On Nonlinearities in QML Paper Implementation

This repository provides implementations of basic nonlinear quantum subroutines proposed in On nonlinear transformations in quantum computation, including the Quantum Hadamard Product for element-wise multiplication of quantum states and the Generalized Quantum Transpose for weighted transposition.

In contrast to the paper which uses Qiskit, these implementations are done using the PennyLane quantum machine learning library, providing an alternative implementation. Additionally, could be building block for pennylane custom operator in the future. More broadly, advancing techniques to perform nonlinear transformations on quantum states, this research aims to enable more powerful quantum algorithms to impact quantum machine learning.

Note: This repository is an attempt to build in public, as such it is not perfect but the goal is to improve and if anyone sees this and has constructive feedback, I am welcome to it!

Paper Reference

Holmes, Z., Coble, N. J., Sornborger, A. T., & Subaşı, Y. (2023). Nonlinear transformations in quantum computation. Phys. Rev. Res., 5(1), 013105. DOI: 10.1103/PhysRevResearch.5.013105

Quantum Hadamard Product (QHP.ipynb)

This Jupyter notebook implements the Quantum Hadamard Product algorithm from the paper "On nonlinear transformations in quantum computation".

The Quantum Hadamard Product takes two input quantum states $\rho(0)$ and $\rho(1)$ and outputs a weighted state $\rho(0) \odot \rho(1)$ where the matrix elements are multiplied element-wise.

Key points:

• Implements the circuit shown in Fig. 2 of the paper using PennyLane.
• Can be iterated to take powers of a state, as shown in Fig. 3 of the paper.
• Convergence analysis shows absolute error remains constant but relative error increases with power.

Generalized Quantum Transpose (GQT.ipynb)

This Jupyter notebook implements the Generalized Quantum Transpose algorithm from the same paper.

The Generalized Quantum Transpose takes an input state $\rho$ and ancilla state $\sigma$ and outputs the weighted state $\sigma \dot \rho^{T}$ where $\rho^{T}$ is the transpose of $\rho$.

Key points:

• Implements the circuit shown in Fig. 4 of the paper using PennyLane.
• SWAP operator measurement implemented as shown in Fig. 5 of the paper.
• Can be used to implement powers of a state, but convergence is slower than direct QHP algorithm.
• Related to quantum teleportation algorithm as discussed in Appendix C of the paper.

The two notebooks demonstrate implementations of key quantum algorithms from the paper for performing nonlinear transformations on quantum states. The Paper provides additional details on the algorithms, theory, and analysis.

Results and Comparisons (WIP)

This is still a WIP as I learn how to properly leverage PennyLane's library to output measurements of the weighted states (i.e. using qml.density_matrix which doesn't support mixed state usage with finite shots vs qml.counts which I am unsure of how to calculate variance from)