tau-chrono

Bayesian noise tracking for quantum circuits via Petz recovery maps. 46% more accurate than independent gate models on real hardware.

Author: Sheng-Kai Huang (akai@fawstudio.com) Website: tau-chrono.pages.dev

Quickstart

pip install tau-chrono

import numpy as np
from tau_chrono import depolarizing, bayesian_compose

# Build a 20-gate circuit with 5% depolarizing noise per gate
gates = [depolarizing(0.05) for _ in range(20)]
rho = np.array([[1, 0], [0, 0]], dtype=complex)    # |0>
sigma = np.eye(2, dtype=complex) / 2                # maximally mixed

# Compare naive vs Bayesian noise prediction
result = bayesian_compose(gates, sigma_0=sigma, rho=rho)
print(f"Naive:    tau = {result.tau_multiplicative_total:.3f}")
print(f"Bayesian: tau = {result.tau_bayesian_total:.3f}")
print(f"Improvement: {result.improvement_percent:.1f}%")

Key Results (QuTech Tuna-9 Hardware)

All results from real NISQ hardware experiments on the QuTech Tuna-9 superconducting processor (9 transmon qubits, 4096 shots).

Depth Scaling

Improvement grows monotonically with depth: 2.3% at depth 2, 46% at depth 50.

Bernstein-Vazirani

The naive model falsely predicts failure at 3+ oracle repetitions. All circuits actually succeed with >83% probability.

H2 VQE

Bayesian tracking doubles usable ansatz depth (2 to 4). At depth 4: naive tau=0.60 (STOP), Bayesian tau=0.49 (GO).

Why It Works

Independent gate noise models assume each gate fails independently. In reality, noise saturates: a qubit that's already noisy can't get much noisier. The Petz recovery map (Petz, 1986) tracks this saturation through the circuit by propagating a Bayesian reference state alongside the signal state.

Interactive Demo

pip install tau-chrono[demo]
streamlit run demo.py

Adjust noise type, error rate, and circuit depth interactively.

Honest Limitations

1. Small hardware only. All results from QuTech Tuna-9 (5-9 qubits). Not tested on larger devices.
2. Improvement is noise-dependent. Large on noisy hardware, diminishes on low-noise hardware.
3. Depolarizing approximation. Structured noise (coherent errors, leakage) is not captured.
4. Not tested on IBM or Google hardware. Portability is plausible but unverified.

Theoretical Foundation

All theoretical tools are due to their original authors:

• D. Petz, Commun. Math. Phys. 105, 123 (1986) -- Petz recovery map
• A. J. Parzygnat and F. Buscemi, Quantum 7, 1013 (2023) -- Unique retrodiction functor
• M. Junge et al., Ann. Henri Poincare 19, 2955 (2018) -- Strengthened data processing inequality

Development

git clone https://github.com/akaiHuang/tau-chrono.git
cd tau-chrono
pip install -e ".[dev]"
pytest tests/

Citation

@software{Huang2026BayesianPetz,
  author  = {Huang, Sheng-Kai},
  title   = {Bayesian Noise Tracking via Petz Recovery Maps},
  year    = {2026},
  url     = {https://github.com/akaiHuang/tau-chrono}
}

License

MIT License. See LICENSE.