Quantum Reservoir Computing for Chaotic System Prediction

156-Qubit QRC on IBM Heron r2 (ibm_fez) — a sample-efficiency study

Authors: Daniel Mo Houshmand Affiliation: QDaria — $\mathbb{Q}|\mathcal{D}\partial\mathfrak{r}\imath\alpha\rangle$ Contact: mo@qdaria.com

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Hardware provenance

Every IBM run executed on the Heron r2 backend ibm_fez via the IBM Quantum Platform open plan (4,000 shots per circuit). Job IDs, dates and sample counts: hardware/PROVENANCE.md. Result files: hardware/jobs/; execution scripts: hardware/scripts/; reservoir states: data/reservoir_states_ibm_4q.npy (50×10) and data/reservoir_states_hero_156q_200.npy (200×156).

Abstract

We present a quantum reservoir computing (QRC) study on IBM superconducting hardware, comparing 4-qubit and 156-qubit reservoirs executed on the 156-qubit Heron r2 processor ibm_fez (open plan) — a qubit count that, to our knowledge, exceeds previously reported QRC hardware experiments, although the larger reservoir did not translate into better prediction alongside high-fidelity 9-qubit Rigetti simulation employing the Steinegger-Räth (2025) feature engineering methodology.

Key Results:

• IBM 4Q: R² = 0.764 ± 0.018 (50 samples, 10 features)
• IBM 156Q: R² = 0.723 ± 0.022 (200 samples, 156 features) — 156-qubit hardware run (ibm_fez)
• Rigetti 9Q (simulation): R² = 0.959 ± 0.012 (800 samples, 3,375 features)

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Highlights

Contribution	Details
Hardware scale	Full-processor QRC on 156-qubit IBM Heron r2 (ibm_fez); a larger qubit count than prior reports (120Q preliminary, Yasuda et al. 2023; 108 atoms, Kornjača et al. 2024), not better learning
Sample Efficiency Crisis	156Q (1.28 samples/feature) performs comparably to 4Q (5.0 samples/feature) — diminishing returns identified
Multi-System Validation	Average R² = 0.908 across Lorenz-63, Rössler, and turbulence (13× range in Lyapunov exponents)
Feature Engineering Dominance	9Q with polynomial features outperforms 156Q hardware by ΔR² = 0.236 (p < 0.001)

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Key Figures

Figure 1: System Performance Comparison

Turbulence Prediction Performance. (A) Test R² scores showing IBM 4Q (0.764), IBM 156Q (0.723), and Rigetti 9Q simulation (0.959). The 9Q system with Steinegger-Räth polynomial feature engineering exceeds the classical LSTM baseline (0.85). (B) Sample efficiency analysis reveals the critical samples-per-feature ratios: IBM 4Q operates in the marginal zone (5.0), IBM 156Q in the critical zone (1.28), while Rigetti 9Q achieves excellent performance at 0.19 samples/feature through ridge regularization.

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Figure 2: Forecast Trajectories

Forecast trajectories across quantum systems on spectral turbulence data. (A) IBM 4Q maintains valid predictions for 1.7 Lyapunov times. (B) IBM 156Q achieves similar 1.8τ forecast horizon despite 39× more qubits, demonstrating the sample efficiency bottleneck. (C) Rigetti 9Q simulation extends valid forecasting to 23.9τ (14× improvement), showing that feature engineering dominates raw qubit count.

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Figure 3: Sample Efficiency Analysis

Sample Efficiency vs. Performance. Scatter plot showing the relationship between samples per feature (log scale) and R² performance. Despite operating with 26× fewer samples per feature than IBM 4Q, Rigetti 9Q achieves 96% variance explanation through polynomial feature engineering and ridge regularization.

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Figure 4: Hardware Topology Comparison

Quantum processor topologies. (A) IBM Heron r2 heavy-hex lattice (156Q fragment showing 3 unit cells) with maximum degree 3 and reduced crosstalk. (B) Rigetti Novera 3×3 square lattice with full nearest-neighbor connectivity (33.3%) enabling dense coupling maps without SWAP gates.

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Figure 5: Feature Engineering Ablation Study

Progressive Feature Engineering Impact. (A) R² progression from baseline 9-qubit measurements (0.12) through correlations (0.34), temporal multiplexing V=5 (0.62), spatial multiplexing r=3 (0.79), to full polynomial expansion G=4 (0.959). (B) Feature count explosion vs. samples-per-feature decline, illustrating the regularization requirement.

