GHZ States

qc/entanglement

Generalizes Bell States to N qubits — all qubits maximally correlated:

$|{\text{GHZ}}_N⟩=\frac{1}{\sqrt{2}}(|00\dots 0⟩+|11\dots 1⟩)$

Measuring gives all-0s or all-1s, 50/50, never mixed.

Two naive constructions, both depth N (no advantage over each other):

1. Fan-out: H(0), then CX(0,1), CX(0,2), …, CX(0,N-1) — all share qubit 0 as control, so none parallelize.
2. Chain: H(0), then CX(0,1), CX(1,2), … — each CX depends on the previous, purely sequential.

Reducing depth is the whole story of this topic — see Circuit Depth, Start From the Middle, and Recursive Fan-Out.

Related

• Bell States
• Circuit Depth
• Bridge Gate Identity
• Heavy-Hex Topology — building GHZ states that respect real hardware connectivity
• Dynamic GHZ via Qubit Reuse — a third construction: mid-circuit measurement + feedforward instead of more connectivity or more depth-optimization

Self-Check

• Why do fan-out and chain constructions both have depth N, despite looking structurally different?
• What outcome do you always get when measuring a GHZ state, and why never a mix?
• Name the three different ways this vault builds a GHZ state, and what each one optimizes for.