Subsystem Composition in Qiskit

Question

I am reading Subsystem Composition in Qiskit Tutorial "Operators"

https://qiskit.org/documentation/tutorials/circuits_advanced/02_operators_overview.html#Subsystem-Composition

# Compose XZ with an 3-qubit identity operator
op = Operator(np.eye(2 ** 3))
XZ = Operator(Pauli(label='XZ'))
op.compose(XZ, qargs=[0, 2])

Why the above code will have the following output? Could anyone explain and provide math derivation? Thank you very much!

Operator([[ 0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j,
        0.+0.j],
      [ 0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j,  0.+0.j,
        0.+0.j],
      [ 0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j,
        0.+0.j],
      [ 0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,
       -1.+0.j],
      [ 1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,
        0.+0.j],
      [ 0.+0.j, -1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,
        0.+0.j],
      [ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,
        0.+0.j],
      [ 0.+0.j,  0.+0.j,  0.+0.j, -1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,
        0.+0.j]],
     input_dims=(2, 2, 2), output_dims=(2, 2, 2))

Answer 1

Your resulting operator is essentially $X \otimes I \otimes Z$ and is represented by your matrix. Pauli operators are applied separately to the subsystems, that is single qubits, of your 3-qubit system. In your case, you create this operator by composing a 3 qubit identity operator op = $I \otimes I \otimes I$ with a smaller operator XZ = $X \otimes Z$, whereby XZ is applied to the qubits 0 and 2. This is only allowed in qiskit if you compose a smaller operator with a larger one, whereby the smaller operator is applied to some subsystem(s) of the larger operator.

If you create the following circuit and translate it into an Operator by op = Operator(qc), you will have the exact same matrix representation.

Notice how the operators in $X \otimes I \otimes Z$ are applied to qubits 2, 1 and 0 in reversed order. To understand why the matrix looks like it looks I suggest you read about the Kronecker product. For example, for two $2\times 2$ it is defined in the following way:

$ A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B\\ a_{21}B & a_{22}B \end{pmatrix} $

which means, for example, for two Pauli matrices:

$ X \otimes Z = \begin{pmatrix} 0Z & 1Z\\ 1Z & 0Z \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} $

You can derive the matrix representation for $X \otimes I \otimes Z$ by hand on the similar manner.

Finally, here are some alternative ways, in which you could create this operator:

IZ= I.tensor(Z)
X.tensor(IZ)

XI = X.tensor(I)
XI.tensor(Z)

ZI = Z.expand(I)
ZI.expand(X)