QT01 — Bell State Preparation

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

What this circuit does

Bell state preparation is the canonical “Hello World” of quantum computing: it takes two qubits in the ground state |00⟩ and produces the maximally entangled Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 using exactly two gates (Hadamard + CNOT, or equivalently one FLIP and one BIND).

This is the simplest genuinely H² computation: it cannot be done without a BIND opcode. A circuit with only ORBIT and TWIST opcodes (H⁰ + H¹) can never produce entanglement from a product state.

ISA programme (hardware-independent) 

INIT:     ORBIT[|00⟩]              -- ground state (both qubits)
FLIP:     FLIP[qubit 0]            -- H⁰→H¹: |0⟩ → (|0⟩+|1⟩)/√2
BIND:     BIND[qubits 0, 1]        -- H² holonomy: generates entanglement
LABEL:    LABEL[qubit 0, qubit 1]  -- measure in computational basis

Programme length: 2 opcodes (1 FLIP + 1 BIND). This is the canonical length — any implementation using fewer than 1 BIND cannot produce an entangled state.

Resource content:

  • BIND count: 1 (1 MS gate / CNOT / CZ — all equivalent)
  • Mana: 0 (Bell states are stabiliser states; TV = 1)
  • H^k tier: H² — the output state is an H² object; it has no H¹ description

Interpretation functor

Opcode Meaning in 2-qubit Hilbert space
FLIP Hadamard H: rotates Bloch vector from Z-axis to X-axis; takes |0⟩ to the H⁰/H¹ boundary
BIND Controlled-NOT (or MS gate): generates entanglement via shared interaction; H² holonomy; cannot be factored as a product of single-qubit operations
LABEL Computational basis measurement σ_z; collapses to |00⟩ or |11⟩ with equal probability

Gate circuit translations

This is the Rosetta Stone for this ISA programme: the same 2-opcode programme compiles to three physically different gate circuits, all producing the same state.

IonQ / Quantinuum (MS gate native) 

FLIP[q0]  →  R(π/2, 0) on ion 0          -- single laser pulse
BIND[0,1] →  U_MS(π/4) on ions (0,1)     -- bichromatic laser; phonon bus
              + R_z(−π/2) on ion 0        -- virtual frame correction
              + R_z(−π/2) on ion 1        -- virtual frame correction

MS gate count: 1 (= BIND count). All corrections are virtual Z-rotations (TWIST, zero physical time). Total physical pulses: 2.

IBM / Google (CZ or CR gate native)

OpenQASM 3:

h q[0];           // FLIP
cx q[0], q[1];    // BIND (CNOT = CZ + local H)

Alternatively with native CZ:

h q[0];
h q[1]; cz q[0],q[1]; h q[1];   // CZ version: H·CZ·H = CNOT

CZ / CNOT count: 1 (= BIND count). Total gates: 2 (H + CNOT).

Neutral atom (Rydberg CZ) 

Rx(π/2) on atom 0                    // FLIP
Rydberg-CZ on (atom 0, atom 1)       // BIND
+ local single-qubit corrections      // TWIST

CZ (Rydberg) count: 1. Same BIND count as other backends.

Qiskit code 

from qiskit import QuantumCircuit

# ISA programme → Qiskit circuit
qc = QuantumCircuit(2, 2)
qc.h(0)       # FLIP
qc.cx(0, 1)  # BIND (1 CNOT = 1 MS gate on IonQ)
qc.measure_all()  # LABEL

# Compile to IonQ backend
from qiskit_ionq import IonQProvider
provider = IonQProvider(token="...")
backend = provider.get_backend("ionq_qpu")
job = backend.run(qc)  # automatically maps CNOT → U_MS

PennyLane code 

import pennylane as qml

dev_ionq    = qml.device("ionq.qpu", wires=2)
dev_ibm     = qml.device("qiskit.ibmq", wires=2, backend="ibm_brisbane")
dev_default = qml.device("default.qubit", wires=2)

@qml.qnode(dev_ionq)   # swap device to change backend; ISA programme unchanged
def bell_state():
    qml.Hadamard(wires=0)     # FLIP
    qml.CNOT(wires=[0, 1])    # BIND (PennyLane maps to U_MS on IonQ backend)
    return qml.probs(wires=[0, 1])

# Identical ISA programme, three physical backends:
# dev_ionq    → U_MS gate (Mølmer-Sørensen, phonon bus)
# dev_ibm     → CX gate (cross-resonance, microwave)
# dev_default → classical simulation

The ISA point: the programme FLIP · BIND is identical for all three device declarations. Only the dev argument changes. This is the ARM analogy: same assembly, different silicon.

BIND count analysis

Backend Physical gate for BIND Time per BIND Fidelity per BIND
IonQ Forte U_MS(π/4) ~200 μs 99.5%
Quantinuum H2-1 U_MS(π/4) ~100 μs 99.9%
IBM Heron CX (cross-resonance) ~100 ns 99.5%
Neutral atom (QuEra) CZ (Rydberg) ~1 μs 99.5%

For a 1-BIND circuit, all backends produce fidelity > 99%. The BIND count analysis becomes decisive for deep circuits (k ≥ 20 BINDs); see Paper 604 §5 and x604a results.

Speedup classification

Bell state preparation is not a speedup — it solves no classically hard problem. But it is the unit H² operation: the minimal computation that cannot be done without one BIND.

The speedup enabled by Bell pairs comes later:

  • Quantum teleportation (Q05): 1 Bell pair + 1 classical bit → teleport qubit
  • Superdense coding (Q06): 1 Bell pair → 2 classical bits over 1 qubit channel
  • CHSH violation (Q07): 1 Bell pair → win cooperative game at rate 85.4% > 75%

All three speedups have exactly 1 BIND in their resource account. The Bell pair is the H² token that is spent in each of them.

Zoo neighbours

  • Q05 — Quantum teleportation (uses 1 Bell pair as resource)
  • Q06 — Superdense coding (uses 1 Bell pair as resource)
  • Q07 — CHSH game (1 Bell pair → quantum advantage)
  • QH01 — Trapped-ion MS gate (the physical BIND for this circuit)
  • QT02 — GHZ state (n-qubit generalisation; n-1 BINDs)

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