Picturing a Qubit

A qubit is the basic unit of quantum information. Like a classical bit it can be measured as 0 or 1, but before measurement it lives in a richer mathematical space. Any single-qubit pure state can be written

where $\alpha$ and $\beta$ are complex amplitudes satisfying $|\alpha|^2 + |\beta|^2 = 1$. The real-valued probabilities $|\alpha|^2$ and $|\beta|^2$ say how likely each outcome is when you measure, but the amplitudes themselves can interfere — a feature that has no classical counterpart.

The Bloch sphere

Because of the normalization constraint and the fact that an overall complex phase carries no physical meaning, the state space of a single qubit is actually a two-dimensional surface, not a four-dimensional one. A convenient way to see this is to write the general state in terms of two real angles $\theta \in [0, \pi]$ and $\phi \in [0, 2\pi)$:

These two angles parameterize every point on a unit sphere — the Bloch sphere. The north and south poles correspond to the computational basis states:

• North pole $(\theta = 0)$: $\cos 0\,|0\rangle + e^{i\phi}\sin 0\,|1\rangle = |0\rangle$.
• South pole $(\theta = \pi)$: $\cos\tfrac{\pi}{2}|0\rangle + e^{i\phi}\sin\tfrac{\pi}{2}|1\rangle = e^{i\phi}|1\rangle$, which is physically the same as $|1\rangle$ (global phase).

States along the equator ($\theta = \pi/2$) are equal superpositions. The most familiar ones sit on the $\pm x$ and $\pm y$ axes:

| State | $\theta$ | $\phi$ | Bloch direction | |---|---|---|---| | $|0\rangle$ | $0$ | — | $+z$ | | $|1\rangle$ | $\pi$ | — | $-z$ | | $|{+}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ | $\pi/2$ | $0$ | $+x$ | | $|{-}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$ | $\pi/2$ | $\pi$ | $-x$ | | $|{+i}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$ | $\pi/2$ | $\pi/2$ | $+y$ | | $|{-i}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle-i|1\rangle)$ | $\pi/2$ | $3\pi/2$ | $-y$ |

Single-qubit gates as rotations

Every single-qubit unitary gate corresponds to a rotation of the Bloch sphere. For example:

• The X gate (bit-flip) swaps $|0\rangle \leftrightarrow |1\rangle$, which is a $180°$ rotation about the $x$-axis, sending the north pole to the south pole and vice versa.
• The Z gate (phase-flip) leaves $|0\rangle$ alone and flips the sign of $|1\rangle$, which is a $180°$ rotation about the $z$-axis.
• The Hadamard gate $H$ swaps the $z$- and $x$-axes, sending $|0\rangle \mapsto |{+}\rangle$ — the north pole moves to the $+x$ equator.

This rotational picture makes it easy to reason about what a sequence of gates will do: you trace a path of rotations on the sphere rather than multiplying matrices step by step.

Understanding the Bloch sphere is a foundation for the entire module. Future lessons will use it to build intuition for phase kickback, gate decompositions, and eventually the structure of multi-qubit entanglement.

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