Lie algebra-assisted quantum simulation and quantum optimal control via high-order Magnus expansions

TL;DR

This work presents a scalable polynomial Magnus expansion for time-dependent quantum dynamics with Hamiltonians $H(t)=A+d(t)B$, where $d(t)$ is represented polynomially. By framing the dynamics in a dynamical Lie algebra and precomputing structure-constant-derived coefficients, the authors obtain a fast, analytic ME up to high orders (up to 12), enabling rapid quantum simulation and gradient-based optimal control. They demonstrate the approach on neutral-atom/Rydberg platforms, designing multi-qubit gates with Hermite-spline control pulses and joint optimization of pulse shape and duration, achieving significant speedups over prior methods. The methodology reduces computational complexity to the control degrees of freedom, offering a practical tool for scalable quantum engineering on classical hardware and informing experimental pulse design.

Abstract

The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and non-commutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short time-scales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach that reduces the computational effort to depend only on the degrees of freedom defining the time-dependent control function. We focus specifically on Hamiltonians consisting of a constant drift term and a controllable term. Our method provides a polynomial expression for the Magnus expansion which can be evaluated several orders of magnitude faster than previous techniques, enabling broad applications in the realms of quantum simulation and quantum optimal control. We showcase an application of the method by designing control pulses for the 5-qubit phase gate on a neutral-atom platform utilizing Rydberg atoms.

Figures (10)

• Figure 1: In this work we propose a method to efficiently evaluate the Magnus expansion. (a) Diagram of the generation of the the dynamical Lie algebra that stems from the single-control Hamiltonian terms $A,B$. (b) The control function is represented as a polynomial with maximum degree $m$, which allow us to solve high-order integrals analytically. (c) The high-order integrals and commutators from the Magnus expansion are simplified into a polynomial expression. (d) This enables high-order simulation of continuous-controlled quantum systems. (e) The analytically-differentiable expression enables quantum optimal control applications.
• Figure 2: Left axis, blue color: Scaling of the size of the Lie algebra with system size, for the two models and $\mathfrak{su}(n)$, using ${k_{\text{M}}=10}$ and ${\Gamma=12}$. Right axis, yellow color: Scaling of the calculation time, in hours, of the dynamical coefficients as system size increases.
• Figure 3: Median evaluation time of the ME polynomial with truncation order $k_M$ and $\Gamma=12$, for both models and various system sizes $n$.
• Figure 4: Comparison of state errors at different orders. The control coefficients $d_\gamma$ are sampled from a uniform distribution $d_\gamma\in[-1,1]$. Used sparse model with $n=3$ and time truncation $\Gamma=14$. Average of 20 samples (random initial state and random controls) is plotted in a line, the minimum and maximum samples are plotted as a band of the respective color. A plateau at around $\sim 2\cdot 10^{-16}$ shows up due to floating-point error.
• Figure 5: Scaling laws for models sparse and dense with $n=10$, where for the sparse model we also compare the order scaling with $\Gamma$. Dashed lines indicate the time-symmetry threshold at ${k_{app}=k_{\text{M}}+3}$ (black) and the threshold ${k_{app}=k_{\text{M}}+1}$ (grey). Solid lines connect between markers of the color-corresponding model for even-valued $k_{\text{M}}$. Most models considered follow the time-symmetry scaling $k_{app} \simeq k_{\text{M}}+3$.