Symmetric derivatives of parametrized quantum circuits

TL;DR

This paper introduces symmetry-aware training methods for parametrized quantum circuits using projected derivatives, linking them to quantum Fisher information and extending natural gradient techniques to continuous symmetry groups.

Contribution

It presents the concept of equivariant and covariant derivatives for quantum circuits, enabling symmetry-aware optimization without modifying circuit design.

Findings

Covariant derivative relates to quantum Fisher information.

Extended quantum natural gradient to all continuous symmetry groups.

Connected covariant derivative to physical gauge theory.

Abstract

Symmetries are crucial for tailoring parametrized quantum circuits to applications, due to their capability to capture the essence of physical systems. In this work, we shift the focus away from incorporating symmetries in the circuit design and towards symmetry-aware training of variational quantum algorithms. For this, we introduce the concept of projected derivatives of parametrized quantum circuits, in particular the equivariant and covariant derivatives. We show that the covariant derivative gives rise to the quantum Fisher information and quantum natural gradient. This provides an operational meaning for the covariant derivative, and allows us to extend the quantum natural gradient to all continuous symmetry groups. Connecting to traditional particle physics, we confirm that our covariant derivative is the same as the one introduced in physical gauge theory. This work provides tools for tailoring variational quantum algorithms to symmetries by incorporating them locally in derivatives, rather than into the design of the circuit.