inZOR-ND: Hardware-Native Quantum Error Correction on IBM Quantum (7-qubit, Heron)

Abstract

We apply the inZOR-ND bio-adaptive genomic discovery engine to quantum error correction (QEC) on real IBM Quantum hardware, without implementing any known QEC code structure. Two versions are presented: v1 (21 Ry parameters, X-only correction, 4 test states) and v2 (42 Ry+Rz parameters, Full Pauli correction with 22 syndromes, 6 test states, multi-noise training).

Both versions discover hardware-native 7-qubit encoding circuits optimized for the IBM Heron processor's tree topology. Across 7 hardware runs on ibm_fez and ibm_marrakesh, inZOR-ND outperforms Steane [[7,1,3]] in all observed hardware runs, with gains ranging from +0.2 pp to +5.1 pp. The v1 LUPA precision zoom achieves simulation fidelity 0.999843, while v2 reaches 0.990244 (multi-noise average over 4 noise levels, Full Pauli correction, 6 test states).

inZOR-ND Discovered Ansatz (7-qubit, IBM Heron native)

v1: Ry(7) → CZ → Ry(7) → CZ → Ry(7) | v2: [Ry(7)+Rz(7)] → CZ → [Ry+Rz] → CZ → [Ry+Rz]

v1: 21 params · v2: 42 params · 12 CZ gates (native IBM Heron) · Rz = virtual gate (zero error)

7/7

Hardware runs where inZOR-ND beats Steane

0.9998

v1 sim fidelity (LUPA final)

0.9902

v2 sim fidelity (multi-noise avg)

+5.1 pp

Max gain vs Steane (v1, noisy)

~58

Avg transpiled depth (inZOR-ND)

~115

Transpiled depth (Steane)

v2 parameters (Ry+Rz)

v2 syndromes (Full Pauli)

How to read this result

QEC only helps in noisy regimes. When hardware is very clean (no-code fidelity >0.9), both QEC codes perform worse than the unprotected qubit — circuit overhead exceeds error correction benefit.
inZOR-ND outperforms Steane in all 7 observed runs — in both noisy and clean regimes. The advantage comes from shallower native circuits (CZ vs CNOT decomposition).
v2 adds Full Pauli correction, Rz gates, and multi-noise training — a stricter, more comprehensive evaluation than v1.

Circuit Architecture — IBM Heron Tree Topology

Fig 0 — IBM Fez tree topology used as inZOR-ND ansatz. 7 qubits mapped to physical IBM Fez qubits (3, 2, 4, 16, 1, 5, 23). CZ gates on native hardware edges.

Component	v1	v2
Rotation gates per layer	Ry(7) = 7 params	Ry(7) + Rz(7) = 14 params
Total parameters	21 (3 layers × 7)	42 (3 layers × 14)
CZ gates	12 (6 pairs × 2 layers)
Native circuit depth	10	13
Error correction	X-only (8 syndromes)	Full Pauli X+Y+Z (22 syndromes)
Test states	4 (\|0⟩, \|1⟩, \|+⟩, \|−⟩)	6 (+ \|+i⟩, \|−i⟩)
Noise training	Single p_cz=0.002	Multi-noise: p_cz ∈ {0.001, 0.002, 0.005, 0.01}
Rz gate cost on IBM Heron	—	Virtual (zero error, zero depth)

1. Real Hardware Results — IBM Quantum (7 runs)

Seven hardware runs were performed on IBM Heron processors across two backends and two code versions. inZOR-ND outperforms Steane [[7,1,3]] in all observed hardware runs.

Fig 1 — Average fidelity on real IBM Quantum hardware, 7 runs (8000 shots each). v1 runs (left 4) and v2 runs (right 3, purple shading). inZOR-ND (blue) beats Steane (orange) in all cases.

#	Backend	Version	inZOR-ND	Steane	No-code	Gain vs Steane
1	ibm_fez	v1	0.8302	0.6035	0.9939	+0.2267
2	ibm_marrakesh	v1	0.6982	0.6470	0.6316	+0.0512
3	ibm_fez	v1	0.8791	0.8773	0.9934	+0.0018
4	ibm_marrakesh	v1	0.8874	0.8808	0.9278	+0.0066
5	ibm_fez	v2	0.8770	0.8618	0.9926	+0.0153
6	ibm_marrakesh	v2	0.8889	0.8868	0.9461	+0.0021
7	ibm_marrakesh	v2	0.8717	0.8657	0.9200	+0.0060

VERDICT: inZOR-ND outperforms Steane in all 7 observed runs.
Average gain vs Steane: +0.044 pp | Best gain (noisy regime): +5.12 pp
Consistent advantage across both backends, both versions, all noise regimes.

