QT02 — GHZ State (n qubits)

Field Value
Domain Quantum Computing
System n-qubit register
Group SU(2)^⊗n
H^k tier H²
ISA Meld (β = it)
Status Validated
Opcodes FLIP · BIND^{n−1} · LABEL
Papers Paper 604, Paper 605
Tags Tutorial · QC circuit · Rosetta

What this circuit does

The Greenberger-Horne-Zeilinger (GHZ) state is the n-qubit generalisation of the Bell state:

\[\vert\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}\bigl(\vert 0\rangle^{\otimes n} + \vert 1\rangle^{\otimes n}\bigr)\]

It is the canonical multipartite entangled state: every qubit is maximally correlated with every other, yet measuring any single qubit yields a random outcome. GHZ states are the resource for quantum secret sharing, multi-party Bell tests, and distributed quantum sensing.

ISA programme (hardware-independent) 

INIT:       ORBIT[|00…0⟩]                    -- n qubits in ground state
FLIP:       FLIP[qubit 0]                    -- |0⟩ → (|0⟩+|1⟩)/√2
BIND^{n-1}: BIND[0→1] · BIND[0→2] · … · BIND[0→n-1]
                                             -- propagate entanglement
LABEL:      LABEL[all n qubits]              -- always outputs 00…0 or 11…1

Programme length: 1 FLIP + (n−1) BINDs. This is optimal: one FLIP to create superposition; n−1 BINDs because each BIND can entangle at most one new qubit, and GHZ requires all n qubits to be entangled.

Resource accounting:

Metric Value What it measures
BIND count n−1 algorithm entanglement cost (hardware-independent)
BIND depth (all-to-all) 1 sequential BIND layers on IonQ/Quantinuum
BIND depth (nearest-neighbour) n−1 sequential BIND layers on IBM
Mana / TV 0 / 1 GHZ is a stabiliser state; no magic content
H^k tier H² genuinely entangling; no H¹ circuit can produce GHZ

The key distinction: BIND count (n−1) is fixed by the algorithm. BIND depth (1 or n−1) is fixed by the hardware connectivity graph. The ISA separates these; a gate-count analysis conflates them.

What the ISA lens adds here

The n−1 depth advantage of trapped ions for GHZ is real, but it comes from all-to-all connectivity, not from the ISA. A textbook comparing CNOT chains on IBM vs MS gates on IonQ reaches the same conclusion without any ISA machinery.

What the ISA contributes:

  1. BIND count is the algorithm resource — the irreducible entanglement cost of GHZ is n−1, stated independently of gate set or connectivity. This is the quantity that should appear in algorithm analyses, not “CNOT count on device X.”

  2. BIND depth = fidelity-relevant cost — since hardware fidelity decays as F^{BIND_depth} (sequential layers, not total count), BIND depth is the metric that actually predicts circuit fidelity. For GHZ, BIND depth on all-to-all hardware = 1 because all n−1 BINDs are mutually independent (qubit 0 is the only shared wire, but the BINDs fan out in parallel). The ISA programme graph makes this independence explicit.

  3. The same programme compiles to any backend — the ISA separates what the circuit computes from how hardware executes it. The depth difference is a compiler/topology fact, not an algorithmic one.

The honest claim: the ISA gives a clean name (BIND depth) to the metric that the field already uses informally. For GHZ, BIND depth = 1 on all-to-all is a topology statement. The ISA makes it derivable from the programme graph rather than requiring hardware knowledge.

Gate circuit translations

IonQ / Quantinuum (MS gate native)

All-to-all connectivity: every BIND[0→k] is a direct MS gate, and all n−1 are mutually independent, so the hardware executes them as a single parallel layer.

# PennyLane — device determines depth
import pennylane as qml

def ghz_isa(n):
    qml.Hadamard(wires=0)           # FLIP
    for k in range(1, n):
        qml.CNOT(wires=[0, k])      # BIND[0→k] — all independent

# IonQ Forte: executes all CNOT[0→k] as one parallel MS layer (depth 1)
dev_ionq = qml.device("ionq.qpu", wires=10)

# IBM: CNOT chain, depth grows with n
dev_ibm = qml.device("qiskit.ibmq", wires=10, backend="ibm_brisbane")

# Same ISA programme; depth difference is a hardware property

IonQ fidelity at n=10: F ≈ (0.999)^9 × 0.998 ≈ 0.990 (Quantinuum H2-1, one BIND layer).

IBM (CZ / CR gate native)

Nearest-neighbour layout: propagation chain. The CNOT chain is not an ISA requirement — it is one valid compilation of the ISA programme onto a linear topology.

h q[0];
cx q[0], q[1];     // depth 1
cx q[1], q[2];     // depth 2
…
cx q[n-2], q[n-1]; // depth n-1

IBM fidelity at n=10: F ≈ (0.995)^9 × 0.990 ≈ 0.856 (IBM Heron, no SWAP needed on linear layout, n−1 sequential BIND layers).

The fidelity gap (0.990 / 0.856 = 1.16× at n=10, growing to 1.49× at n=20) is a consequence of BIND depth difference, not BIND count difference. Both backends execute exactly n−1 BINDs.

Neutral atom / QuEra (Rydberg CZ)

Reconfigurable connectivity: with hub layout, achieves depth 1 like IonQ. Atom sorting adds latency not counted in gate depth.

BIND count vs BIND depth

n BIND count BIND depth (all-to-all) BIND depth (nearest-neighbour)
2 1 1 1
4 3 1 3
8 7 1 7
10 9 1 9
20 19 1 19
n n−1 1 n−1

BIND count is the same for both hardware families. BIND depth diverges as O(n). Fidelity scales with depth, not count. This table is a topology statement, not an ISA statement — the ISA provides the vocabulary to express it cleanly.

Zoo neighbours

  • QT01 — Bell state (n=2 special case of GHZ; BIND count = 1)
  • QT03 — QFT (H¹ circuit; BIND count = 0; no topology advantage to unlock)
  • QT04 — QAOA (also uses parallel BINDs on all-to-all; depth analysis similar)
  • Q07 — CHSH game (2-qubit H² resource; generalises to GHZ-based Bell tests)

Part of the ISA Zoo. QC Tutorial series: Paper 605.