ParametricDFT.jl

Learning parametric quantum Fourier transforms via Riemannian optimization on manifolds

Overview

ParametricDFT.jl is a Julia package for learning parametric quantum Fourier transforms using manifold optimization. It implements a variational approach to approximate the Discrete Fourier Transform (DFT) using parameterized quantum circuits, with applications to image compression and signal processing.

The core idea: optimize quantum circuit parameters on Riemannian manifolds (products of U(2) and U(1) groups) to learn frequency-domain representations that are more compressible than the standard DFT.

Circuit Architectures

The package supports three parametric circuit designs:

Comparison of Standard 2D QFT and Entangled QFT architectures for a 3×3 system. The entangled variant adds learnable cross-dimensional gates (red) that couple row and column qubits.
  • Standard QFT Basis — Full quantum Fourier transform circuit with Hadamard and controlled-phase gates
  • Entangled QFT Basis — QFT with additional learnable 2-qubit entanglement gates between row and column qubits
  • TEBD Basis — Time-Evolving Block Decimation with ring topology for 2D separable transforms

Training & Optimization

The package provides Riemannian optimization on the unitary group, with support for multiple solvers:

  • Riemannian Gradient Descent
  • Riemannian Conjugate Gradient
  • Riemannian L-BFGS
  • Riemannian Adam

All optimizers support both CPU and GPU execution via a custom CUDA-compatible backend.

GPU Acceleration

A key contribution is GPU-accelerated Riemannian optimization that bypasses limitations of existing manifold optimization libraries. Batched operations on (2,2,n) tensor arrays minimize kernel launches for significant speedups at scale.

Key Dependencies

Package Role
Yao.jl Quantum circuit construction
OMEinsum.jl Tensor network contractions
Manifolds.jl Riemannian geometry abstractions
Manopt.jl Riemannian optimization solvers
Zygote.jl Automatic differentiation
CUDA.jl GPU acceleration

References