2. v1 vs v2 — Head-to-Head Comparison

Runs #4 (v1) and #6/#7 (v2) were performed on ibm_marrakesh on the same day (same calibration window), enabling a direct comparison of the two code versions.

Fig 7 — Left: Fidelity comparison on ibm_marrakesh same-day calibration. v2 has slightly higher absolute fidelity. Right: Both versions beat Steane, but v1 has a larger margin. v2 features: Rz gates, Full Pauli correction, 6 test states, multi-noise training.

Metric	v1 (run #4)	v2 (run #6)	v2 (run #7)
inZOR-ND fidelity	0.8874	0.8889	0.8717
Steane fidelity	0.8808	0.8868	0.8657
No-code fidelity	0.9278	0.9461	0.9200
Gain vs Steane	+0.0066	+0.0021	+0.0060
Parameters	21	42	42
Test states	4	6	6
Syndromes	8 (X-only)	22 (Full Pauli)	22 (Full Pauli)

3. Per-State Analysis — v2 (6 test states)

Fig 3 — Per-state fidelity on ibm_marrakesh (v2 run #7). inZOR-ND wins on 4/6 states (|0⟩, |1⟩, |+i⟩, |−i⟩). Steane is stronger on superposition states |+⟩ and |−⟩.

State	inZOR-ND v2	Steane	No-code	Winner
\|0⟩	0.8864	0.8469	0.8956	inZOR-ND
\|1⟩	0.8738	0.8450	0.9476	inZOR-ND
\|+⟩	0.8688	0.9159	0.8900	Steane
\|−⟩	0.8601	0.8852	0.9511	Steane
\|+i⟩	0.8714	0.8589	0.8894	inZOR-ND
\|−i⟩	0.8699	0.8421	0.9461	inZOR-ND
Average	0.8717	0.8657	0.9200	inZOR-ND +0.006

4. Circuit Depth Comparison

Fig 2 — Left: Native circuit properties. Right: After transpilation on IBM Heron — inZOR-ND circuits are ~2x shallower than Steane, resulting in less decoherence.

Code	Native gates	Native depth	2Q gates	Transpiled depth
inZOR-ND v1	33	10	12 CZ (native)	~53
inZOR-ND v2	54	13	12 CZ (native)	~58
Steane [[7,1,3]]	14	8	6 CNOT → 12 CZ*	~115

*CNOT requires decomposition on Heron: CNOT ≈ Rz + CZ + Rz. v2's Rz gates are virtual on IBM Heron (zero depth, zero error), so v2 depth increase is minimal (+5 vs v1).

5. Simulation Results & Cross-Validation

v2 Cross-validation (Qiskit AerSimulator)

Fig 6 — v2 cross-validation: NumPy vs Qiskit AerSimulator at all 4 noise levels + multi-noise average. Maximum difference: 1.1e-16 (machine epsilon). Perfect match confirmed.

Noise level	NumPy Fidelity	Qiskit AerSim	\|Diff\|	Status
p_cz = 0.001	0.997296	0.997296	0.00e+00	MATCH
p_cz = 0.002	0.995268	0.995268	1.1e-16	MATCH
p_cz = 0.005	0.989212	0.989212	2.2e-16	MATCH
p_cz = 0.010	0.979200	0.979200	0.00e+00	MATCH
Multi-noise avg	0.990244	0.990244	1.1e-16	MATCH

v2 Simulation: inZOR-ND vs Steane (Full Pauli)

Multi-noise simulation and noise analysis

Fig 5 — Left: Simulation fidelity vs noise level — Steane (Full Pauli) beats inZOR-ND in ideal simulation. Right: On real hardware, inZOR-ND wins consistently due to shallower circuits.

p_cz	inZOR-ND v2	Steane (Full Pauli)	Diff
0.001	0.9973	0.9999	−0.0026
0.002	0.9953	0.9996	−0.0044
0.005	0.9892	0.9979	−0.0087
0.010	0.9792	0.9924	−0.0132
Average	0.9902	0.9974	−0.0072

In simulation (ideal depolarizing noise only), Steane's [[7,1,3]] code with Full Pauli correction outperforms inZOR-ND at all noise levels. However, on real hardware, inZOR-ND's shallower native circuits (CZ vs decomposed CNOT) give it a decisive advantage.

6. LUPA Precision Zoom — v1 Refinement

Fig 4 — v1 LUPA precision zoom: 3 rounds refine simulation fidelity from ~0.973 to 0.999843 (p_cz=0.002, X-only correction).

