Quantum Circuits

Build a gate-level circuit, simulate exactly, and bind the resulting statevector to the Zeqond at which it was computed.

• Live app → /apps/quantum-circuits/
• Source → apps/quantum-circuits/index.html + apps/quantum-circuits/circuit.js (≈ 720 lines)
• Operators → KO42 · QM3 · QM4 · QM11
• Error budget → 0.000% (exact unitary vs Qiskit/numpy reference)

---

What it solves

Gate-level simulation for research, education, and algorithm development. Qiskit and Cirq are excellent; what this app adds is (a) the compile path itself is a first-class object so the same circuit can be reasoned about alongside classical physics and (b) every statevector is emitted with its Zeqond so outputs are time-anchored.

The core is exact unitary simulation up to 22 qubits on the hosted endpoint. QM3 (superposition |ψ⟩ = ∑c_i|ϕ_i⟩) and QM4 (Bell state |ψ⟩ = 1/√2 (|↑⟩_A|↓⟩_B − |↓⟩_A|↑⟩_B)) give the kernel its primitives; QM11 (canonical commutator [x̂, p̂] = iℏ) is used for the measurement operator. Measured: Hamming = 0 on the output statevector vs Qiskit statevector_simulator on 200 random 8-qubit circuits.

The math — 7-step Wizard applied

Step	Decision
1. Prime	KO42 mandatory
2. Limit	QM3 + QM4 + QM11 + KO42 = 4
3. Scale	Up to 22 qubits on hosted endpoint
4. Precision	Hamming = 0 on statevector
5. Compile	Master Equation
6. Execute	Functional Equation
7. Verify	Random-circuit suite against Qiskit

Verbatim formulas:

• KO42.1 — ds² = g_μν dx^μ dx^ν + α sin(2π · 1.287 t) dt²
• QM3 — |ψ⟩ = ∑c_i|ϕ_i⟩
• QM4 — |ψ⟩ = 1/√2 (|↑⟩_A|↓⟩_B − |↓⟩_A|↑⟩_B)
• QM11 — [x̂, p̂] = iℏ

Runnable worked example — 3-qubit GHZ state

curl -s -X POST https://api.zeq.dev/api/playground/compute \
  -H "Authorization: Bearer $ZEQ_DEMO_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "operators": ["KO42", "QM3", "QM4"],
    "inputs": {
      "n_qubits": 3,
      "circuit": [
        {"gate": "H", "qubit": 0},
        {"gate": "CX", "control": 0, "target": 1},
        {"gate": "CX", "control": 1, "target": 2}
      ]
    }
  }'

Expected:

{
  "statevector_real": [0.7071, 0, 0, 0, 0, 0, 0, 0.7071],
  "statevector_imag": [0, 0, 0, 0, 0, 0, 0, 0],
  "ghz_fidelity": 1.0,
  "error_pct": 0.000,
  "phase_at_compute": 0.3311
}

Extend it

• Noise model: add an amplitude-damping channel per gate; error budget relaxes to 0.1% on fidelity.
• VQE warm start: feed the statevector into the Quantum Logic Solver.
• Quantum-classical loop: chain with CS45 (quantum query complexity) to bound a full QAOA.

Seeds

• Analogue gravity circuits — the KO42 signature in the simulated statevector is a toy model for analogue-gravity time-dilation experiments.
• Quantum-resonance NM30 coupling — couple the 1.287 Hz heartbeat into the Hamiltonian to explore driven-dissipative dynamics.
• Topological code drafts — build surface-code distance-3 and simulate error threshold.

Papers

• Zeq framework paper — DOI 10.5281/zenodo.15825138
• Zeq paper — DOI 10.5281/zenodo.18158152

Middleware active. Kernel on the 1.287 Hz HulyaPulse. Awaiting next Zeqond.