# Quellcode für qiskit.algorithms.amplitude_amplifiers.grover

```
# This code is part of Qiskit.
#
# (C) Copyright IBM 2018, 2022.
#
# 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.
"""Grover's search algorithm."""
import itertools
import operator
from typing import Iterator, List, Optional, Union
import warnings
import numpy as np
from qiskit import ClassicalRegister, QuantumCircuit
from qiskit.algorithms.exceptions import AlgorithmError
from qiskit.primitives import BaseSampler
from qiskit.providers import Backend
from qiskit.quantum_info import partial_trace
from qiskit.utils import QuantumInstance
from qiskit.utils.deprecation import deprecate_function
from .amplification_problem import AmplificationProblem
from .amplitude_amplifier import AmplitudeAmplifier, AmplitudeAmplifierResult
[Doku]class Grover(AmplitudeAmplifier):
r"""Grover's Search algorithm.
.. note::
If you want to learn more about the theory behind Grover's Search algorithm, check
out the `Qiskit Textbook <https://qiskit.org/textbook/ch-algorithms/grover.html>`_.
or the `Qiskit Tutorials
<https://qiskit.org/documentation/tutorials/algorithms/07_grover_examples.html>`_
for more concrete how-to examples.
Grover's Search [1, 2] is a well known quantum algorithm that can be used for
searching through unstructured collections of records for particular targets
with quadratic speedup compared to classical algorithms.
Given a set :math:`X` of :math:`N` elements :math:`X=\{x_1,x_2,\ldots,x_N\}`
and a boolean function :math:`f : X \rightarrow \{0,1\}`, the goal of an
unstructured-search problem is to find an element :math:`x^* \in X` such
that :math:`f(x^*)=1`.
The search is called *unstructured* because there are no guarantees as to how
the database is ordered. On a sorted database, for instance, one could perform
binary search to find an element in :math:`\mathbb{O}(\log N)` worst-case time.
Instead, in an unstructured-search problem, there is no prior knowledge about
the contents of the database. With classical circuits, there is no alternative
but to perform a linear number of queries to find the target element.
Conversely, Grover's Search algorithm allows to solve the unstructured-search
problem on a quantum computer in :math:`\mathcal{O}(\sqrt{N})` queries.
To carry out this search a so-called oracle is required, that flags a good element/state.
The action of the oracle :math:`\mathcal{S}_f` is
.. math::
\mathcal{S}_f |x\rangle = (-1)^{f(x)} |x\rangle,
i.e. it flips the phase of the state :math:`|x\rangle` if :math:`x` is a hit.
The details of how :math:`S_f` works are unimportant to the algorithm; Grover's
search algorithm treats the oracle as a black box.
This class supports oracles in form of a :class:`~qiskit.circuit.QuantumCircuit`.
With the given oracle, Grover's Search constructs the Grover operator to amplify the
amplitudes of the good states:
.. math::
\mathcal{Q} = H^{\otimes n} \mathcal{S}_0 H^{\otimes n} \mathcal{S}_f
= D \mathcal{S}_f,
where :math:`\mathcal{S}_0` flips the phase of the all-zero state and acts as identity
on all other states. Sometimes the first three operands are summarized as diffusion operator,
which implements a reflection over the equal superposition state.
If the number of solutions is known, we can calculate how often :math:`\mathcal{Q}` should be
applied to find a solution with very high probability, see the method
`optimal_num_iterations`. If the number of solutions is unknown, the algorithm tries different
powers of Grover's operator, see the `iterations` argument, and after each iteration checks
if a good state has been measured using `good_state`.
The generalization of Grover's Search, Quantum Amplitude Amplification [3], uses a modified
version of :math:`\mathcal{Q}` where the diffusion operator does not reflect about the
equal superposition state, but another state specified via an operator :math:`\mathcal{A}`:
.. math::
\mathcal{Q} = \mathcal{A} \mathcal{S}_0 \mathcal{A}^\dagger \mathcal{S}_f.
