Demo: Solving 3-SAT Using Grover's Algorithm
Introduction
Grover's algorithm [1,2] is a quantum search algorithm that enables searching a dataset of \(N\) items using \(O(\sqrt{N})\) queries, in contrast to classical algorithms which require \(\Omega(N)\) queries in the worst case. Using the algorithm allows us to speed-up various computations that are considered classically intractable. For example, the 3-SAT problem [3] is a famous \(\text{NP-Complete}\) problem, a solution of which allows solving any problem in the complexity class \(\text{NP}\).
In this demo, we will show how the Classiq platform enables us to solve an instance of 3-SAT using Grover's algorithm at a higher functional level, avoiding the implementation details one is usually exposed to when using other platforms.
The 3-SAT Formula
We now specify the 3-CNF [4] formula that we wish to find a solution for:
def _not(y):
return y ^ 1
def sat_formula(x1, x2, x3):
return (
((x1) | (x2) | (x3))
& ((_not(x1)) | (x2) | (x3))
& ((_not(x1)) | (_not(x2)) | (_not(x3)))
& ((_not(x1)) | (_not(x2)) | (x3))
& ((x1) | (x2) | (_not(x3)))
& ((_not(x1)) | (x2) | (_not(x3)))
)
NUM_VARIABLES = 3
import itertools
def print_truth_table(num_variables, boolean_func):
variables = [f"x{i}" for i in range(num_variables)]
combinations = list(itertools.product([0, 1], repeat=len(variables)))
header = " ".join([f"{var:<5}" for var in variables]) + " | Result"
print(header)
print("-" * len(header))
for combination in combinations:
result = boolean_func(*combination)
values_str = " ".join([f"{str(val):<5}" for val in combination])
print(f"{values_str} | {str(result):<5}")
We can see that the formula has 2 possible solutions:
print_truth_table(NUM_VARIABLES, sat_formula)
x0 x1 x2 | Result
----------------------------
0 0 0 | 0
0 0 1 | 0
0 1 0 | 1
0 1 1 | 1
1 0 0 | 0
1 0 1 | 0
1 1 0 | 0
1 1 1 | 0
Note that since there are multiple satisfying assignments for the formula, the goal of the Grover quantum algorithm is to find one of them. This is contrast to the usual description of the search problem (found in 2), in which there is a single solution (out of \(N\) possibilities).
Loading the Grover search model
Next, we load the Grover search model which we use to find the solution. To specify the model, we need to define the oracle it uses. Each oracle call is analogous to a classical query. In this case, we define an oracle based on the 3-CNF formula defined earlier. We use the standard phase_oracle that transforms 'digital' oracle, i.e. \(|x\rangle|0\rangle \rightarrow |x\rangle|f(x)\rangle\) to a phase oracle \(|x\rangle \rightarrow (-1)^{f(x)}|x\rangle\).
In addition, we define the number of grover operator repetitions in the model (which is based on the frequency of solutions in the search space), this time 1 repetition is all needed.
from classiq import *
@qfunc
def sat_oracle(x: QArray[QBit], res: QBit):
res ^= sat_formula(*[x[i] for i in range(NUM_VARIABLES)])
@qfunc
def main(x: Output[QArray[QBit, NUM_VARIABLES]]):
allocate(x.len, x)
grover_search(
reps=1,
oracle=lambda vars: phase_oracle(sat_oracle, vars),
packed_vars=x,
)
qmod = create_model(
main, constraints=Constraints(max_width=20), out_file="3_sat_grover"
)
Synthesizing the Circuit
We proceed by synthesizing the circuit using Classiq's synthesis engine. The synthesis should take approximately several seconds:
qprog = synthesize(qmod)
Showing the Resulting Circuit
After Classiq's synthesis engine has finished the job, we can show the resulting circuit in the interactive GUI:
show(qprog)
Opening: https://platform.classiq.io/circuit/7046ee83-a959-4bef-a15d-743320efa8d0?version=0.0.0
Executing the circuit
Lastly, we can execute the resulting circuit with Classiq's execute interface, using the execute function.
