Basic Operators and Functions for Vectors

Basically, operators and functions for vectors operate element-wise.

Basic operators for vectors

The basic operators +, -, *, and / work for vectors of variables and expressions.

In QUBO++, these operators are applied element-wise.

Vector–Scalar Operations

When you combine a vector and a scalar, the scalar is applied to each element of the vector. For example, if x is a vector of size 3, then:

• 2 * x produces {2*x[0], 2*x[1], 2*x[2]}
• x + 1 produces {x[0] + 1, x[1] + 1, x[2] + 1}

The following program illustrates this behavior:

#define MAXDEG 2
#include <qbpp/qbpp.hpp>

int main() {
  auto x = qbpp::var("x", 3);
  auto f = 2 * x + 1;

  std::cout << "f = " << f << std::endl;
  for (size_t i = 0; i < f.size(); ++i) {
    std::cout << "f[" << i << "] = " << f[i] << std::endl;
  }
}

This program creates a vector x = {x[0], x[1], x[2]} of binary variables. Then 2 * x multiplies each element by 2, and + 1 adds 1 to each element, so f becomes: {1 + 2*x[0], 1 + 2*x[1], 1 + 2*x[2]}. This program produces the following output:

f = {1 +2*x[0],1 +2*x[1],1 +2*x[2]}
f[0] = 1 +2*x[0]
f[1] = 1 +2*x[1]
f[2] = 1 +2*x[2]

Vector–Vector Operations

When you combine two vectors of the same size, the operation is performed element-wise at each index.

The following example uses two vectors x and y, both of size 3:

#define MAXDEG 2
#include <qbpp/qbpp.hpp>

int main() {
  auto x = qbpp::var("x", 3);
  auto y = qbpp::var("y", 3);
  auto f = 2 * x + 3 * y + 1;

  std::cout << "f = " << f << std::endl;
  for (size_t i = 0; i < f.size(); ++i) {
    std::cout << "f[" << i << "] = " << f[i] << std::endl;
  }
}

Here:

• 2 * x becomes {2*x[0], 2*x[1], 2*x[2]}
• 3 * y becomes {3*y[0], 3*y[1], 3*y[2]}
• adding them is element-wise, so the i-th element is 2*x[i] + 3*y[i]
• + 1 is again applied element-wise

Therefore, f becomes: {1 + 2*x[0] + 3*y[0], 1 + 2*x[1] + 3*y[1], 1 + 2*x[2] + 3*y[2]} which matches the output:

f = {1 +2*x[0] +3*y[0],1 +2*x[1] +3*y[1],1 +2*x[2] +3*y[2]}
f[0] = 1 +2*x[0] +3*y[0]
f[1] = 1 +2*x[1] +3*y[1]
f[2] = 1 +2*x[2] +3*y[2]

Vector–vector operations require the same vector size.

The next example demonstrates a more complex element-wise expression involving vector–scalar operations, vector–vector operations, unary minus, and parentheses:

#define MAXDEG 2
#include <qbpp/qbpp.hpp>

int main() {
  auto x = qbpp::var("x", 3);
  auto y = qbpp::var("y", 3);
  auto f = 6 * -(x + 1) * (y - 1);
  auto g = f / 3;

  std::cout << "f = " << f << std::endl;
  for (size_t i = 0; i < x.size(); ++i) {
    std::cout << "f[" << i << "] = " << f[i] << std::endl;
  }

  std::cout << "g = " << g << std::endl;
  for (size_t i = 0; i < x.size(); ++i) {
    std::cout << "g[" << i << "] = " << g[i] << std::endl;
  }
}

In this example, all operations are still applied element-wise.

• First, x + 1 and y - 1 add/subtract the scalar to/from each element, producing two vectors {x[i]+1} and {y[i]−1}.
• The unary minus -(x + 1) also works element-wise, so it becomes {−(x[i]+1)}.
• The multiplication 6 * -(x + 1) * (y - 1) is then performed element-wise as well, so for each index i, f[i]=6⋅(−(x[i]+1))⋅(y[i]−1). Expanding this expression yields f[i]=6−6x[i]y[i]+6x[i]−6y[i], which matches the printed form in the output.
• Finally, g = f / 3 divides each element by 3, so g[i]=f[i]/3=2−2x[i]y[i]+2x[i]−2y[i], again matching the output.