For more information, see the :class:`~qiskit.circuit.library.GroverOperator` in the
circuit library.
References:
[1]: L. K. Grover (1996), A fast quantum mechanical algorithm for database search,
`arXiv:quant-ph/9605043 <https://arxiv.org/abs/quant-ph/9605043>`_.
[2]: I. Chuang & M. Nielsen, Quantum Computation and Quantum Information,
Cambridge: Cambridge University Press, 2000. Chapter 6.1.2.
[3]: Brassard, G., Hoyer, P., Mosca, M., & Tapp, A. (2000).
Quantum Amplitude Amplification and Estimation.
`arXiv:quant-ph/0005055 <http://arxiv.org/abs/quant-ph/0005055>`_.
"""
def __init__(
self,
iterations: Optional[Union[List[int], Iterator[int], int]] = None,
growth_rate: Optional[float] = None,
sample_from_iterations: bool = False,
quantum_instance: Optional[Union[QuantumInstance, Backend]] = None,
sampler: Optional[BaseSampler] = None,
) -> None:
r"""
Args:
iterations: Specify the number of iterations/power of Grover's operator to be checked.
* If an int, only one circuit is run with that power of the Grover operator.
If the number of solutions is known, this option should be used with the optimal
power. The optimal power can be computed with ``Grover.optimal_num_iterations``.
* If a list, all the powers in the list are run in the specified order.
* If an iterator, the powers yielded by the iterator are checked, until a maximum
number of iterations or maximum power is reached.
* If ``None``, the :obj:`AmplificationProblem` provided must have an ``is_good_state``,
and circuits are run until that good state is reached.
growth_rate: If specified, the iterator is set to increasing powers of ``growth_rate``,
i.e. to ``int(growth_rate ** 1), int(growth_rate ** 2), ...`` until a maximum
number of iterations is reached.
sample_from_iterations: If True, instead of taking the values in ``iterations`` as
powers of the Grover operator, a random integer sample between 0 and smaller value
than the iteration is used as a power, see [1], Section 4.
quantum_instance: Pending deprecation: A Quantum Instance or Backend to run the circuits.
sampler: A Sampler to use for sampling the results of the circuits.
Raises:
ValueError: If ``growth_rate`` is a float but not larger than 1.
ValueError: If both ``iterations`` and ``growth_rate`` is set.
References:
[1]: Boyer et al., Tight bounds on quantum searching
`<https://arxiv.org/abs/quant-ph/9605034>`_
"""
# set default value
if growth_rate is None and iterations is None:
growth_rate = 1.2
if growth_rate is not None and iterations is not None:
raise ValueError("Pass either a value for iterations or growth_rate, not both.")
if growth_rate is not None:
# yield iterations ** 1, iterations ** 2, etc. and casts to int
self._iterations = map(lambda x: int(growth_rate**x), itertools.count(1))
elif isinstance(iterations, int):
self._iterations = [iterations]
else:
self._iterations = iterations
if quantum_instance is not None and sampler is not None:
raise ValueError("Only one of quantum_instance or sampler can be passed, not both!")
# check positionally passing the sampler in the place of quantum_instance
# which will be removed in future
if isinstance(quantum_instance, BaseSampler):
sampler = quantum_instance
quantum_instance = None
self._quantum_instance = None
if quantum_instance is not None:
warnings.warn(
"The quantum_instance argument has been superseded by the sampler argument. "
"This argument will be deprecated in a future release and subsequently "
"removed after that.",
category=PendingDeprecationWarning,
stacklevel=2,
)
with warnings.catch_warnings():
warnings.simplefilter("ignore", category=PendingDeprecationWarning)
self.quantum_instance = quantum_instance
self._sampler = sampler
self._sample_from_iterations = sample_from_iterations
self._iterations_arg = iterations
@property
@deprecate_function(
"The Grover.quantum_instance getter is pending deprecation. "
"This property will be deprecated in a future release and subsequently "
"removed after that.",
category=PendingDeprecationWarning,
)
def quantum_instance(self) -> Optional[QuantumInstance]:
r"""Pending deprecation\; Get the quantum instance.
