Dynamical decoupling of interacting spins through group factorization

TL;DR

The paper develops a symmetry-based framework for dynamical decoupling in interacting spin systems, exploiting inaccessible symmetries of the undesired Hamiltonian via factorization of decoupling groups into subgroups. It combines Majorana constellations with group-theoretic constructions to design nested, Eulerian DD sequences that selectively suppress different decoherence channels, recovering Lee–Goldburg as a special case and introducing novel protocols such as TEDD and TEDDY. The approach enables hierarchical decoupling across fast and slow error dynamics and extends to dephasing qudits, offering compact, robust sequences with improved performance over state-of-the-art methods in relevant regimes. Overall, the framework provides a flexible, geometry-driven toolkit for Hamiltonian engineering and coherence protection across quantum platforms.

Abstract

Dynamical decoupling (DD) is a well-known open-loop protocol for suppressing unwanted interactions in a quantum system, thereby drastically extending the coherence time of useful quantum states. In the original framework of evolution symmetrization, a DD sequence was shown to enforce a symmetry on the unwanted Hamiltonian, thereby suppressing it if the symmetry was inaccessible. In this work, we show how symmetries already present in the undesired Hamiltonian can be harnessed to reduce the complexity of decoupling sequences and to construct nested protocols that correct dominant errors at shorter timescales, using the factorization of DD symmetry groups into a product of its subgroups. We provide many relevant examples in various spin systems, using the Majorana constellation and point-group factorization to identify and exploit symmetries in the interaction Hamiltonian. Our framework recovers tailored pulse sequences developed in the context of NMR, including the classical Lee-Goldburg protocol, and further produces novel short and robust sequences capable of suppressing on-site disorder, dipole-dipole interactions, and more exotic many-body interactions in spin ensembles.

Figures (11)

• Figure 1: Majorana constellation of an operator $H$. The diagram also illustrates the effect of an $\mathrm{SU(2)}$ transformation $g^\dagger$ on the operator and its corresponding $\mathrm{SO(3)}$ rotation $\mathrm{R}(\hat{n},\theta)$ acting on the constellations associated with $H$. In this illustrative example, the constellation is rotated by an angle $\theta=-\pi/4$ about the $z$ axis. Figure reproduced from Ref. Read2025platonicdynamical.
• Figure 2: Majorana constellations of Hamiltonians $H\propto \hat{m}_i\boldsymbol{\cdot} \mathbf{S}^i$ (left) and $H\propto 3(\hat{e}_{ij}\boldsymbol{\cdot} \mathbf{S}^i)(\hat{e}_{ij}\boldsymbol{\cdot} \mathbf{S}^j) - \mathbf{S}^i\boldsymbol{\cdot}\mathbf{S}^j$ (right). The label $2$ next to the black star in the right-hand constellation indicates a degeneracy where two pairs of antipodal stars occupy the same position on the sphere.
• Figure 3: Visual representation of the nesting process for multisymmetrization. Each stick represents a pulse ($P_k$) and the different colours represent the different generators ($\gamma_\lambda$). The waiting time between successive pulses is set to $\tau_0$ and the duration of each pulse to $\tau_p$, so that the total pulse interval is $\tau = \tau_0 + \tau_p$.
• Figure 4: (a) Majorana constellation of the dipole-dipole Hamiltonian $H_{\mathrm{dd}}^{\mathrm{RWA}}$ in Eq. \ref{['eq:Dip.Ham.']}. (b) Tetrahedron whose $\mathrm{D}_2$ symmetry axes coincide with those of the Majorana constellation. (c) Cayley graph of the $\mathrm{C}_3$ symmetry group.
• Figure 5: (a) Majorana constellations of the Hamiltonian \ref{['eq:Dip.Dis.Ham.']}. (b) Tetrahedron whose $\mathrm{C}_3$ symmetry axis ($z$ axis) coincides with that of the Majorana constellation. The axes shown in blue are associated with the $\mathrm{D}_2$ symmetry. (c) Cayley graph of the $\mathrm{D}_2$ group. An undirected edge on a Cayley graph should be understood as two directed edges with opposite directions.