12.2 From bits to qubits

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §12.2. Rewritten in original wording. Continues 12.1b. Next: 12.2b.

Classical memory stores bits $0$ and $1$. QC stores qubits: normalized superpositions of $|0\rangle$ and $|1\rangle$, manipulated by quantum gates. Hardware may be superconducting circuits (Josephson junctions), not necessarily neutral-atom levels — but the math is the same spin-½ language.

Bloch sphere and tensor product

A normalized qubit $|\psi\rangle=u|0\rangle+v|1\rangle$ ($|u|^{2}+|v|^{2}=1$) is a point on the Bloch sphere. Multi-qubit states use Kronecker products. Pick the Bloch parametrization and $[a,b]^{T}\otimes[c,d]^{T}$ among lookalikes.

$\theta=0$ → north pole $|0\rangle$; $\theta=\pi$ → south pole $|1\rangle$; generic pure states live on the sphere’s surface. Two-qubit computational basis: $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ as the standard $4$-D unit vectors.

Takeaway

1. Qubit = normalized $u|0\rangle+v|1\rangle$.
2. Bloch angles $(\theta,\phi)$ picture pure states.
3. $n$ qubits live in $\mathbb{C}^{2^n}$ via $\otimes$.

Quiz

Qubits

1 / 4

Normalization for one qubit requires…

Pick an answer to see if you are right.

Next: 12.2b Qubit labs.