Square Root

This example demonstrates how to compute the square root of $c=2$ using large integers represented by qbpp::cpp_int. Let $s = 10 ^{20}$ be a fixed integer. Since QUBO++ cannot handle real numbers directly, we compute $\sqrt{cs^2}$ instead of $\sqrt{c}$. From the following relation,

\[\begin{aligned} \sqrt{c} &= \sqrt{cs^2}/s \end{aligned}\]

we can obtain an approximation of $\sqrt{c}$ with 20 decimal-digit precision.

HUBO formulation of the square root computation

We define an integer variable $x$ that takes values in the range $[s, 2s]$. We then formulate the problem using the following equation:

\[\begin{aligned} x ^ 2 &= cs ^ 2 \end{aligned}\]

In QUBO++, this equality constraint is converted into the following HUBO expression:

\[(x ^ 2 -cs^2)^2\]

By finding the value of $x$ that minimizes this expression, we obtain an approximation of the square root of $c$ with 20 decimal-digit precision. Since $x$ is internally represented as a linear expression of binary variables, this objective function becomes quartic in those binary variables.

QUBO++ parogram

The following QUBO++ program constructs a HUBO expression based on the above idea and solves it using the Easy Solver:

#define INTEGER_TYPE_CPP_INT

#include <qbpp/qbpp.hpp>
#include <qbpp/easy_solver.hpp>

int main() {
  const int c = 2;
  auto s = qbpp::integer("100000000000000000000");
  auto x = s <= qbpp::var_int("x") <= c * s;
  auto f = x * x == c * s * s;
  f.simplify_as_binary();
  auto solver = qbpp::EasySolver(f);
  auto sol = solver.search({{"time_limit", 10.0}});
  auto xv = sol(x);
  std::cout << "sqrt(" << c << ") ≈ " << xv << " / " << s << std::endl;
  std::cout << "       = " << (xv / s) << "." << (xv % s) << std::endl;
  std::cout << "Energy = " << sol.energy() << std::endl;
}

Since very large coefficients are used, we define INTEGER_TYPE_CPP_INT before including the header, which sets both coeff_t and energy_t to cpp_int (arbitrarily large integers). The constant s, the integer variable x, and the HUBO expression f are defined according to the formulation described above. The Easy Solver is executed with a time limit of 10 seconds.

The integer solution xv is split into its quotient xv / s and remainder xv % s, which are joined by a decimal point to obtain the decimal representation. Only cpp_int integer arithmetic is used, so no precision is lost by converting to double.

This program produces the following output:

sqrt(2) ≈ 141421356237309504880 / 100000000000000000000
       = 1.41421356237309504880
Energy = 2281431565136320033809509291861647360000

We can confirm that the Easy Solver outputs the correct approximation:

\[\sqrt{2}\approx 1.41421356237309504880\]

Note that the reported energy value is not zero, and the equality constraint is not satisfied exactly. This is simply because there is no exact integer solution to the equality. Instead, the solver finds a solution that minimizes the error of the equality constraint. The energy value shown in the output corresponds to the square of this error. Since the error is minimized, the resulting value of $x$ represents an approximation of the square root.