Sub-Riemannian geometry of measurement based quantum computation

TL;DR

This paper uncovers a geometric framework based on sub-Riemannian geometry to optimize measurement-based quantum computation, linking resource minimization to geodesic problems in quantum phases of matter.

Contribution

It introduces a novel geometric approach to optimize measurement-based quantum computation by formulating resource minimization as a sub-Riemannian geodesic problem.

Findings

Resource minimization corresponds to solving a sub-Riemannian geodesic problem.

Reveals geometric structure underlying measurement-based quantum computation.

Provides a principled method to optimize quantum processing in computational phases.

Abstract

The computational power of quantum phases of matter with symmetry can be accessed through local measurements, but what is the most efficient way of doing so? In this work, we show that minimizing operational resources in measurement-based quantum computation on subsystem symmetric resource states amounts to solving a sub-Riemannian geodesic problem between the identity and the target logical unitary. This reveals a geometric structure underlying MBQC and offers a principled route to optimize quantum processing in computational phases.