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Figure 6: Lyapunov Forecast Horizon Comparison

Chaotic System Predictability. Horizontal bar chart showing forecast horizons in Lyapunov times (τ). The 9-qubit simulated system achieves 14× longer forecast horizon (23.9τ) compared to hardware implementations (1.7-1.8τ), demonstrating superior predictability through polynomial feature engineering.

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Multi-System Validation Results

System	Lyapunov λ	Test R²	Horizon (τ)	Best α
Lorenz-63	0.906	0.796	4.4	0.001
Rössler	0.071	0.969	31.7	0.100
Turbulence	0.245	0.959	23.9	0.1
Average	—	0.908	20.0	—

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Methodology: Steinegger-Räth Feature Engineering

Following Steinegger & Räth (Scientific Reports 15, 6201, 2025), we employ:

1. Temporal Multiplexing (V=5): 5 virtual nodes per physical qubit
2. Spatial Multiplexing (r=3): 3 independent reservoir initializations
3. Polynomial Expansion (G=4): Degree-4 polynomial feature transformation
4. Ridge Regularization: Cross-validated α optimization

Total features: 9 qubits × 5 temporal × 3 spatial = 135 base → 3,375 polynomial features

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Repository Structure

qrc/
├── README.md                 # This file
├── paper/
│   ├── qrc_paper.tex        # LaTeX source (IEEE format)
│   ├── qrc_paper.pdf        # Compiled paper
│   └── figures/             # All publication figures (PNG + PDF)
├── data/
│   ├── validation_results.json
│   ├── novera_9q_results.json
│   ├── multi_system_comparison.json
│   └── *.json               # Experimental results
└── scripts/
    ├── simulate_rigetti_novera_9q.py   # 9Q turbulence simulation
    ├── simulate_9q_lorenz63.py         # Lorenz-63 simulation
    ├── simulate_9q_rossler.py          # Rössler simulation
    └── generate_figure*.py             # Figure generation scripts

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Quick Start

Prerequisites

pip install qiskit qiskit-aer numpy scipy scikit-learn matplotlib

Run Simulations

# 9-qubit turbulence simulation (Steinegger methodology)
cd scripts
python simulate_rigetti_novera_9q.py

# Lorenz-63 and Rössler simulations
python simulate_9q_lorenz63.py
python simulate_9q_rossler.py

Build Paper

cd paper
pdflatex qrc_paper.tex
pdflatex qrc_paper.tex  # Run twice for references

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Hardware Specifications

IBM Heron r2 (156Q) — ibm_fez

• Access: IBM Quantum Platform open plan; Qiskit Runtime SamplerV2, 4,000 shots per circuit
• Jobs: d4kdprd74pkc7386uov0 (4Q), d4m7kg10i6jc73dgg1n0 + d4m87dl74pkc7388nll0 (156Q), 27–30 Nov 2025 — see hardware/PROVENANCE.md
• Run-time calibration not archived

Rigetti Novera (9Q) — Simulated

• Architecture: Ankaa 4th generation
• Topology: 3×3 square lattice
• 1Q Fidelity: 99.9% | 2Q Fidelity: 99.4% (iSWAP)
• T1: 46 μs | T2 echo: 26 μs
• Price: $900,000

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Citation

@article{houshmand2025qrc,
  title={Sample Efficiency Crisis in Quantum Reservoir Computing:
         Scaling Analysis on 156-Qubit IBM Hardware and Rigetti Simulation},
  author={Houshmand, Daniel Mo},
  journal={arXiv preprint arXiv:2412.XXXXX},
  year={2025},
  doi={10.5281/zenodo.17910992},
  url={https://doi.org/10.5281/zenodo.17910992}
}

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Related Work

• Steinegger & Räth (2025): Quantum reservoir computing with polynomial feature engineering
• Kubota et al. (2023): 120-qubit QRC demonstration
• Senanian et al. (2024): 108-qubit QRC with dynamical decoupling

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Acknowledgments

• IBM Quantum — hardware access through the IBM Quantum Platform open plan; all runs on ibm_fez
• Rigetti Computing — Continuous support for quantum simulations
• qBraid — Platform for running Rigetti simulations

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License

This project is licensed under the MIT License — see LICENSE for details.

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Contact

Daniel Mo Houshmand QDaria — $\mathbb{Q}|\mathcal{D}\partial\mathfrak{r}\imath\alpha\rangle$ Email: mo@qdaria.com GitHub: @MoHoushmand