaR corresponds to using the best observed sample.
Intermediate values of :math:`\alpha` interpolate between these two objective functions.
References:
[1]: Barkoutsos, P. K., Nannicini, G., Robert, A., Tavernelli, I., and Woerner, S.,
"Improving Variational Quantum Optimization using CVaR"
`arXiv:1907.04769 <https://arxiv.org/abs/1907.04769>`_
"""
[Doku] def __init__(self, alpha: float, expectation: Optional[ExpectationBase] = None) -> None:
"""
Args:
alpha: The alpha value describing the quantile considered in the expectation value.
expectation: An expectation object to compute the expectation value. Defaults
to the PauliExpectation calculation.
Raises:
NotImplementedError: If the ``expectation`` is an AerPauliExpecation.
"""
self.alpha = alpha
if isinstance(expectation, AerPauliExpectation):
raise NotImplementedError('AerPauliExpecation currently not supported.')
if expectation is None:
expectation = PauliExpectation()
self.expectation = expectation
[Doku] def convert(self, operator: OperatorBase) -> OperatorBase:
"""Return an expression that computes the CVaR expectation upon calling ``eval``.
Args:
operator: The operator to convert.
Returns:
The converted operator.
"""
expectation = self.expectation.convert(operator)
# replace OperatorMeasurements by CVaRMeasurement
def replace_with_cvar(operator):
if isinstance(operator, OperatorStateFn) and operator.is_measurement:
return CVaRMeasurement(operator.primitive, alpha=self.alpha)
elif isinstance(operator, ListOp):
return operator.traverse(replace_with_cvar)
return operator
return replace_with_cvar(expectation)
[Doku] def compute_variance(self, exp_op: OperatorBase) -> Union[list, float]:
"""Returns the variance of the CVaR calculation
Args:
exp_op: The operator whose evaluation yields an expectation
of some StateFn against a diagonal observable.
Returns:
The variance of the CVaR estimate corresponding to the converted
exp_op.
Raises:
ValueError: If the exp_op does not correspond to an expectation value.
"""
def cvar_variance(operator):
if isinstance(operator, ComposedOp):
sfdict = operator.oplist[1]
measurement = operator.oplist[0]
return measurement.eval_variance(sfdict)
elif isinstance(operator, ListOp):
return operator.combo_fn([cvar_variance(op) for op in operator.oplist])
raise ValueError("Input operator does not correspond to a value "
"expectation value.")
cvar_op = self.convert(exp_op)
return cvar_variance(cvar_op)
```