Learning QSP Phase Angles via Gradient Descent

A PennyLane QML Community Demo that demonstrates how Quantum Signal Processing (QSP) phase angles can be trained from random initialization using JAX automatic differentiation and Optax's Adam optimizer — no analytic angle solver required.

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What This Is

Quantum Signal Processing encodes polynomial transformations of a signal into a quantum circuit via a sequence of phase-shifted oracle calls. The standard approach computes the required phase angles analytically given a target polynomial. This demo takes the opposite route: it starts from random phase angles and trains them using gradient-based optimization (JAX + Optax) to recover a target polynomial transformation.

The result is a working recipe for practitioners who need to optimize QSP angles for novel polynomials where no analytic solver exists or where the polynomial is implicitly defined by a loss function.

Setup

git clone https://github.com/rosspeili/qsp-pennylane-demo
cd qsp-pennylane-demo
pip install -r requirements.txt

Running the Demo

jupyter notebook demo.ipynb

Or run the tests:

pytest tests/ -v

Structure

qsp-pennylane-demo/
├── demo.ipynb          # Main community demo notebook
├── qsp_jax/
│   ├── __init__.py
│   └── circuit.py      # Circuit construction and loss function
├── tests/
│   └── test_circuit.py # Unit tests
├── requirements.txt
├── LICENSE             # Apache 2.0
└── README.md

Key Concepts

• Signal oracle: W(x) = H @ RZ(-2*arccos(x)) @ H, encoding signal x ∈ (-1, 1) in the top-left matrix element
• QSP sequence: Flat alternating circuit — one phase rotation RZ(-2*phi_k) per signal query W(x)
• Polynomial encoding: The expectation value <X> encodes a degree-d polynomial in x determined by the phase angles
• Training: Adam optimizer (Optax) minimizes MSE between circuit output and target polynomial via jax.grad
• Note: The circuit is implemented as inline qp.RZ + qp.Hadamard gates, not qp.QSVT, to preserve JAX traceability

Target Polynomial

The default target is a degree-5 Chebyshev approximation of sin(x) on [-1, 1] — an odd polynomial bounded in [-1, 1], consistent with QSP conventions for odd-degree transformations. It has a maximum deviation of ~0.174 from the true sin(x).

Results

After 500 Adam steps (lr=0.05, 64-point signal grid), the trained phase angles reproduce the target polynomial with:

• Final MSE: 4.82×10⁻⁴
• Max pointwise error: 9.83×10⁻²

Loss drops from ~1.44 (random initialization) to ~6.4×10⁻⁴, converging quickly within the first 100 steps.

References

• Martyn et al., A Grand Unification of Quantum Algorithms, PRX Quantum 2021
• Gilyen et al., Quantum singular value transformation, STOC 2019

License

Apache 2.0. See LICENSE.

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Press Release

📄 View the Press Release