Measurement-based quantum computation in finite one-dimensional systems: string order implies computational power

TL;DR

This paper introduces a new framework for evaluating measurement-based quantum computation in one-dimensional systems, linking the presence of string order to the ability to perform high-fidelity quantum gates.

Contribution

It develops a less assumption-dependent formalism for MBQC in finite, non-translation-invariant 1D systems, connecting string order to computational power.

Findings

Non-zero string order implies near-perfect realization of certain unitaries.

Framework applies to finite systems, not just thermodynamic limit.

Strengthens the link between topological order and quantum computational capability.

Abstract

We present a new framework for assessing the power of measurement-based quantum computation (MBQC) on short-range entangled symmetric resource states, in spatial dimension one. It requires fewer assumptions than previously known. The formalism can handle finitely extended systems (as opposed to the thermodynamic limit), and does not require translation-invariance. Further, we strengthen the connection between MBQC computational power and string order. Namely, we establish that whenever a suitable set of string order parameters is non-zero, a corresponding set of unitary gates can be realized with fidelity arbitrarily close to unity.