Function QComplement_V2

QAlg/ArithmeticUnit/ArithmeticUnit_V2.cpp:119–341

* @note circuits is reversible, so it's one to one * input positive 0 or negative 0, output are both postivie 0, so ancil is different for reversibility */

 * input positive 0 or negative 0, output are both postivie 0, so ancil is different for reversibility 
 */
QCircuit QComplement_V2(QVec &a, QVec &aux)
{
  /* 
    we proposed circute for two's complement based on QCLA adder
    filp a then add one bit b_0 which is |1>, if a is negative, otherwise keep not changed
  */
  /* aux should be |0...0>, not checked here */
  /* highest bit of a is sign */
  QVec unsign_a(a.begin(), a.end() - 1);
  Qubit *sign = a.back();

  int n = a.size();

  /* sizeof(z) = n, sizeof(x) = n - log n, total auxiliary size at least  2 * n + 1 - log2(n) */
  std::stringstream error_msg("auxiliary bits size should at leat 2 * n + 1 - std::floor(std::log2(n)) = ");
  error_msg << int(2 * n - std::floor(std::log2(n)));
  QPANDA_ASSERT(aux.size() < int(2 * n - std::floor(std::log2(n))), error_msg.str());

  /*
    z[i+1] = g[i, i + 1]
    a[i] = p[i, i + 1] for i > 0
    x for ancillary space to save p[i,j], j>i+1
  */
  QVec z(aux.begin(), aux.begin() + n);
  QVec x(aux.begin() + n, aux.begin() + int(2 * n - std::floor(std::log2(n))));

  /* resue z[0] as b_0 */
  Qubit *b_0 = z[0];

  DraperQCLAAdder::Propagate p(a, x);

  QCircuit qc;
  /*
    step 0: 
    1. set b_0 |1>, if sign negative
    2. flip bits of unsign_a, if sign negative
  */
  qc << X(b_0).control(sign);
  for (int i = 0; i < unsign_a.size(); i++)
  {
    qc << X(unsign_a[i]).control(sign);
  }

  /*
    QCLA algorithm
    step 1:
    g[0,1] = (a[0] * b_0) ⊕ g[0,1]
  */
  if (1 < n)
    qc << X(z[1]).control({a[0], b_0});

  /* 
    step 2:
    add b_0 to a_0
    this sets a[i] = p[i, i+1]
  */
  qc << CNOT(b_0, a[0]);