Cluster-State Quantum Computer: Measurement-Based Computing

Cluster State Lattice

Grid width (columns) 6

Measurement basis angle θ 0°

6 columns of |+⟩ qubits ready.

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Columns done

0 / 6

Entanglement entropy at cut (edges crossing)

5 ebits

Output bits: (none yet)

How it works

This is the one-way (measurement-based) quantum computer. A 2D lattice of qubits is first prepared as a cluster state: every qubit starts in |+⟩ = (|0⟩+|1⟩)/√2, then a CZ (controlled-Z) gate is applied along every edge of the grid:

|+⟩ per qubit  →  CZ on every lattice edge  →  |cluster⟩
CZ|00⟩=|00⟩, CZ|01⟩=|01⟩, CZ|10⟩=|10⟩, CZ|11⟩=−|11⟩

No further gates are ever applied. Instead the whole computation happens by measuring each qubit, column by column, in a basis rotated by an angle θ in the X–Y plane of the Bloch sphere:

|±θ⟩ = (|0⟩ ± e^(iθ)|1⟩)/√2
P(outcome m) depends on θ and on the qubit's entangled neighbours

Measuring a qubit in this basis teleports the logical state one step to the right and applies an effective single-qubit rotation R_z(θ) to it — exactly the identity Raussendorf and Briegel used to show that measurement alone is computationally universal. Each random outcome bit m must be fed forward: it flips the effective angle used for the next column by π (a byproduct/Pauli-frame correction), which is why the angle used per row here depends on the previous column's result.

Grid width — how many columns of entangled qubits make up the resource state.
θ slider — the measurement basis for the next column; different θ implements a different logical rotation as information teleports through.
Measure Next Column — consumes the leftmost unmeasured column: each qubit collapses to 0 or 1, and its lattice edges (its entanglement with neighbours) are destroyed in the process — the resource state shrinks with every step.
Entanglement entropy at cut — for a graph/cluster state, the entanglement across a straight cut equals the number of edges crossing it; here that is exactly the row count until the lattice is fully consumed.

Real-world relevance: this is the architecture behind leading photonic quantum computing efforts (PsiQuantum, Xanadu), where generating large cluster states is easy but two-qubit gates are hard — so all the "computing" is pushed into cheap single-photon measurements instead.

Unmeasured qubit Active column Measured → 0 Measured → 1

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