Can I change the byte ordering from little endian to big endian in qiskit? - endianness

Question

The regular Matrix representation of a CNOT gate as found in literature is:
CNOT =
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0
\end{bmatrix}
However in Qiskit, the matrix is represented as
CNOT =
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0
\end{bmatrix}
Is this something related to the Big-endian/Little-endian issue? Is there a way to represent my matrix the same way it is recovered in literature?

Answer 1

Yes, as you mentioned this has to do with the little endian bit-order in Qiskit. Most textbooks (and the first matrix you showed) are in big endian order.
If you want to know more you could check out these posts/documentation:
https://qiskit.org/documentation/tutorials/circuits/3_summary_of_quantum_operations.html#Basis-vector-ordering-in-Qiskit
https://quantumcomputing.stackexchange.com/questions/8244/big-endian-vs-little-endian-in-qiskit
If you want to convert your Qiskit circuit to big endian you can just use the reverse_bits method:
from qiskit import QuantumCircuit
from qiskit.quantum_info import Operator
circuit = QuantumCircuit(2)
circuit.cx(0, 1)
print('Little endian:')
print(Operator(circuit))
print('Big endian:')
print(Operator(circuit.reverse_bits()))
gives:
Little endian:
Operator([[1.+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, 1.+0.j, 0.+0.j],
[0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]],
input_dims=(2, 2), output_dims=(2, 2))
Big endian:
Operator([[1.+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, 1.+0.j],
[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j]],
input_dims=(2, 2), output_dims=(2, 2))