Returns:
The quantum instance used to run this algorithm.
"""
return self._quantum_instance
@quantum_instance.setter
@deprecate_function(
"The Grover.quantum_instance setter is pending deprecation. "
"This property will be deprecated in a future release and subsequently "
"removed after that.",
category=PendingDeprecationWarning,
)
def quantum_instance(self, quantum_instance: Union[QuantumInstance, Backend]) -> None:
r"""Pending deprecation\; Set quantum instance.
Args:
quantum_instance: The quantum instance used to run this algorithm.
"""
if isinstance(quantum_instance, Backend):
quantum_instance = QuantumInstance(quantum_instance)
self._quantum_instance = quantum_instance
@property
def sampler(self) -> Optional[BaseSampler]:
"""Get the sampler.
Returns:
The sampler used to run this algorithm.
"""
return self._sampler
@sampler.setter
def sampler(self, sampler: BaseSampler) -> None:
"""Set the sampler.
Args:
sampler: The sampler used to run this algorithm.
"""
self._sampler = sampler
[Doku] def amplify(self, amplification_problem: AmplificationProblem) -> "GroverResult":
"""Run the Grover algorithm.
Args:
amplification_problem: The amplification problem.
Returns:
The result as a ``GroverResult``, where e.g. the most likely state can be queried
as ``result.top_measurement``.
Raises:
ValueError: If a quantum instance or sampler is not set.
AlgorithmError: If a sampler job fails.
TypeError: If ``is_good_state`` is not provided and is required (i.e. when iterations
is ``None`` or a ``list``)
"""
if self._sampler is None and self._quantum_instance is None:
raise ValueError("A quantum instance or sampler must be provided.")
if self._quantum_instance is not None and self._sampler is not None:
raise ValueError("Only one of quantum_instance or sampler can be passed, not both!")
if isinstance(self._iterations, list):
max_iterations = len(self._iterations)
max_power = np.inf # no cap on the power
iterator = iter(self._iterations)
else:
max_iterations = max(10, 2**amplification_problem.oracle.num_qubits)
max_power = np.ceil(
2 ** (len(amplification_problem.grover_operator.reflection_qubits) / 2)
)
iterator = self._iterations
result = GroverResult()
iterations = []
top_measurement = "0" * len(amplification_problem.objective_qubits)
oracle_evaluation = False
all_circuit_results = []
max_probability = 0
shots = 0
for _ in range(max_iterations): # iterate at most to the max number of iterations
# get next power and check if allowed
power = next(iterator)
if power > max_power:
break
iterations.append(power) # store power
# sample from [0, power) if specified
if self._sample_from_iterations:
power = np.random.randint(power)
