Concept 1 of 11 · State

The qubit

Established

A two-level quantum system. Its pure states are points on a sphere; every single-qubit gate is a rotation of that sphere.

Try this

Apply H then T repeatedly: the state walks around the equator in steps of π/4.
Apply H, Z, H. The result is the same as X: a change of basis turns a phase flip into a bit flip.

Why QNACA cares

Unknown states cannot be copied, so the runtime must treat a quantum state as a resource that is prepared, consumed and measured, never duplicated (blueprint, page 3).

Theory

A qubit's state is a unit vector in $\mathbb{C}^2$ . Removing the unobservable global phase leaves two real parameters, a polar angle $\theta$ and an azimuth $\varphi$ : the Bloch sphere. The poles are the computational basis states; the equator holds equal superpositions that differ only in relative phase.

Any single-qubit unitary is a rotation $R_{\hat n}(\alpha)=e^{-i\alpha\,\hat n\cdot\vec\sigma/2}$ . On hardware, $\hat n$ is set by the phase of a microwave pulse and $\alpha$ by its area. A rotation about $z$ needs no pulse at all: it is a change of the reference phase for every later pulse (a 'virtual Z').

|\psi\rangle=\cos\tfrac{\theta}{2}\,|0\rangle+e^{i\varphi}\sin\tfrac{\theta}{2}\,|1\rangle

H=\tfrac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix},\quad X=\begin{pmatrix}0&1\\1&0\end{pmatrix}

T=\mathrm{diag}\big(1,\,e^{i\pi/4}\big),\qquad S=T^{2},\qquad Z=S^{2}

Sources