qsvt-pennylane 0.1.0

Educational utilities and examples for Quantum Singular Value Transformation in PennyLane.

Quantum Singular Value Transformation (QSVT)

This repository provides both:

• a notebook-first introduction to Quantum Singular Value Transformation (QSVT)
• a lightweight PyPI package implementing reusable utilities for QSVT experiments in PennyLane

The focus is on small, explicit examples showing how bounded polynomials can be applied to the singular values / eigenvalues of an operator via (simulator-friendly) block-encodings.

The included Python package:

pip install qsvt-pennylane

provides helpers for:

• Chebyshev and polynomial utilities
• bounded polynomial approximation
• small Hermitian matrix construction
• spectral matrix-function reference calculations
• explicit QSVT matrix extraction
• comparison between classical and QSVT polynomial transforms

The notebooks use PennyLane’s high-level qml.qsvt interface (with block_encoding="embedding") to emphasise mathematical structure, spectral intuition, and polynomial design rather than circuit engineering.

For theoretical background see THEORY.md.

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Repository structure

Quantum_Singular_Value_Transformation
├── LICENSE
├── README.md
├── THEORY.md
├── pyproject.toml
├── src/qsvt
│   ├── polynomials.py
│   ├── approximation.py
│   ├── matrices.py
│   ├── spectral.py
│   ├── qsvt.py
│   └── __main__.py
├── tests
│   └── test_qsvt_smoke.py
└── notebooks
    ├── 01_QSVT_Scalar_and_Diagonal_Matrix.ipynb
    ├── 02_QSVT_Singular_Value_Filter.ipynb
    ├── 03_QSP_Polynomial_Demo.ipynb
    ├── 04_QSVT_Linear_Solver_2x2.ipynb
    ├── 04_QSVT_Linear_Solver_4x4.ipynb
    ├── 05_QSVT_Linear_Solver_Approximate.ipynb
    ├── 06_QSVT_Polynomial_Design_and_Approximation.ipynb
    ├── 07_QSVT_Matrix_Functions_Powers_and_Roots.ipynb
    └── 08_QSVT_Sign_Function_and_Projectors.ipynb

---

01 — QSVT on a scalar and a diagonal matrix

01_QSVT_Scalar_and_Diagonal_Matrix.ipynb

• Minimal “single singular value” intuition: treat a scalar $a \in [-1,1]$ as a block-encoded operator.
• Applies the polynomial $f(x)=x^2$ and verifies the QSVT output matches the classical curve.
• Extends the same idea to diagonal matrices, showing that QSVT transforms eigenvalues independently.

---

02 — QSVT as a (soft) singular value filter

02_QSVT_Singular_Value_Filter.ipynb

• Introduces filtering and suppression using bounded even polynomials.
• Shows how small singular values are attenuated more strongly than large ones.
• Highlights the key admissibility constraint $|f(x)| \le 1$ on $[-1,1]$.

---

03 — QSP demo (two perspectives)

03_QSP_Polynomial_Demo.ipynb

• Part A: QSP via QSVT-on-a-scalar (QSVT reduces to QSP in the scalar case).
• Part B: manual single-qubit construction of a Chebyshev polynomial: $$ T_3(x) = 4x^3 - 3x. $$
• Builds intuition for why Chebyshev polynomials are natural in QSP/QSVT.

---

04 — QSVT linear solver (exact inverse on a special spectrum)

04_QSVT_Linear_Solver_2x2.ipynb and 04_QSVT_Linear_Solver_4x4.ipynb

• Demonstrates QSVT-based inversion for involutory matrices with eigenvalues $\pm1$.
• Uses the polynomial $P(x)=x$ to exactly reproduce the inverse.
• These examples are intentionally spectrally trivial but mathematically exact.

---

05 — QSVT linear solver (approximate / inverse-like polynomial)

05_QSVT_Linear_Solver_Approximate.ipynb

• A non-trivial example where $A \ne A^{-1}$.
• Uses the bounded odd Chebyshev polynomial $T_3(x)$ to reproduce inverse-like behaviour.
• Shows that matching relative eigencomponent scaling is sufficient after normalization.

