Universal compiling and (No-)Free-Lunch theorems for continuous variable quantum learning

TL;DR

This paper introduces new algorithms for continuous variable quantum compiling, analyzes their trainability, demonstrates their effectiveness on Gaussian and Kerr operations, and establishes theoretical bounds on their resource efficiency.

Contribution

It presents novel continuous variable quantum compiling algorithms, analyzes their trainability, and derives No-Free-Lunch theorems for resource bounds in quantum learning.

Findings

Algorithms successfully learn Gaussian operations and Kerr nonlinearities.

Cost functions analyzed for trainability in continuous variable quantum compiling.

Derived resource reduction bounds for learning unitaries with entangled states.

Abstract

Quantum compiling, where a parameterized quantum circuit is trained to learn a target unitary, is an important primitive for quantum computing that can be used as a subroutine to obtain optimal circuits or as a tomographic tool to study the dynamics of an experimental system. While much attention has been paid to quantum compiling on discrete variable hardware, less has been paid to compiling in the continuous variable paradigm. Here we motivate several, closely related, short depth continuous variable algorithms for quantum compilation. We analyse the trainability of our proposed cost functions and numerically demonstrate our algorithms by learning arbitrary Gaussian operations and Kerr non-linearities. We further make connections between this framework and quantum learning theory in the continuous variable setting by deriving No-Free-Lunch theorems. These generalization bounds demonstrate a linear resource reduction for learning Gaussian unitaries using entangled coherent-Fock states and an exponential resource reduction for learning arbitrary unitaries using Two-Mode-Squeezed states.