Gauge invariants of eigenspace and eigenvalue anholonomies: Examples in hierarchical quantum circuits

TL;DR

This paper introduces gauge invariants for quantum anholonomies, explores their behavior in large hierarchical quantum circuits, and reveals that analyzing these circuits often involves NP-complete problems.

Contribution

It defines gauge invariants for quantum anholonomies and applies them to hierarchical quantum circuits, highlighting computational complexity issues.

Findings

Gauge invariants unify Berry phase and eigenspace anholonomies.

Hierarchical quantum circuits can be arbitrarily large.

Analyzing these circuits involves NP-complete problems.

Abstract

A set of gauge invariants are identified for the gauge theory of quantum anholonomies, which comprise both the Berry phase and an exotic anholonomy in eigenspaces. We examine these invariants for hierarchical families of quantum circuits whose qubit size can be arbitrarily large. It is also found that a hierarchical family of quantum circuits generally involves an NP-complete problem.