from qiskit.quantum_info import Statevector

from util import zero, one

Appendix: Binary Notation

The notation

\[ \overline{j} = j_{n-1} \dots j_0 \]

converts the number \(j\) into its binary expansion (in little Endian) in \(n\) bits, i.e.,

\[\begin{align*} j & = j_{n-1} 2^{n-1} + j_{n-2} 2^{n-2} + \dots + j_0 2^0 \,. \end{align*}\]

def to_binary(num_qubits: int, j: int) -> list[int]:
    return [int(x) for x in f"{j:0{num_qubits}b}"]

The notation

\[ |\overline{j}\rangle = |j_{n-1} \dots j_0 \rangle \]

thus indicates a computational basis in \(n\) qubits. As an example,

\[ |\overline{6}\rangle = |110\rangle \]

for \(n = 3\).

def idx_to_basis(num_qubits: int, j: int) -> Statevector:
    vecs = [zero if x == 0 else one for x in to_binary(num_qubits, j)]
    acc = vecs[-1]
    for v in vecs[-2::-1]:
        acc = v ^ acc
    return acc

idx_to_basis(3, 0).draw("latex")

\[ |000\rangle\]

idx_to_basis(3, 1).draw("latex")

\[ |001\rangle\]

idx_to_basis(3, 2).draw("latex")

\[ |010\rangle\]

idx_to_basis(3, 3).draw("latex")

\[ |011\rangle\]

idx_to_basis(3, 4).draw("latex")

\[ |100\rangle\]

idx_to_basis(3, 5).draw("latex")

\[ |101\rangle\]

idx_to_basis(3, 6).draw("latex")

\[ |110\rangle\]

idx_to_basis(3, 7).draw("latex")

\[ |111\rangle\]