Source code for qiskit_algorithms.amplitude_amplifiers.grover

# This code is part of a Qiskit project.
# (C) Copyright IBM 2018, 2023.
# 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
# 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."""
from __future__ import annotations

import itertools
from import Iterator, Generator
from typing import Any

import numpy as np

from qiskit import ClassicalRegister, QuantumCircuit
from qiskit.primitives import BaseSampler
from qiskit.quantum_info import Statevector

from qiskit_algorithms.exceptions import AlgorithmError
from qiskit_algorithms.utils import algorithm_globals

from .amplification_problem import AmplificationProblem
from .amplitude_amplifier import AmplitudeAmplifier, AmplitudeAmplifierResult

[docs]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 <>`_. or the `Qiskit Tutorials <>`_ 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 <>`_. [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 <>`_. """ def __init__( self, iterations: list[int] | Iterator[int] | int | None = None, growth_rate: float | None = None, sample_from_iterations: bool = False, sampler: BaseSampler | None = 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. 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 `<>`_ """ # 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: Generator[int, None, None] | list[int] = ( int(growth_rate**x) for x in itertools.count(1) ) elif isinstance(iterations, int): self._iterations = [iterations] else: self._iterations = iterations # type: ignore[assignment] self._sampler = sampler self._sample_from_iterations = sample_from_iterations self._iterations_arg = iterations @property def sampler(self) -> BaseSampler | None: """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
[docs] 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 sampler is not set. AlgorithmError: If 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: raise ValueError("A sampler must be provided.") if isinstance(self._iterations, list): max_iterations = len(self._iterations) max_power = np.inf # no cap on the power iterator: Iterator[int] = iter(self._iterations) else: max_iterations = max(10, 2** 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 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 = algorithm_globals.random.integers(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 =[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: dict[str, Any] | Statevector | np.ndarray = { 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] # type: ignore[union-attr] ) 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 # type: ignore[assignment] result.max_probability = max_probability return result
[docs] @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)))
[docs] def construct_circuit( self, problem: AmplificationProblem, power: int | None = 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: # type: ignore[arg-type] raise ValueError("Please pass ``power`` if the iterations are not an integer.") power = self._iterations[0] # type: ignore[index] qc = QuantumCircuit(, 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
[docs]class GroverResult(AmplitudeAmplifierResult): """Grover Result.""" def __init__(self) -> None: super().__init__() self._iterations: list[int] | None = 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