---

06 — Polynomial design and approximation

06_QSVT_Polynomial_Design_and_Approximation.ipynb

• Explains *how- QSVT polynomials are designed.
• Introduces Chebyshev approximation as the natural basis for bounded polynomials.
• Shows how polynomial degree controls accuracy and circuit depth.
• Connects approximation quality to linear-system behaviour.

---

07 — QSVT as matrix functions (powers and roots)

07_QSVT_Matrix_Functions_Powers_and_Roots.ipynb

• Presents QSVT as functional calculus for matrices.
• Demonstrates matrix powers, square roots, and fractional powers.
• Shows how eigenvalues are transformed while eigenvectors are preserved.
• Unifies inversion, filtering, and simulation under a single spectral viewpoint.

---

08 — Sign function and spectral projectors

08_QSVT_Sign_Function_and_Projectors.ipynb

• Approximates the sign function using bounded odd polynomials.
• Builds approximate spectral projectors: $$ \Pi_\pm = \frac{I \pm \mathrm{sgn}(A)}{2}. $$
• Demonstrates subspace selection and basis-independence.
• Bridges QSVT toward ground-state filtering and Hamiltonian algorithms.

---

Scope and limitations (important)

This repository is educational by design.

• All examples are small, explicit, and simulator-friendly.
• Block encodings are constructed using PennyLane’s "embedding" mode.
• Linear solvers are shown as polynomial mechanisms, not full algorithms.
• Topics such as state preparation, success probability, amplitude amplification, condition number scaling, and fault-tolerant resource estimates are intentionally excluded.

The goal is conceptual understanding, not algorithmic optimization.

---

Python package

The repository now includes a reusable Python package:

pip install qsvt-pennylane

Example usage

from qsvt.qsvt import qsvt_scalar_output
from qsvt.polynomials import chebyshev_t3

# apply polynomial x^2 via QSVT
qsvt_scalar_output(0.5, [0,0,1], encoding_wires=[0])

from qsvt.qsvt import qsvt_diagonal_transform

vals = qsvt_diagonal_transform(
    [1.0, 0.7, 0.3, 0.1],
    [0,0,1],
    encoding_wires=[0,1,2],
)

CLI

After installation:

qsvt scalar --x 0.5 --poly "0,0,1"

qsvt diag \
  --values "1.0,0.7,0.3,0.1" \
  --poly "0,0,1" \
  --wires 3

qsvt cheb --degree 3 --x 0.5

Modules

module	purpose
qsvt.polynomials	Chebyshev utilities and polynomial helpers
qsvt.approximation	bounded polynomial approximation
qsvt.matrices	small Hermitian test matrices
qsvt.spectral	classical spectral matrix functions
qsvt.qsvt	PennyLane QSVT wrappers

---

Installation

Install from PyPI

pip install qsvt-pennylane

Install from source

git clone https://github.com/SidRichardsQuantum/Quantum_Singular_Value_Transformation.git
cd Quantum_Singular_Value_Transformation

pip install -e .

Dependencies

• Python $\geq$ 3.10
• PennyLane $\geq$ 0.36
• NumPy $\geq$ 1.23
• Matplotlib $\geq$ 3.7

---

Quickstart (notebooks)

git clone https://github.com/SidRichardsQuantum/Quantum_Singular_Value_Transformation.git
cd Quantum_Singular_Value_Transformation

python -m venv .venv
source .venv/bin/activate

pip install -e .

jupyter lab

Open the notebooks in order:

1. scalar QSVT intuition
2. singular value filtering
3. QSP polynomials
4. linear solvers
5. polynomial approximation
6. matrix functions
7. spectral projectors

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Author

Sid Richards

LinkedIn: https://www.linkedin.com/in/sid-richards-21374b30b/

GitHub: https://github.com/SidRichardsQuantum

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License

MIT License — see LICENSE