result = execute(qprog).result_value()
Printing out the result, we see that our execution of Grover's algorithm successfully found the satisfying assignments for the input formula:
result.parsed_counts
[{'x': [0, 1, 0]}: 510, {'x': [0, 1, 1]}: 490]
assert sat_formula(*result.parsed_counts[0].state["x"])
Large 3-SAT example
We now domenstrate on a larger example:
def sat_formula_large(x1, x2, x3, x4):
return (
(x2 | x3 | x4)
& (_not(x1) | x2 | x3)
& (_not(x1) | x2 | _not(x3))
& (_not(x1) | _not(x2) | x3)
& (x1 | _not(x2) | _not(x3))
& (x1 | _not(x2) | x3)
& (_not(x1) | _not(x2) | _not(x4))
& (_not(x1) | _not(x2) | x4)
& (_not(x2) | _not(x3) | _not(x4))
& (x2 | _not(x3) | x4)
& (x1 | _not(x3) | x4)
& (x1 | _not(x2) | _not(x4))
& (_not(x1) | _not(x2) | _not(x3))
)
NUM_VARIABLES_LARGE = 4
print_truth_table(NUM_VARIABLES_LARGE, sat_formula_large)
x0 x1 x2 x3 | Result
-----------------------------------
0 0 0 0 | 0
0 0 0 1 | 1
0 0 1 0 | 0
0 0 1 1 | 1
0 1 0 0 | 0
0 1 0 1 | 0
0 1 1 0 | 0
0 1 1 1 | 0
1 0 0 0 | 0
1 0 0 1 | 0
1 0 1 0 | 0
1 0 1 1 | 0
1 1 0 0 | 0
1 1 0 1 | 0
1 1 1 0 | 0
1 1 1 1 | 0
@qfunc
def sat_oracle_large(x: QArray[QBit], res: QBit):
res ^= sat_formula_large(*[x[i] for i in range(NUM_VARIABLES_LARGE)])
@qfunc
def main(x: Output[QArray[QBit, NUM_VARIABLES_LARGE]]):
allocate(x.len, x)
grover_search(
reps=2,
oracle=lambda vars: phase_oracle(sat_oracle_large, vars),
packed_vars=x,
)
qmod_large = create_model(
main, constraints=Constraints(max_width=24), out_file="3_sat_grover_large"
)
qprog_large = synthesize(qmod_large)
show(qprog_large)
Opening: https://platform.classiq.io/circuit/7f95d865-c8d2-43f8-a21c-1d06ab720f19?version=0.0.0
result_large = execute(qprog_large).result_value()
result_large.parsed_counts
[{'x': [0, 0, 0, 1]}: 492,
{'x': [0, 0, 1, 1]}: 458,
{'x': [1, 1, 0, 0]}: 6,
{'x': [0, 1, 1, 0]}: 5,
{'x': [0, 1, 1, 1]}: 5,
{'x': [0, 0, 0, 0]}: 5,
{'x': [1, 0, 0, 1]}: 4,
{'x': [1, 1, 0, 1]}: 4,
{'x': [0, 1, 0, 0]}: 4,
{'x': [0, 1, 0, 1]}: 3,
{'x': [1, 1, 1, 1]}: 3,
{'x': [1, 0, 0, 0]}: 3,
{'x': [1, 0, 1, 0]}: 3,
{'x': [1, 1, 1, 0]}: 3,
{'x': [1, 0, 1, 1]}: 2]
assert sat_formula_large(*result_large.parsed_counts[0].state["x"])
References
[1]: Grover, Lov K. "Quantum mechanics helps in searching for a needle in a haystack." Physical review letters 79.2 (1997): 325.
[2]: Grover's algorithm (Wikipedia)