Part III. Circuit Model

One-Qubit Gates and the Bloch Sphere

Every one-qubit pure state is a point on a sphere, and every one-qubit gate is a rotation of that sphere — but the sphere is the illustration and the matrices are the authority. This chapter teaches you to move between the two without mistaking one for the other.

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Relative phase is the most underpriced idea in one-qubit quantum computing: no single measurement can see it, and it decides what every later gate will do.

By the end you will compute X, Z, and H against any one-qubit state by hand, say exactly which phase changes matter and which do not, and use the Bloch sphere as a map without confusing the map for the territory.

Core concepts: Bloch sphere, Pauli gates, Hadamard, global versus relative phase.

Notice that |+⟩ and |−⟩ sit at opposite ends of the equator: identical Z-basis statistics, opposite relative phases. The sphere keeps phase visible in a way a probability table never can.

Two views of one qubit

A one-qubit state is:

α ∣ 0 ⟩ + β ∣ 1 ⟩, with ∣ α ∣^{2} + ∣ β ∣^{2} = 1

A one-qubit gate is a $2 \times 2$ unitary that rewrites those amplitudes while preserving the normalization. The Bloch sphere visualizes pure one-qubit states up to global phase: the poles are the basis states, the equator holds the equal superpositions, and every gate is a rotation. It is excellent intuition for rotations, basis changes, and phase — but the calculations still come from matrix-vector multiplication. The sphere summarizes; it does not compute.

The distinction that runs the chapter is global versus relative phase. Multiply the whole state by $e^{i θ}$ and nothing measurable changes. Change the phase between $∣ 0 ⟩$ and $∣ 1 ⟩$ and you have changed what later gates will produce — a relative phase is a physical fact waiting for a basis change to expose it.

The working set

Three gates cover most of the territory:

state	$X$	$Z$	$H$
$∣0 ⟩$	$∣1 ⟩$	$∣0 ⟩$	$(∣0 ⟩ + ∣1 ⟩) / 2$
$∣1 ⟩$	$∣0 ⟩$	$- ∣1 ⟩$	$(∣0 ⟩ - ∣1 ⟩) / 2$

Two Hadamard identities earn their keep constantly: $H^{2} = I$ — Hadamard is its own inverse — and $H Z H = X$ — conjugating by H swaps the X and Z bases.

And the global-phase rule sets the boundary of what matters:

∣ ψ ⟩ and e^{i θ} ∣ ψ ⟩ give the same measurement probabilities

That rule does not make all phase irrelevant. A sign change on one component is relative phase, and a later gate can convert it into a different measurement distribution — as the worked example shows with three gates you already own.

Worked example: how HZH catches a phase

Start with $∣ 0 ⟩$ and apply H, Z, H:

∣ 0 ⟩ H \frac{∣ 0 ⟩ + ∣ 1 ⟩}{2} Z \frac{∣ 0 ⟩ - ∣ 1 ⟩}{2} H ∣ 1 ⟩

The middle Z created a relative phase — invisible to any measurement made at that moment, where the odds still read fifty-fifty. The final H changed basis and made the phase visible as a bit flip. This is the identity $H Z H = X$ experienced from the inside: phase became amplitude, and amplitude became certainty.

Where the intuition fails

The first trap is saying H makes a qubit random. H is a basis-change gate. That it produces equal computational-basis probabilities from $∣ 0 ⟩$ is a side effect; its real power is changing which phase differences will interfere downstream. A random source cannot be uncomputed — H can, with another H.

The second trap is overusing the sphere. It models one pure qubit beautifully. It does not model multi-qubit entanglement or circuit cost, and mixed states live in its interior — a story Chapter 15 (Density Matrices and Mixed States) tells properly. Use the sphere as a view, never as the source of truth.

The engineering view

For a software engineer, one-qubit gates are primitive instructions with exact semantics. The simulator implements them as matrix-vector multiplication. The compiler may replace a requested gate with native rotations the device prefers. The hardware implements those rotations with pulses, each carrying fidelity and calibration limits.

You should be able to move freely among four representations of the same operation:

Gate name — the symbol on the diagram.
Matrix action — the authoritative semantics.
Statevector update — the concrete effect on α and β.
Bloch-sphere movement — the geometric summary.

If the four ever disagree, trust the algebra. The diagram and the sphere are interface views over the same mathematical operation, and views can lie in ways matrices cannot.

What this buys you in diligence

When a company or library reports strong one-qubit gates, ask what basis, pulse model, calibration cycle, and error metric sit behind the number. Single-qubit quality is necessary but nowhere near sufficient: it proves nothing about two-qubit operations, readout quality, useful circuit depth, or application advantage. Still, weak single-qubit control is a serious warning sign, because every larger circuit is built from these primitives — a platform that cannot rotate one qubit cleanly will not compose rotations into anything better.

Exercise

Rotate by hand. Compute the H-Z-H sequence on $∣ 0 ⟩$ with column vectors, then check it against the identity.

Trace: every vector step of H, then Z, then H on $∣ 0 ⟩$ , with no skipped arithmetic.
Submit: the full trace, the final computational-basis probabilities, and one sentence on the role of relative phase in the result.
Check: mark which changes in your trace were global phase only and which affected later interference — then confirm the net effect matches $H Z H = X$ .
Repair: if the final answer was a guess, or phase kept turning into probability, redo Chapter 11 (Matrices as Gates) before moving to multi-qubit circuits.

Check your understanding

Answer without notes: compute the action of X, Z, H, and a phase gate on basis states and on $∣ + ⟩$ , by hand.

A passing answer shows the vector arithmetic, keeps global and relative phase separate, and states which results a measurement could see immediately and which only a later basis change could expose.

Oral defense: explain $H Z H = X$ twice — once with matrices, once as rotations on the sphere — and say why the two explanations cannot disagree.

If you get stuck

If you cannot keep state, gate, phase, and measurement in separate lanes during a trace, go back to Chapter 11 (Matrices as Gates) and rebuild the multiplication habit. The sphere will still be here when the algebra is solid.