A Dynamical Lie-Algebraic Framework for Hamiltonian Engineering and Quantum Control

Abstract

Determining the physically accessible unitary dynamics of a quantum system under finite Hamiltonian resources is a central problem in quantum control and Hamiltonian engineering. Dynamical Lie algebras (DLAs) provide the fundamental link between available control Hamiltonians and the resulting quantum dynamics. While the structural classification of DLAs is well-established, how to systematically engineer and reshape these algebraic structures under realistic physical constraints remains largely unexplored. In this work, building upon recent results on direct sums of identical DLAs, we develop a unified framework for engineering Hamiltonian-driven quantum dynamics based on DLAs: (i) constructing qubit-efficient direct-sum Hamiltonian structures via spectral decomposition of Hermitian operators, enabling parallel simulation of multiple quantum subsystems; (ii) identifying Hamiltonian modifications that preserve full controllability, including the $\mathfrak{su}(2^N)$ algebra, even when additional physically motivated control terms are introduced; and (iii) engineering restricted Hamiltonian sets that confine quantum dynamics to target subalgebras through irreducible Lie-algebra decompositions, providing a principled approach to symmetry-based dynamical reduction. By bridging these Lie-algebraic insights with practical control objectives, our framework provides a systematic pathway for engineering expressive and resource-efficient unitary evolutions, thus unlocking greater structural flexibility of Hamiltonian-driven quantum systems.

Figures (6)

• Figure 1: Illustration of three questions on modifying the generating set $\mathcal{A}$ of a DLA $\mathfrak{g}_\mathcal{A}$: DLA composition, DLA invariance, and DLA reduction.
• Figure 2: Lie-theoretic background. The evolution of a quantum state $\rho$ is governed by unitary transformations $U(t)$ belonging to the DLG $G$ generated by the Hamiltonians $H_l$. These evolutions are generated by the DLA $\mathfrak{g}$, which is mathematically defined as the tangent space of the smooth manifold $G$ at its identity $e$. The exponential map provides the fundamental link between $\mathfrak{g}$ and $G$.
• Figure 3: Illustration of Theorem 1. (a) shows the composition of generator sets for the dipole–field coupling and Heisenberg-type interaction DLAs, while (b) illustrates the process of evolution using Hamiltonian simulation. For example, the evolution for $H = A_1 \otimes \Pi_1$ is $U_{A_1}(t) = e^{-iHt} = e^{-i(A_1 \otimes \Pi_1)t}.$ Since $\Pi_1$ is a projector, $U_{A_1}(t) = e^{-iA_1 t} \otimes \Pi_1 + \mathbb{I} \otimes \Pi_2$. Its qubit-efficiency realization with $2+\lceil \log 2 \rceil = 3$ qubits, compared with $4$ qubits in naive way. Here $|\mathcal{A}'| =|\mathcal{A}|+|\mathcal{B}|= 3$ cannot be further reduced, otherwise the structure of $\mathfrak{g}_{\mathcal{A}_m}$ will be destroyed.
• Figure 4: Comparison of dynamical evolution versus algebraic structural stability under different perturbations.
• Figure 5: Schematic illustration of the generator sets for $\mathfrak{g}_{LTFIM}$ and $\mathfrak{g}_{TFIM}$, highlighting the reduction process. The chaotic color scheme depicts the complexity and irregularity inherent to the full LTFIM, in contrast to TFIM.

Theorems & Definitions (19)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Proposition 1
• Theorem 2
• Proposition 2
• Theorem 3
• Theorem A1
• proof