Computing by Measurement

We now make precise the central mechanism of the one-way model: how measuring one qubit performs a unitary gate on another. The elementary unit is a two-qubit gadget called one-bit teleportation, and understanding it once is enough to understand the whole model.

The elementary gadget

Take an input qubit in an unknown state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ on wire $1$ , and an ancilla prepared in $|+\rangle$ on wire $2$ . Entangle them with a controlled- $Z$ — exactly the cluster-state construction on a single edge — and then measure wire $1$ in the $X$ basis, i.e. in the basis $\{|+\rangle, |-\rangle\}$ . We claim wire $2$ is left holding a Hadamard-rotated copy of the input.

Track the state. After the controlled- $Z$ ,

CZ\,\big(|\psi\rangle \otimes |+\rangle\big) = \tfrac{1}{\sqrt2}\big(\alpha|0\rangle(|0\rangle + |1\rangle) + \beta|1\rangle(|0\rangle - |1\rangle)\big).

Now expand wire $1$ in the $X$ basis using $|0\rangle = \tfrac{1}{\sqrt2}(|+\rangle + |-\rangle)$ and $|1\rangle = \tfrac{1}{\sqrt2}(|+\rangle - |-\rangle)$ . Collecting terms,

= \tfrac{1}{\sqrt2}\Big( |+\rangle_1 \otimes H|\psi\rangle \;+\; |-\rangle_1 \otimes XH|\psi\rangle \Big),

where $H$ is the Hadamard gate. (Verifying this collection of terms by hand is the best possible exercise for building intuition.)

Reading the gadget

The displayed form says everything:

The measurement on wire $1$ returns outcome $s \in \{0, 1\}$ , corresponding to $|+\rangle$ ( $s=0$ ) or $|-\rangle$ ( $s=1$ ), each with probability $\tfrac12$ .
Whatever the outcome, wire $2$ ends up in $X^{s}\,H\,|\psi\rangle$ . The intended gate $H$ is always applied; the only outcome-dependence is a known Pauli byproduct $X^{s}$ .

So a single $X$ -basis measurement teleports $|\psi\rangle$ from wire $1$ to wire $2$ and applies a Hadamard along the way. No unitary gate was ever applied to wire $2$ — only an entangling step and a measurement on wire $1$ .

Tilting the basis chooses the gate

Measuring in the plain $X$ basis gives a Hadamard. Measuring in a tilted basis dials in a rotation first. Define the angle- $\xi$ basis

|{\pm}_\xi\rangle = \tfrac{1}{\sqrt2}\big(|0\rangle \pm e^{i\xi}|1\rangle\big),

which is the $X$ basis pre-rotated by $R_z(\xi) = \mathrm{diag}(1, e^{i\xi})$ . Repeating the calculation with this basis, the gadget now leaves wire $2$ in

X^{s}\, H\, R_z(\xi)\, |\psi\rangle .

The measurement angle $\xi$ is a programmable gate parameter. Sweeping $\xi$ over $[0, 2\pi)$ realises the whole one-parameter family $H R_z(\xi)$ , and chaining gadgets along a cluster composes them. Since $H$ and $R_z(\xi)$ together generate all single-qubit unitaries, a short chain of measurements suffices to apply any one-qubit gate to the teleported information.

From one gadget to a computation

A linear cluster is a sequence of these gadgets sharing wires: the output of one is the input of the next. Each measured qubit applies its $H R_z(\xi_k)$ and hands the state to its successor, so after $n$ measurements the surviving qubit holds

(\text{Pauli byproduct}) \cdot H R_z(\xi_n) \cdots H R_z(\xi_1)\,|\psi\rangle .

Two-qubit gates between different logical wires come from the cross-edges of a 2D cluster, where a shared controlled- $Z$ couples two chains. The full algorithm is therefore nothing but a list of measurement angles $\{\xi_k\}$ on a fixed cluster — which is exactly the measurement pattern of the next lesson.

What we have established

One $X$ -basis measurement on an edge teleports a qubit and applies a Hadamard, up to a Pauli byproduct $X^{s}$ .
Tilting the measurement basis by $\xi$ inserts an $R_z(\xi)$ rotation: the basis angle programs the gate.
The byproduct $X^{s}$ is the only trace of measurement randomness, and it is a known Pauli — the thing feed-forward corrections will later absorb.

The next lessons formalise these gadgets into reusable measurement patterns and then show precisely how the byproducts are managed.