This page was generated from tutorials/algorithms/08_grover_examples.ipynb.
Run interactively in the IBM Quantum lab.
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 BasicAer from qiskit.tools.visualization import plot_histogram from qiskit.aqua import QuantumInstance from qiskit.aqua.algorithms import Grover from qiskit.aqua.components.oracles import LogicalExpressionOracle, TruthTableOracle
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
LogicalExpressionOracle component, which supports parsing DIMACS-CNF format strings and constructing the corresponding oracle circuit.
oracle = LogicalExpressionOracle(input_3sat_instance)
oracle can now be used to create an Grover instance:
grover = Grover(oracle)
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/algorithms/amplitude_amplifiers/grover.py:215: DeprecationWarning: The package qiskit.aqua.algorithms.amplitude_amplifiers is deprecated. It was moved/refactored to qiskit.algorithms.amplitude_amplifiers (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide> warn_package('aqua.algorithms.amplitude_amplifiers',
We can then configure the backend and run the Grover instance to get the result:
backend = BasicAer.get_backend('qasm_simulator') quantum_instance = QuantumInstance(backend, shots=1024) result = grover.run(quantum_instance) print(result.assignment)
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/quantum_instance.py:135: DeprecationWarning: The class qiskit.aqua.QuantumInstance is deprecated. It was moved/refactored to qiskit.utils.QuantumInstance (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide> warn_class('aqua.QuantumInstance',
[1, -2, 3]
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
'qasm_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.
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
LogicalExpressionOracle can actually be configured using arbitrary Boolean logical expressions, as demonstrated below.
expression = '(w ^ x) & ~(y ^ z) & (x & y & z)' oracle = LogicalExpressionOracle(expression) grover = Grover(oracle) result = grover.run(QuantumInstance(BasicAer.get_backend('qasm_simulator'), shots=1024)) plot_histogram(result.measurement)
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.
Oracles can also be constructed from truth tables, meaning we can also perform Quantum Search on truth tables. Even though this might seem like a moot point as we would be essentially searching for entries of a truth table with the \(1\) value, it’s a good example for demonstrative purpose.
truthtable = '1000000000000001'
As shown, the
truthtable is specified with a bitstring containing values of all entries in the table. It has length \(16\), so the corresponding truth table is of \(4\) input bits. Since the very first and last values are \(1\), the corresponding truth table target entries are
Next, we can setup the
Grover objects to perform Quantum Search as usual.
oracle = TruthTableOracle(truthtable) grover = Grover(oracle) result = grover.run(QuantumInstance(BasicAer.get_backend('qasm_simulator'), shots=1024)) plot_histogram(result.measurement)
As seen in the above plot the search result coincides with our expectation.
import qiskit.tools.jupyter %qiskit_version_table %qiskit_copyright
|IBM Q Provider||0.12.3|
|Python||3.8.10 (default, May 4 2021, 07:16:51) [GCC 9.3.0]|
|Wed May 05 19:46:13 2021 UTC|
This code is a part of Qiskit
© Copyright IBM 2017, 2021.
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.