Greatest Common Divisor (GCD)

Let $P$ and $Q$ be two positive integers. The computation of the greatest common divisor (GCD) can be formulated as a HUBO problem.

Let $p$, $q$, and $r$ be positive integers satisfying the following constraints:

\[\begin{aligned} p\cdot r &= P \\ q\cdot r &=Q \end{aligned}\]

Clearly, $r$ is a common divisor of $P$ and $Q$. Therefore, the maximum value of $r$ satisfying these constraints is the GCD of $P$ and $Q$. To find such an $r$, we use $-r$ as the objective function in the HUBO formulation.

PyQBPP program

Based on the idea above, the following PyQBPP program computes the GCD of two integers, P = 858 and Q = 693:

import pyqbpp as qbpp

P = 858
Q = 693
p = qbpp.between(qbpp.var_int("p"), 1, 1000)
q = qbpp.between(qbpp.var_int("q"), 1, 1000)
r = qbpp.between(qbpp.var_int("r"), 1, 1000)

constraint = (p * r == Q) + (q * r == P)
f = -r + constraint * 1000

f.simplify_as_binary()

solver = qbpp.EasySolver(f)
sol = solver.search({"time_limit": 1.0})

print(f"GCD = {sol(r)}")
print(f"{sol(p)} * {sol(r)} = {P}")
print(f"{sol(q)} * {sol(r)} = {Q}")

In this program, p, q, and r are defined as integer variables in the range $[1,1000]$. The expression constraint is constructed so that it evaluates to zero when both constraints are satisfied.

The objective function -r is combined with the constraint term multiplied by a penalty factor of 1000, and the resulting expression is stored in f.

The EasySolver searches for a solution that minimizes f. The resulting values of p, q, and r are printed as follows:

GCD = 33
21 * 33 = 858
26 * 33 = 693

This output confirms that the GCD of 858 and 693 is correctly obtained as 33.