code: def _pauli_decompose(matrix, num_wires):
      r"""Compute the coefficients of a matrix or a batch of matrices (batch dimension(s) in the
      leading axes) in the Pauli basis.
  
      Args:
          matrix (tensor_like): Matrix or batch of matrices to decompose into the Pauli basis.
          num_wires (int): Number of wires the matrices act on.
  
      Returns:
          tensor_like: Coefficients of the input ``matrix`` in the Pauli basis.
  
      For a matrix :math:`M`, these coefficients are defined via
      :math:`M = \sum_\ell c_\ell P_\ell` and they can be computed using the (Frobenius) inner
      product of :math:`M` with the corresponding Pauli word :math:`P_\ell`:
      :math:`c_\ell = \frac{1}{2^N}\operatorname{Tr}\left\{P_\ell M\right\}` where the prefactor
      is the normalization that makes the standard Pauli basis orthonormal, and :math:`N`
      is the number of qubits.
      That is, the normalization is such that a single Pauli word :math:`P_k` has
      coefficients ``c_\ell = \delta_{k\ell}``.
  
      Note that this implementation takes :math:`\mathcal{O}(16^N)` operations per input
      matrix but there is a more efficient method taking only :math:`\mathcal{O}(N4^N)`
      operations per matrix.
      """
      # Create the dense Pauli basis tensor (shape (d^2-1, d, d)) with d = 2**num_wires
      basis = pauli_basis_matrices(num_wires)
      # Contract the Pauli basis tensor with the input, which has shape (*batch_dims, d, d)
      # We are interested in the traces of the matrix products, for each Pauli basis. Both
      # contractions (mult and trace) can be executed by providing all axes of size d to
      # ``tensordot``, which gives us a vectorized way to compute the coefficients.
      coefficients = qml.math.tensordot(
          basis, matrix, axes=[[1, 2], [-1, -2]]
      )
      # Finally, cast to the original data type and renormalize
      return qml.math.cast(coefficients, matrix.dtype) / 2**num_wires
  
imports: from functools import lru_cache, reduce
  from itertools import product
  import numpy
  import pennylane
  from pennylane.operation import AnyWires, Operation
  from pennylane.ops.qubit.parametric_ops_multi_qubit import PauliRot
label_desc: Use image node_id 3 for calling a global function with example usage: _pauli_decompose(matrix, num_wires) and returns: unknown