通过脉冲的哈密顿量工程：超越群平均

中文摘要 AI 辅助

我们为有限维量子系统中使用理想控制脉冲的哈密顿量工程建立了一个几何框架。从具有无界控制幅度的双线性薛定谔方程出发，我们构造了闭脉冲群，并利用控制系统的扩展和Filippov松弛定理，获得了一族有效哈密顿量，该族由漂移的伴随轨道的凸包加上脉冲群的李代数给出。该轨道体的几何结构描述了超越群平均的可能性。利用伴随表示的同型分解，我们刻画了其仿射包，并证明了群平均位于其内部。这产生了围绕漂移不变部分的有效哈密顿量的局部可达族。我们将该框架应用于恢复从有限维浴的任意相互作用进行动力学解耦的充分必要条件。对于连通的阿贝尔脉冲群，我们通过限制根描述了相关的表示分解。最后，对于具有相同成对耦合的量子比特网络，我们给出了使用我们的框架可生成的有效哈密顿量的定性表征和定量估计。

英文摘要

We develop a geometric framework for Hamiltonian engineering in finite-dimensional quantum systems using ideal control pulses. Starting from a bilinear Schrödinger equation with unbounded control amplitudes, we construct the closed pulse group and use extensions of control systems and Filippov's relaxation theorem to obtain a family of effective Hamiltonians given by the convex hull of the drift's adjoint orbit plus the Lie algebra of the pulse group. The geometry of this orbitope describes possibilities beyond group averaging. Using the isotypic decomposition of the adjoint representation, we characterize its affine hull and show that the group average lies in its interior. This yields locally accessible families of effective Hamiltonians around the invariant part of the drift. We apply the framework to recover the necessary and sufficient condition for dynamical decoupling from arbitrary interactions with a finite-dimensional bath. For connected abelian pulse groups, we describe the relevant representation decomposition through restricted roots. Finally, for qubit networks with identical pairwise couplings, we give a qualitative characterization and quantitative estimation of effective Hamiltonians that can be generated using our framework.

发表机构

• INRIA, MCTAO（法国国家信息与自动化研究所，MCTAO）

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