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Grover’s algorithm examples#
This notebook has examples demonstrating how to use the Qiskit Algorithms Grover search algorithm, with different oracles.
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 from qiskit.circuit.library.phase_oracle import PhaseOracle 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 ImportError as ex: print(ex) finally: os.remove(file_name)
oracle can now be used to create an Grover instance:
from qiskit_algorithms import AmplificationProblem 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:
from qiskit_algorithms import Grover from qiskit.primitives import Sampler grover = Grover(sampler=Sampler()) 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
Sampler, 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.
from qiskit.tools.visualization import plot_histogram 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(sampler=Sampler()) 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
|Python compiler||GCC 11.4.0|
|Python build||default, Aug 28 2023 08:27:22|
|Wed Nov 29 12:08:37 2023 UTC|
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