Authors: Daniel Mo Houshmand
Affiliation: QDaria —
We present the largest quantum reservoir computing (QRC) demonstration on real quantum hardware to date, comparing 4-qubit and 156-qubit experimental IBM systems (Heron r3) 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) — Largest real QRC hardware
- Rigetti 9Q (simulation): R² = 0.959 ± 0.012 (800 samples, 3,375 features)
| Contribution | Details |
|---|---|
| Scale Record | First experimental QRC on 156-qubit real hardware, surpassing prior records of 120Q and 108Q |
| 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) |
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.
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.
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.
Quantum processor topologies. (A) IBM Heron r3 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.
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.
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.
| 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 | — |
Following Steinegger & Räth (Scientific Reports 15, 6201, 2025), we employ:
- Temporal Multiplexing (V=5): 5 virtual nodes per physical qubit
- Spatial Multiplexing (r=3): 3 independent reservoir initializations
- Polynomial Expansion (G=4): Degree-4 polynomial feature transformation
- Ridge Regularization: Cross-validated α optimization
Total features: 9 qubits × 5 temporal × 3 spatial = 135 base → 3,375 polynomial features
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
pip install qiskit qiskit-aer numpy scipy scikit-learn matplotlib# 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.pycd paper
pdflatex qrc_paper.tex
pdflatex qrc_paper.tex # Run twice for references- Released: July 2025
- T1: 300 μs | T2: 370 μs
- 2Q Gate Error: 5×10⁻⁴ (99.95% fidelity)
- Topology: Heavy-hex lattice
- 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
@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}
}- 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
- IBM Quantum — Hardware access through IBM Quantum Network
- Rigetti Computing — Continuous support for quantum simulations
- qBraid — Platform for running Rigetti simulations
This project is licensed under the MIT License — see LICENSE for details.
Daniel Mo Houshmand
QDaria —