# Run a grover experiment for a given power of the Grover operator.
if self._sampler is not None:
qc = self.construct_circuit(amplification_problem, power, measurement=True)
job = self._sampler.run([qc])
try:
results = job.result()
except Exception as exc:
raise AlgorithmError("Sampler job failed.") from exc
num_bits = len(amplification_problem.objective_qubits)
circuit_results = {
np.binary_repr(k, num_bits): v for k, v in results.quasi_dists[0].items()
}
top_measurement, max_probability = max(circuit_results.items(), key=lambda x: x[1])
else:
if self._quantum_instance.is_statevector:
qc = self.construct_circuit(amplification_problem, power, measurement=False)
circuit_results = self._quantum_instance.execute(qc).get_statevector()
num_bits = len(amplification_problem.objective_qubits)
# trace out work qubits
if qc.width() != num_bits:
indices = [
i
for i in range(qc.num_qubits)
if i not in amplification_problem.objective_qubits
]
rho = partial_trace(circuit_results, indices)
circuit_results = np.diag(rho.data)
max_amplitude = max(circuit_results.max(), circuit_results.min(), key=abs)
max_amplitude_idx = np.where(circuit_results == max_amplitude)[0][0]
top_measurement = np.binary_repr(max_amplitude_idx, num_bits)
max_probability = np.abs(max_amplitude) ** 2
shots = 1
else:
qc = self.construct_circuit(amplification_problem, power, measurement=True)
circuit_results = self._quantum_instance.execute(qc).get_counts(qc)
top_measurement = max(circuit_results.items(), key=operator.itemgetter(1))[0]
shots = sum(circuit_results.values())
max_probability = (
max(circuit_results.items(), key=operator.itemgetter(1))[1] / shots
)
all_circuit_results.append(circuit_results)
if (isinstance(self._iterations_arg, int)) and (
amplification_problem.is_good_state is None
):
oracle_evaluation = None # cannot check for good state without is_good_state arg
break
# is_good_state arg must be provided if iterations arg is not an integer
if (
self._iterations_arg is None or isinstance(self._iterations_arg, list)
) and amplification_problem.is_good_state is None:
raise TypeError("An is_good_state function is required with the provided oracle")
# only check if top measurement is a good state if an is_good_state arg is provided
oracle_evaluation = amplification_problem.is_good_state(top_measurement)
if oracle_evaluation is True:
break # we found a solution
result.iterations = iterations
result.top_measurement = top_measurement
result.assignment = amplification_problem.post_processing(top_measurement)
result.oracle_evaluation = oracle_evaluation
result.circuit_results = all_circuit_results
result.max_probability = max_probability
return result
[Doku] @staticmethod
def optimal_num_iterations(num_solutions: int, num_qubits: int) -> int:
"""Return the optimal number of iterations, if the number of solutions is known.
Args:
num_solutions: The number of solutions.
num_qubits: The number of qubits used to encode the states.
Returns:
The optimal number of iterations for Grover's algorithm to succeed.
"""
amplitude = np.sqrt(num_solutions / 2**num_qubits)
return round(np.arccos(amplitude) / (2 * np.arcsin(amplitude)))
[Doku] def construct_circuit(
self, problem: AmplificationProblem, power: Optional[int] = None, measurement: bool = False
) -> QuantumCircuit:
"""Construct the circuit for Grover's algorithm with ``power`` Grover operators.
Args:
problem: The amplification problem for the algorithm.
power: The number of times the Grover operator is repeated. If None, this argument
is set to the first item in ``iterations``.
measurement: Boolean flag to indicate if measurement should be included in the circuit.
Returns:
QuantumCircuit: the QuantumCircuit object for the constructed circuit
Raises:
ValueError: If no power is passed and the iterations are not an integer.
"""
if power is None:
if len(self._iterations) > 1:
raise ValueError("Please pass ``power`` if the iterations are not an integer.")
power = self._iterations[0]
qc = QuantumCircuit(problem.oracle.num_qubits, name="Grover circuit")
qc.compose(problem.state_preparation, inplace=True)
if power > 0:
qc.compose(problem.grover_operator.power(power), inplace=True)
if measurement:
measurement_cr = ClassicalRegister(len(problem.objective_qubits))
qc.add_register(measurement_cr)
qc.measure(problem.objective_qubits, measurement_cr)
return qc
[Doku]class GroverResult(AmplitudeAmplifierResult):
"""Grover Result."""
def __init__(self) -> None:
super().__init__()
self._iterations = None
@property
def iterations(self) -> List[int]:
"""All the powers of the Grover operator that have been tried.
Returns:
The powers of the Grover operator tested.
"""
return self._iterations
@iterations.setter
def iterations(self, value: List[int]) -> None:
"""Set the powers of the Grover operator that have been tried.
Args:
value: A new value for the powers.
"""
self._iterations = value
```