Platonic dynamical decoupling sequences for qudits

In the NISQ era, where quantum information processing is hindered by the
decoherence and dissipation of elementary quantum systems, developing new
protocols to extend the lifetime of quantum states is of considerable practical
and theoretical importance. A prominent method, called dynamical decoupling,
uses a carefully designed sequence of pulses applied to a quantum system, such
as a qudit (a d-level quantum system), to suppress the coupling Hamiltonian
between the system and its environment, thereby mitigating dissipation. While
dynamical decoupling of qubit systems has been widely studied, the decoupling
of qudit systems has been far less explored and often involves complex
sequences and operations. In this work, we design efficient decoupling
sequences composed solely of SU(2) rotations and based on tetrahedral,
octahedral, and icosahedral point groups, which we call Platonic sequences. We
use a generalization of the Majorana representation for Hamiltonians to develop
a simple framework that establishes the decoupling properties of each Platonic
sequence and show its efficiency on many examples. These sequences are
universal in their ability to cancel any type of interaction with the
environment for single qudits with up to 6 levels, and they are capable of
decoupling up to 5-body interactions in an ensemble of interacting qubits with
only global pulses, provided that the interaction Hamiltonian has no isotropic
component, with the exception of the global identity. We also discuss their
inherent robustness to finite pulse duration and a wide range of pulse errors,
as well as their potential application as building blocks for dynamically
corrected gates.