code: def probs_to_unitary(probs):
      r"""Calculates the unitary matrix corresponding to an input probability distribution.
  
      For a given distribution :math:`p(i)`, this function returns the unitary :math:`\mathcal{A}`
      that transforms the :math:`|0\rangle` state as
  
      .. math::
  
          \mathcal{A} |0\rangle = \sum_{i} \sqrt{p(i)} |i\rangle,
  
      so that measuring the resulting state in the computational basis will give the state
      :math:`|i\rangle` with probability :math:`p(i)`. Note that the returned unitary matrix is
      real and hence an orthogonal matrix.
  
      Args:
          probs (array): input probability distribution as a flat array
  
      Returns:
          array: unitary
  
      Raises:
          ValueError: if the input array is not flat or does not correspond to a probability
              distribution
  
      **Example:**
  
      >>> p = np.ones(4) / 4
      >>> probs_to_unitary(p)
      array([[ 0.5       ,  0.5       ,  0.5       ,  0.5       ],
             [ 0.5       , -0.83333333,  0.16666667,  0.16666667],
             [ 0.5       ,  0.16666667, -0.83333333,  0.16666667],
             [ 0.5       ,  0.16666667,  0.16666667, -0.83333333]])
      """
  
      if not qml.math.is_abstract(
          sum(probs)
      ):  # skip check and error if jitting to avoid JAX tracer errors
          if not qml.math.allclose(sum(probs), 1) or min(probs) < 0:
              raise ValueError(
                  "A valid probability distribution of non-negative numbers that sum to one "
                  "must be input"
              )
  
      # Using the approach discussed here:
      # https://quantumcomputing.stackexchange.com/questions/10239/how-can-i-fill-a-unitary-knowing-only-its-first-column
      psi = qml.math.sqrt(probs)
      overlap = psi[0]
      denominator = qml.math.sqrt(2 + 2 * overlap)
      psi = qml.math.set_index(
          psi, 0, psi[0] + 1
      )  # psi[0] += 1, but JAX-JIT compatible
      psi /= denominator
  
      dim = len(probs)
      return 2 * qml.math.outer(psi, psi) - np.eye(dim)
  
file_path: pennylane/pennylane/templates/subroutines/qmc.py
imports: import numpy
  import pennylane
  from pennylane.operation import AnyWires, Operation
  from pennylane.ops import QubitUnitary
label_desc: Use image node_id 1 for calling a global function with example usage: probs_to_unitary(probs) and returns: unknown