Dynamical decoupling and quantum error correction with SU(d) symmetries

TL;DR

This paper introduces a Lie group theory-based framework for dynamical decoupling and quantum error correction in qudit systems, enabling systematic design of protocols using SU(d) symmetries.

Contribution

It extends the group theory approach to dynamical decoupling for qudits, providing a systematic method to identify decoupling groups within SU(d) and linking symmetry sectors to error-correcting codes.

Findings

Constructed new pulse sequences for qutrit systems based on SU(3) subgroups.

Demonstrated shorter, more practical protocols for spin-1 systems with large zero-field splitting.

Unified dynamical decoupling and quantum error correction via symmetry sectors.

Abstract

Dynamical decoupling is a long-established and effective way to suppress unwanted interactions in qubit systems, enabling advances in fields ranging from quantum metrology to quantum computing. For general qudit systems, however, comparable protocols remain rare, mainly because Hamiltonian engineering in higher dimensions lacks the geometric intuition available for qubits. Here we present a general framework for dynamical decoupling in qudit systems, based on Lie group representation theory. By extending the group theory approach to dynamical decoupling, we show how decoupling groups can be systematically identified among the finite subgroups of SU(d) by analyzing their access to the irreducible components of the operator space. As an application, we construct new pulse sequences for interacting qutrit systems based on finite subgroups of SU(3), and show how subgroup factorizations and group orientations can be exploited to obtain shorter and more experimentally practical protocols for spin-1 systems with large zero-field splitting. We further show that the same symmetry-based framework yields quantum error-correcting codes: whenever a finite subgroup of SU(d) acts as a decoupling group for the relevant set of operators, the associated one-dimensional symmetry sectors define codespaces satisfying the Knill-Laflamme conditions, thereby unifying dynamical decoupling and quantum error correction in multi-level quantum systems.