Note

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)

The oracle can now be used to create an Grover instance:

grover = Grover(oracle)

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)

[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 000, 011, and 101 (note the bit order in each string), corresponding to the three satisfying solutions all have high probabilities associated with them.

plot_histogram(result.measurement)

Boolean Logical Expressions

Qiskit’s 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 ^, ~, and & 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 w and x taking different values; ~(y ^ z) requires y and 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.

TruthTable Oracles

With Qiskit, 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 0000 and 1111.

Next, we can setup the Oracle and 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

Version Information

Qiskit Software	Version
Qiskit	0.24.1
Terra	0.16.4
Aer	0.7.6
Ignis	0.5.2
Aqua	0.8.2
IBM Q Provider	0.12.2
System information
Python	3.7.7 (default, Apr 22 2020, 19:15:10) [GCC 9.3.0]
OS	Linux
CPUs	32
Memory (Gb)	125.71903228759766
Tue May 25 15:04:26 2021 EDT

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