Gradient-Based Learning of Quantum Signal Processing Phase Angles

A reproducible benchmark and JAX implementation note

Train QSP phase angles with Adam instead of analytic solvers — on a flat, JAX-traceable circuit (QSP · QSVT · variational / differentiable quantum programming). We measure when gradient learning beats mapped analytic baselines (PennyLane poly_to_angles, Chao/pyqsp) on a shared flat-circuit MSE, and release code, sweeps, convention docs, and a reproducible Chebyshev benchmark.

For: variational / QML pipelines with implicit targets · solver comparisons · anyone embedding differentiable QSP blocks · reproducibility audits. PennyLane is the reference frontend only; the flat-circuit pattern ports to Qiskit, Cirq, TFQ, etc. (docs/FRAMEWORKS.md).

Outcome (headline): learned train MSE $\approx10^{-4}$ vs mapped analytic $\approx10^{-3}$ on the same circuit at $d=5$ — with a documented phase-map protocol, not a claim of universal superiority.

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Paper

Read or download: Zenodo — Version 1.1 (PDF + results ZIP)

LaTeX sources for audit and reuse: manuscript.tex, manuscript_numbers.tex, references.bib.

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Problem & scope

Classical QSP computes phase angles analytically from a target polynomial. This repo benchmarks the alternative: optimize angles with automatic differentiation and Adam, compare fairly to analytic solvers on one metric, and document pitfalls (e.g. high-level QSVT templates that break JAX grads through qml.QSVT).

This is not a new QSP algorithm — a reproducible measurement protocol, open implementation, and convention mapping.

When to use which

Situation	Approach
Known target, flat-circuit MSE (this benchmark)	Gradient training
Fast classical angles in native solver convention	Chao / pyqsp or PennyLane
Implicit loss (e.g. energy in a VQA)	End-to-end gradient
Novel polynomial, no closed form	Gradient training

Contributions

• JAX-traceable flat QSP circuit + QSVT-template workaround
• Degree-5 Chebyshev sin(x) benchmark: multi-seed ($d=5,7,15$), scaling, ablation, off-grid probe
• Phase-convention protocol — docs/CONVENTIONS.md (φ_flat = π/2 − φ_chao)
• Experiment pipeline, tests, audit trail, manuscript

Code layout: start in demo.ipynb · library in qsp_jax/circuit.py · CLI in experiments/

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Quick start

git clone https://github.com/rosspeili/trainable-qsp-angles
cd trainable-qsp-angles
py -3.13 -m pip install -r requirements.txt
py -3.13 -m pytest tests/ -v

Try it (~5 min)

Interactive training and plots — no sweeps required:

py -3.13 -m jupyter notebook demo.ipynb

View on nbviewer

Reproduce the paper

Download the results ZIP from Zenodo, or regenerate locally (defaults in experiments/configs/default.json):

py -3.13 -m experiments.train --seed 0 --steps 500
py -3.13 -m experiments.baseline_analytic
py -3.13 -m experiments.sweep multi-seed
py -3.13 -m experiments.sweep scaling
py -3.13 -m experiments.summarize baseline

Full CLI: docs/REPRODUCING.md. Analysis notebooks (after sweeps):

Notebook	nbviewer
notebooks/01_baseline_comparison.ipynb	open
notebooks/02_scaling_study.ipynb	open

Local runs write to results/ (gitignored; see results/schema.json). Committed paper figure values: manuscript_numbers.tex.

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Results (headline)

Default target: degree-5 Chebyshev approximation of sin(x) on $[-1,1]$.

Chebyshev target (solid) vs. sin(x) (dashed). Training minimizes MSE to the polynomial, not to sin(x) directly.

Mapped analytic rows use the same flat circuit after φ_flat = π/2 − φ_chao (docs/CONVENTIONS.md); PennyLane and pyqsp baselines are compared on equal footing.

Metric (seed 0, default protocol)	Value
Train MSE (gradient Adam)	$9.6 \times 10^{-5}$
Train MSE (PennyLane / pyqsp, mapped)	$4.7 \times 10^{-3}$
Multi-seed median ($n=30$, $d=5$)	$6.3 \times 10^{-5}$
Multi-seed success ($d=7$, $d=15$)	30/30 each, below $10^{-3}$
Off-grid max error (learned vs mapped analytic)	$2.9 \times 10^{-2}$ vs $1.8 \times 10^{-1}$

Demo snapshot (seed 0). Full figures and tables: Zenodo PDF.

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Documentation

Doc	What it covers
docs/REPRODUCING.md	Full experiment CLI and sweep commands
docs/CONVENTIONS.md	Phase maps (pyqsp / PennyLane → flat circuit)
docs/FRAMEWORKS.md	PennyLane vs Qiskit / Cirq / TFQ / analytic solvers
docs/AUDIT_TRAIL.md	Failures, fixes, rationale
CHANGELOG.md	Version history
CITATION.cff	Machine-readable cite metadata

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Citation

Peilivanidis, V. (2026). Learning Quantum Signal Processing Phase Angles via Gradient Descent: A Reproducible Benchmark and JAX Implementation Note (Version 1.1). https://doi.org/10.5281/zenodo.20645402

Apache 2.0 — see LICENSE and NOTICE. Attribution required for code, data, figures, or manuscript reuse.

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Related work

Low & Chuang (2017) · Gilyén et al. (2019) · Martyn et al. (2021) · Chao et al. (2020) · Haah (2019) · Cerezo et al. (2021) · Bergholm et al. (2022) — full list in references.bib.

Evolved from the earlier PennyLane community demo.