The One-Way Model

In the circuit model, the engine of computation is a sequence of unitary gates acting on a fixed register; measurement appears only at the end to read out the answer. The one-way model of quantum computation, introduced by Raussendorf and Briegel in 2001, turns this picture inside out. There the only dynamical operations are single-qubit measurements, and the entire algorithm is encoded in the choice of measurement bases and their order. This lesson explains why such a thing can compute at all.

Two resources, cleanly separated

The one-way model splits a computation into two phases that never overlap:

Resource preparation. A large, highly entangled state — a cluster state — is prepared once, independent of the specific algorithm. Every qubit starts in $|+\rangle = \tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ , and a controlled- $Z$ gate is applied across every edge of a chosen lattice (a chain, a square grid). This creates a generic, problem-agnostic web of entanglement.
Adaptive measurement. The qubits are then measured one at a time in single-qubit bases of the form $\{|{+}_\xi\rangle, |{-}_\xi\rangle\}$ with $|{\pm}_\xi\rangle = \tfrac{1}{\sqrt2}(|0\rangle \pm e^{i\xi}|1\rangle)$ . The measurement angle $\xi$ — and sometimes its sign — is what encodes the algorithm.

Because each qubit is measured exactly once and then discarded, the entanglement is consumed: the process runs in one direction only and cannot be undone. That irreversibility is the origin of the name one-way.

Why measurement can drive computation

The surprising claim is that destroying entanglement, qubit by qubit, performs unitary logic on the surviving qubits. The mechanism is gate teleportation. When a qubit entangled with its neighbour is measured in a tilted basis, the act of measurement propagates a rotated copy of its quantum information onto the neighbour — much as standard teleportation moves a state across a Bell pair. Choosing the measurement angle chooses the gate that gets imprinted. Chaining many such measurements along the lattice composes many gates, and a two-dimensional lattice supplies the connectivity needed for entangling gates between logical wires.

The price of measurement: randomness and feed-forward

Measurement outcomes are random: each single-qubit measurement returns $0$ or $1$ with the Born-rule probabilities. A naive worry is that this randomness corrupts the computation. It does not, because the two outcomes differ only by a known Pauli byproduct — an $X$ or $Z$ (or both) applied to the teleported state. The model handles this in two complementary ways:

Feed-forward. The basis angle of a later measurement is adjusted, conditioned on earlier outcomes, so that byproducts are absorbed before they matter. This adaptivity is why the measurements must, in general, be performed in a definite temporal order.
Byproduct bookkeeping. Any leftover Pauli operators are tracked classically (in the Pauli frame) and undone with a final correction on the output qubits.

The classical side-processing is efficient — it is just tracking and propagating Pauli operators — so the quantum cost remains a single round of measurements.

What this model buys us

The one-way model is equivalent in power to the circuit model: anything one can do, the other can do with only polynomial overhead. But the change of perspective is far from cosmetic.

It cleanly separates the (hard) job of building entanglement from the (simple) job of measuring single qubits, which matches certain hardware — photonics and atom arrays especially — where entanglement is easier to mass-produce than to apply on the fly.
Because non-output qubits are measured early, much of the computation can proceed in parallel, reducing the logical depth.
It exposes a deep structural fact: a fixed, universal entangled state plus classically chosen local measurements is already a universal quantum computer.

Looking ahead

The remaining lessons make each piece precise: how to construct cluster states (next lesson), how single measurements implement gates, how to specify an algorithm as a measurement pattern, how feed-forward corrections absorb randomness, and finally how the model connects back to the circuit model and to fault tolerance. Keep the two-phase structure — entangle once, then measure — firmly in mind; every later lesson is a refinement of it.