Phase	Cycles	step_scale	Fidelity
Initial (random)	20	~0.08	~0.973
LUPA Round 1	20	0.005	~0.9965
LUPA Round 2	20	0.005	~0.9985
LUPA Round 3	20	0.005	0.999843

7. Discovered Codes — Best Angles

v1: 21 Ry parameters

Layer 0: [ 2.3111,  0.0413,  1.5849, -1.6710, -0.6804, -0.3224,  2.3266]
Layer 1: [ 1.5807, -1.4940,  3.1331, -3.1416, -0.8384,  1.5683, -2.4826]
Layer 2: [ 0.0209, -0.0165, -3.0455,  1.5756,  1.6865,  1.2787,  0.7795]
# Sim fidelity: 0.999843 (p_cz=0.002, X-only)
    

v2: 42 Ry+Rz parameters

Layer 0 Ry: [ 0.0711,  3.1416, -1.6156,  1.4526, -1.8792,  0.2007,  2.3637]
Layer 0 Rz: [-1.0470,  0.1789, -2.1217,  1.9662,  0.1666, -0.7400,  2.6914]
Layer 1 Ry: [-1.5718, -1.6031, -3.1416, -0.0318, -1.9116, -1.5387,  1.2361]
Layer 1 Rz: [-1.7021,  0.0572,  2.6904,  3.1416,  1.5846,  0.2212, -1.4541]
Layer 2 Ry: [-2.5857,  2.6224, -0.3381, -2.2045,  0.1963,  2.5228,  0.0592]
Layer 2 Rz: [ 2.7810, -2.3226,  0.0457, -0.8940, -2.7580,  1.3710,  2.2129]
# Sim fidelity: 0.990244 (multi-noise avg, Full Pauli)
    

8. Role of inZOR-ND

The entire QEC code discovery was powered by inZOR-ND with zero domain knowledge. The same engine is used across all published tests (power systems, astrophysics, social dynamics, thermal management).

What inZOR-ND does NOT know:

Quantum error correction theory (stabilizer formalism, distance, CSS codes, Knill-Laflamme conditions)
The Steane code or any other known QEC code
What a "good" quantum encoding should look like
Quantum gates semantics — it only sees real-valued parameters and a fidelity score

9. Data, Code & Reproducibility

Hardware: IBM Quantum Open Plan · ibm_fez (Heron r1, 156Q) · ibm_marrakesh (Heron r1, 156Q)
Shots per circuit: 8000 · Total runs: 7 (v1 × 4 + v2 × 3)

# v1:
python3 env_qec_7hw.py            # v1 environment (21 params, X-only)
python3 run_qec_7hw.py            # v1 inZOR-ND + LUPA
python3 validate_qiskit_hw.py     # v1 Qiskit cross-validation
python3 run_ibm_hardware_hw.py --token TOKEN --backend ibm_fez

# v2:
python3 env_qec_7hw_v2.py         # v2 environment (42 params, Full Pauli)
python3 run_seed1_save.py          # v2 inZOR-ND (seed 1, 30 cycles)
python3 validate_qiskit_hw_v2.py   # v2 Qiskit cross-validation
python3 run_ibm_hardware_hw_v2.py --token TOKEN --backend ibm_marrakesh

# Results:
# results_7hw/best_angles_hw.json       (v1: 21 params)
# results_7hw_v2/best_angles_hw_v2.json (v2: 42 params)
    

Disclaimer: Hardware results vary between calibration cycles. inZOR-ND outperforms Steane in all 7 observed runs, but the absolute fidelity depends on the hardware noise level at the time of measurement. In simulation, Steane's Full Pauli correction outperforms inZOR-ND; the real-hardware advantage comes from shallower native circuits that accumulate less decoherence.

10. Conclusion

These results suggest that genomic search can discover hardware-native QEC encoders competitive with known codes on NISQ devices, without implementing any known QEC code structure. The key finding is not that inZOR-ND produces a theoretically superior code — in ideal simulation, Steane's Full Pauli correction remains stronger — but that theoretical optimality is not the same as hardware optimality: a shallower, hardware-native circuit accumulates less decoherence and can outperform a theoretically stronger but hardware-mismatched code on real NISQ processors.

Core finding (7 IBM hardware runs)

Hardware-native circuit depth < theoretical depth → real hardware advantage

Genomic search discovers hardware-adapted QEC encoders competitive with known codes on NISQ devices

inZOR-ND: Hardware-Native Quantum Error Correction Discovered by Genomic Evolution on IBM Quantum