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Grover’s algorithm examples¶
This notebook has examples demonstrating how to use the Qiskit Grover search algorithm, with different oracles.
import pylab import numpy as np from qiskit import Aer from qiskit.utils import QuantumInstance from qiskit.tools.visualization import plot_histogram from qiskit.algorithms import Grover, AmplificationProblem from qiskit.circuit.library.phase_oracle import PhaseOracle
Finding solutions to 3-SAT problems¶
Let’s look at an example 3-Satisfiability (3-SAT) problem and walk-through how we can use Quantum Search to find its satisfying solutions. 3-SAT problems are usually expressed in Conjunctive Normal Forms (CNF) and written in the DIMACS-CNF format. For example:
input_3sat_instance = ''' c example DIMACS-CNF 3-SAT p cnf 3 5 -1 -2 -3 0 1 -2 3 0 1 2 -3 0 1 -2 -3 0 -1 2 3 0 '''
The CNF of this 3-SAT instance contains 3 variables and 5 clauses:
\((\neg v_1 \vee \neg v_2 \vee \neg v_3) \wedge (v_1 \vee \neg v_2 \vee v_3) \wedge (v_1 \vee v_2 \vee \neg v_3) \wedge (v_1 \vee \neg v_2 \vee \neg v_3) \wedge (\neg v_1 \vee v_2 \vee v_3)\)
It can be verified that this 3-SAT problem instance has three satisfying solutions:
\((v_1, v_2, v_3) = (T, F, T)\) or \((F, F, F)\) or \((T, T, F)\)
Or, expressed using the DIMACS notation:
1 -2 3, or
-1 -2 -3, or
1 2 -3.
With this example problem input, we then create the corresponding
oracle for our
Grover search. In particular, we use the
PhaseOracle component, which supports parsing DIMACS-CNF format strings and constructing the corresponding oracle circuit.
import os import tempfile from qiskit.exceptions import MissingOptionalLibraryError fp = tempfile.NamedTemporaryFile(mode='w+t', delete=False) fp.write(input_3sat_instance) file_name = fp.name fp.close() oracle = None try: oracle = PhaseOracle.from_dimacs_file(file_name) except MissingOptionalLibraryError as ex: print(ex) finally: os.remove(file_name)
oracle can now be used to create an Grover instance:
problem = None if oracle is not None: problem = AmplificationProblem(oracle, is_good_state=oracle.evaluate_bitstring)
We can then configure the backend and run the Grover instance to get the result:
backend = Aer.get_backend('aer_simulator') quantum_instance = QuantumInstance(backend, shots=1024) grover = Grover(quantum_instance=quantum_instance) result = None if problem is not None: result = grover.amplify(problem) print(result.assignment)
As seen above, a satisfying solution to the specified 3-SAT problem is obtained. And it is indeed one of the three satisfying solutions.
Since we used the
'aer_simulator', the complete measurement result is also returned, as shown in the plot below, where it can be seen that the binary strings
101 (note the bit order in each string), corresponding to the three satisfying solutions all have high probabilities associated with them.
if result is not None: display(plot_histogram(result.circuit_results))
Boolean Logical Expressions¶
Grover can also be used to perform Quantum Search on an
Oracle constructed from other means, in addition to DIMACS. For example, the
PhaseOracle can actually be configured using arbitrary Boolean logical expressions, as demonstrated below.
expression = '(w ^ x) & ~(y ^ z) & (x & y & z)' try: oracle = PhaseOracle(expression) problem = AmplificationProblem(oracle, is_good_state=oracle.evaluate_bitstring) grover = Grover(quantum_instance=QuantumInstance(Aer.get_backend('aer_simulator'), shots=1024)) result = grover.amplify(problem) display(plot_histogram(result.circuit_results)) except MissingOptionalLibraryError as ex: print(ex)
In the example above, the input Boolean logical expression
'(w ^ x) & ~(y ^ z) & (x & y & z)' should be quite self-explanatory, where
& represent the Boolean logical XOR, NOT, and AND operators, respectively. It should be quite easy to figure out the satisfying solution by examining its parts:
w ^ x calls for
x taking different values;
~(y ^ z) requires
z be the same;
x & y & z dictates all three to be
True. Putting these
together, we get the satisfying solution
(w, x, y, z) = (False, True, True, True), which our
Grover’s result agrees with.
import qiskit.tools.jupyter %qiskit_version_table %qiskit_copyright
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/__init__.py:86: DeprecationWarning: The package qiskit.aqua is deprecated. It was moved/refactored to qiskit-terra For more information see <https://github.com/Qiskit/qiskit-aqua/blob/main/README.md#migration-guide> warn_package('aqua', 'qiskit-terra')
|Python||3.8.12 (default, Sep 13 2021, 08:28:12) [GCC 9.3.0]|
|Thu Oct 21 17:47:24 2021 UTC|
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