  f = {6 -6*x[0]*y[0] +6*x[0] -6*y[0],6 -6*x[1]*y[1] +6*x[1] -6*y[1],6 -6*x[2]*y[2] +6*x[2] -6*y[2]}
  f[0] = 6 -6*x[0]*y[0] +6*x[0] -6*y[0]
  f[1] = 6 -6*x[1]*y[1] +6*x[1] -6*y[1]
  f[2] = 6 -6*x[2]*y[2] +6*x[2] -6*y[2]
  g = {2 -2*x[0]*y[0] +2*x[0] -2*y[0],2 -2*x[1]*y[1] +2*x[1] -2*y[1],2 -2*x[2]*y[2] +2*x[2] -2*y[2]}
  g[0] = 2 -2*x[0]*y[0] +2*x[0] -2*y[0]
  g[1] = 2 -2*x[1]*y[1] +2*x[1] -2*y[1]
  g[2] = 2 -2*x[2]*y[2] +2*x[2] -2*y[2]
  

Compound opertors for vectors

Similarly, the compound operators +=, -=, *=, and /= work for vectors of variables and expressions. The following example demonstrates how these operators work for a vector of size 3:

#define MAXDEG 2
#include <qbpp/qbpp.hpp>

int main() {
  auto x = qbpp::var("x", 3);
  auto y = qbpp::var("y", 3);
  auto f = 6 * x + 4;

  f += 3 * y;
  std::cout << "f = " << f << std::endl;
  f -= 12;
  std::cout << "f = " << f << std::endl;
  f *= 2 * y;
  std::cout << "f = " << f << std::endl;
  f /= 2;
  std::cout << "f = " << f << std::endl;
}

This program produces the following output:

f = {4 +6*x[0] +3*y[0],4 +6*x[1] +3*y[1],4 +6*x[2] +3*y[2]}
f = {-8 +6*x[0] +3*y[0],-8 +6*x[1] +3*y[1],-8 +6*x[2] +3*y[2]}
f = {12*x[0]*y[0] +6*y[0]*y[0] -16*y[0],12*x[1]*y[1] +6*y[1]*y[1] -16*y[1],12*x[2]*y[2] +6*y[2]*y[2] -16*y[2]}
f = {6*x[0]*y[0] +3*y[0]*y[0] -8*y[0],6*x[1]*y[1] +3*y[1]*y[1] -8*y[1],6*x[2]*y[2] +3*y[2]*y[2] -8*y[2]}

Square functions for vectors

Square functions also work for vectors, as demonstrated below:

#define MAXDEG 2
#include <qbpp/qbpp.hpp>

int main() {
  auto x = qbpp::var("x", 3);
  auto f = x + 1;

  std::cout << "f = " << qbpp::sqr(f) << std::endl;
  std::cout << "f = " << f << std::endl;
  f.sqr();
  std::cout << "f = " << f << std::endl;
}

This program produces the following output:

f = {1 +x[0]*x[0] +x[0] +x[0],1 +x[1]*x[1] +x[1] +x[1],1 +x[2]*x[2] +x[2] +x[2]}
f = {1 +x[0],1 +x[1],1 +x[2]}
f = {1 +x[0]*x[0] +x[0] +x[0],1 +x[1]*x[1] +x[1] +x[1],1 +x[2]*x[2] +x[2] +x[2]}

Simplfy functions for vectors

Simplify functions also work for vectors, as demonstrated below:

#define MAXDEG 2
#include <qbpp/qbpp.hpp>

int main() {
  auto x = qbpp::var("x", 3);
  auto f = qbpp::sqr(x - 1);
  std::cout << "f = " << f << std::endl;
  std::cout << "simplified(f) = " << qbpp::simplify(f) << std::endl;
  std::cout << "simplified_as_binary(f) = " << qbpp::simplify_as_binary(f) << std::endl;
  std::cout << "simplified_as_spin(f) = " << qbpp::simplify_as_spin(f) << std::endl;
}

This program produces the following output:

f = {1 +x[0]*x[0] -x[0] -x[0],1 +x[1]*x[1] -x[1] -x[1],1 +x[2]*x[2] -x[2] -x[2]}
simplified(f) = {1 -2*x[0] +x[0]*x[0],1 -2*x[1] +x[1]*x[1],1 -2*x[2] +x[2]*x[2]}
simplified_as_binary(f) = {1 -x[0],1 -x[1],1 -x[2]}
simplified_as_spin(f) = {2 -2*x[0],2 -2*x[1],2 -2*x[2]}

NOTE These operators and functions also work for multi-dimensional arrays.