Topological Quantum Reservoir Computing

A No-Go Theorem: Why Unitarity Opposes the Echo State Property

Daniel Mo Houshmand
QDaria | Oslo, Norway
mo@qdaria.com

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Overview

This repository contains the complete research artifacts for our paper establishing fundamental no-go results for Topological Quantum Reservoir Computing (TQRC) using Fibonacci anyons.

Key Finding: The unitary nature of topological quantum evolution mathematically prevents the Echo State Property (ESP) required for reservoir computing. This is not an engineering limitation but a fundamental incompatibility.

Metric	Pure Unitary TQRC	Classical ESN	Gap
Mackey-Glass NRMSE	0.966	0.015	64x
Lorenz-63 NRMSE	1.01	0.005	202x
NARMA-10 NRMSE	0.75	0.81	~1x

Note: TQRC performs comparably on memory-focused tasks (NARMA-10) but fails catastrophically on chaotic prediction requiring fading memory.

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Quick Start

Installation

# Clone repository
git clone https://github.com/QDaria/tqrc.git
cd tqrc

# Create virtual environment (recommended)
python -m venv .venv
source .venv/bin/activate  # Linux/macOS
# .venv\Scripts\activate   # Windows

# Install dependencies
pip install -r requirements.txt

# Verify installation
python -c "from src.tqrc import FibonacciAnyon; print('OK')"

Run Experiments

# Reproduce main results (Table 1-3)
python scripts/run_benchmarks.py

# ESP violation demonstration (Figure 5)
python scripts/esp_violation.py

# Memory capacity analysis (Figure 6)
python scripts/memory_capacity.py

Build Paper

cd paper/v3
pdflatex -interaction=nonstopmode tqrc_ieee_v3.tex
pdflatex -interaction=nonstopmode tqrc_ieee_v3.tex
pdflatex -interaction=nonstopmode tqrc_ieee_v3.tex

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Key Results

Theorem 1: Unitarity-ESP Incompatibility

No-Go Theorem: Let $\mathcal{E}(\rho) = U\rho U^\dagger$ be a unitary quantum channel. Then $\mathcal{E}$ cannot satisfy the quantum Echo State Property.

Proof: Unitary evolution preserves trace distance: $$D_{\text{tr}}(\mathcal{E}(\rho_1), \mathcal{E}(\rho_2)) = D_{\text{tr}}(\rho_1, \rho_2)$$

ESP requires $D_{\text{tr}} \to 0$ as $t \to \infty$, which is impossible when distance is preserved.

Lemma: Spectral Radius Condition

The quantum ESP is satisfied if and only if the spectral radius of the channel restricted to traceless operators satisfies $\rho(\mathcal{E}|_{\text{traceless}}) < 1$.

For unitary channels, all eigenvalues have magnitude 1, so $\rho = 1$, violating ESP.

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Repository Structure

tqrc/
├── paper/
│   └── v3/
│       ├── tqrc_ieee_v3.tex      # Main paper (IEEE format)
│       ├── tqrc_ieee_v3.pdf      # Compiled PDF
│       └── figures/              # Symlink to ../figures
├── figures/                      # All 18 publication figures
│   ├── fig01_architecture.pdf
│   ├── fig05_esp_violation.pdf
│   ├── fig06_dissipative_results.pdf
│   ├── fig07_root_cause.pdf
│   └── ...
├── src/
│   └── tqrc/                     # Python package
│       ├── __init__.py
│       ├── fibonacci.py          # Fibonacci anyon implementation
│       ├── reservoir.py          # TQRC reservoir class
│       ├── benchmarks.py         # Mackey-Glass, Lorenz, NARMA
│       └── esn.py                # Classical ESN baseline
├── scripts/
│   ├── run_benchmarks.py         # Reproduce all results
│   ├── esp_violation.py          # ESP demonstration
│   └── memory_capacity.py        # Memory scaling analysis
├── results/                      # Cached experimental results
├── requirements.txt              # Python dependencies
├── CITATION.cff                  # Citation metadata
├── LICENSE                       # CC BY 4.0
└── CLAUDE.md                     # AI assistant instructions

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Figures

Figure 5: ESP Violation (click to expand)

Pure unitary evolution (red) maintains constant state distance, violating ESP. Dissipative dynamics (blue) achieve convergence but sacrifice topological protection.

Figure 6: Dissipative Results

NRMSE vs dissipation rate showing optimal performance at Γ ≈ 0.25.

Figure 7: Root Cause Analysis

ESN uses 13/13 dimensions with 0.98 input correlation; TQRC uses only 4/13 with near-zero correlation.

Figure 11: Protection-ESP Tradeoff

Fundamental tradeoff: dissipation enables ESP but destroys topological protection.

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Benchmarks

Mackey-Glass Time Series

Model	Dimension	NRMSE	95% CI
Classical ESN	13	0.015	[0.013, 0.017]
Classical ESN	100	0.004	[0.004, 0.004]
Pure Unitary TQRC	13	0.966	[0.966, 0.966]
Dissipative TQRC	13	1.18	[0.97, 1.45]

Lorenz-63 Attractor

Model	NRMSE	Performance Gap
Classical ESN (13D)	0.005	1x
Pure Unitary TQRC	1.01	202x worse
Dissipative TQRC	1.12	224x worse

NARMA-10 (Memory Task)

Model	NRMSE	Notes
Pure Unitary TQRC	0.75	Competitive
Classical ESN	0.81	Similar

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Citation

@article{houshmand2026tqrc,
  title   = {The Fundamental Tension in Topological Quantum Reservoir
             Computing: Why Unitarity Opposes the Echo State Property},
  author  = {Houshmand, Daniel Mo},
  journal = {arXiv preprint arXiv:2501.XXXXX},
  year    = {2026},
  doi     = {10.22541/au.176549133.31550916/v2}
}

CITATION.cff

This repository includes a CITATION.cff file for automatic citation in GitHub and Zenodo.

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Preprint Links

Platform	Link	Status
arXiv	arXiv:2501.XXXXX	Pending
TechRxiv	10.22541/au.176549133	Published
Zenodo	10.5281/zenodo.17889778	Published

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Related Work

Foundational Papers

• Jaeger (2001). The "echo state" approach to analysing and training recurrent neural networks. GMD Report 148.
• Nayak et al. (2008). Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083.

Recent Experimental Advances

• Xu et al. (2024). Non-Abelian braiding of Fibonacci anyons with a superconducting processor. Nature Physics 20, 1469.
• Iqbal et al. (2024). Non-Abelian topological order and anyons on a trapped-ion processor. Nature 626, 505.

Quantum Reservoir Computing

• Fujii & Nakajima (2017). Harnessing disordered-ensemble quantum dynamics for machine learning. Phys. Rev. Applied 8, 024030.
• Sannia et al. (2024). Dissipation as a resource for quantum reservoir computing. Quantum 8, 1291.

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License

This work is licensed under CC BY 4.0. You are free to share and adapt this material for any purpose with attribution.

Code is licensed under MIT License.

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Acknowledgments

We thank the quantum computing community for valuable discussions on the intersection of topological protection and machine learning dynamics.

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