Topological Characterization with a Twist, Condensation, and Reflection

TL;DR

This paper develops experimentally accessible protocols to characterize and assess the fidelity of topological quantum systems, focusing on twisted gauge theories, topological quasiparticles, and error correction, to advance robust quantum computation.

Contribution

It introduces new logical and scattering protocols for topological codes, extending twist concepts, and provides a unified performance metric for topological lifetime across various systems.

Findings

Twisted qubit codes exhibit doubled and tripled code distances.

Error correction algorithms for $ ext{Z}_4$ double semion condensation are developed.

Topological quasiparticle reflectometry infers system properties and performance.

Abstract

Despite its putative robustness, the realization of and control over topological quantum matter is an ongoing grand challenge. Looking forward, robust characterization protocols are needed to first certify topological substrates before they are utilized in quantum algorithms. We contribute to this grand challenge by providing a series of experimentally accessible near- and medium-term protocols assessing the fidelity of logical processes. To do so we examine logical operators and anyonic quasiparticle excitations in twisted $\mathbb{Z}_{N=2,4}$ gauge theories. Extending the finite twist, a promising route to Ising computing in its own right, to a non-contractible twist fuses prior logical operators together and results in a twisted qubit code. The code is notable for a doubled and tripled code distance for logical $Y$ and $X$ errors respectively. Next, we review the deconfinement properties of a $\mathbb{Z}_4$ double semion condensation and provide an error correction algorithm. Based on this understanding we then present a $\mathbb{Z}_4$ topological quasiparticle reflectometry and scattering protocol. The protocol infers the topological properties of the system and serves as a high-level metric for the performance and lifetime of the interfaced topological codes. Our logical and scattering protocols are suitable for near-term devices where many physical qubits encode few logical qubits. The topological lifetime of a particle within a condensate conjugacy class, previously considered in fabricated and hetero-structured condensed-matter experiments, serves as a unifying performance metric across synthetic, qubit-based, and naturally occurring topological order.