Geometric and Resource-Theoretic Characterisation of Non-Stabiliserness in Quantum Algorithms

TL;DR

This paper introduces a geometric and resource-theoretic framework to quantify and analyze non-stabiliser resources in quantum algorithms, aiding the understanding of their role in quantum advantage.

Contribution

It develops permutation agnostic distance measures to reveal hidden non-stabiliser effects and compares resource usage in different quantum algorithm approaches.

Findings

Structured approaches use non-stabiliser resources more efficiently.

Classical optimization can lead to unnecessary non-stabiliser consumption.

The framework helps analyze quantum resource utilization for algorithmic advantage.

Abstract

While there is strong evidence for advantages of quantum over classical computation, the repertoire of computational primitives with proven or conjectured quantum advantage remains limited. A big challenge of quantum algorithmic design is a still incomplete understanding of the sources of quantum computational power. Advancing towards systematic quantum advantage calls for a better understanding of the efficient use of non-classical resources like non-stabiliser states. We present an approach to track non-classical contributions in the form of non-stabiliserness across various algorithms by pairing resource theory of non-stabiliser entropies with the geometry of quantum state evolution, and introduce permutation agnostic distance measures that reveal and quantify non-stabiliser effects previously hidden by a subset of Clifford operations. We find different efficiency in the use of non-stabiliserness for structured and unstructured variational approaches, and show that greater freedom for classical optimisation in quantum-classical methods increases unnecessary non-stabiliser consumption. Our results open new means of analysing the efficient utilisation of quantum resources, and contribute towards the targeted construction of algorithmic quantum advantage.