Time-dependent Hamiltonian Simulation via Magnus Expansion: Algorithm and Superconvergence

TL;DR

This work develops a quantum algorithm for general time-dependent Hamiltonian simulation using a second-order Magnus expansion, achieving commutator scaling with a logarithmic dependence on time-derivative magnitudes. By discretizing time into segments and using high-precision quadrature implemented via LCU and QSVT with OAA, the method provides accurate short-time propagators and provable long-time error bounds. A central result is a fourth-order superconvergence for unbounded Hamiltonians in the interaction picture, with a preconstant independent of spatial discretization, established through semiclassical microlocal analysis. The authors also construct quantum circuits and input models for both general and interaction-picture Hamiltonians, and derive long-time complexity bounds that demonstrate practical efficiency, especially for simulated Schrödinger equations with real-space discretization. Overall, the paper delivers a rigorous Magnus-based framework with superior scaling and a striking superconvergence phenomenon for a class of physically important unbounded operators.

Abstract

Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion that exhibits both features. Importantly, when applied to unbounded Hamiltonian simulation in the interaction picture, we prove that the commutator in the second-order algorithm leads to a surprising fourth-order superconvergence, with an error preconstant independent of the number of spatial grids. This extends the qHOP algorithm [An, Fang, Lin, Quantum 2022] based on first-order Magnus expansion, and the proof of superconvergence is based on semiclassical analysis that is of independent interest.

Figures (5)

• Figure 1: The unbounded Hamiltonian simulation problem considers the Hamiltonian $-\Delta + \cos(x)$, and $A$ and $B$ are the central finite discretization of the operators $-\Delta$ and $\cos(x)$ respectively with periodic boundary conditions using $N$ spatial grids. Left: Plot of the operator norm of the term $[\alpha, [\beta, \gamma(t)]]$ versus the short time $t\in [0, 1]$ for various grid numbers $N$. This serves as evidence for terms becoming uncontrolled when expanding in the Taylor form. Right: Plot of the operator norm of the key commutator as in \ref{['lem:bound_commutator_realspace_key']}. This demonstrates that the key commutator remains controlled, insensitive to the grid number $N$, in contrast to the term after expanding by Taylor theorems. The numerical evidence agrees with the theoretical result proved in \ref{['lem:bound_commutator_realspace_key']}.
• Figure 2: Quantum circuit of implementing the block-encoding of the Hamiltonian of the second-order Magnus expansion for general Hamiltonian \ref{['equ:second_order_circ']}. The short-time evolution operator can then be implemented according to \ref{['lem:ham_sim_qsvt']} using the circuit here as input block encoding of Hamiltonian. Here HAD is the single qubit Hadamard gate and COMP is a compare oracle defined in \ref{['equ:comp']}. The $\text{HAM-T}_{j}$ is the input model for the time-dependent Hamiltonian at the time step. For the readability of superscripts and subscripts, we use $a$ and $m$ to represent $n_a$ and $n_m$ respectively in this figure.
• Figure 3: Quantum circuit of the HAM-T oracle in the interaction picture Hamiltonian for $H(t)=A+B(t)$.
• Figure 4: Log-log plot of the errors in the operator norm for various time step sizes $h$. The spatial discretization is finite difference. Both our algorithm (q-Mag) and the fourth-order classical Magnus integrator (c-Mag) exhibit fourth-order convergence. However, the error of our algorithm is smaller than that of the fourth-order classical Magnus integrator, and does not grow as the number of the grid points in spatial discretization $N$ increases. The reference line demonstrates the asymptotic scaling.
• Figure 5: Log-log plot of the errors in the operator norm for various time step sizes $h$ of both this algorithm (q-Mag) and qHOP. The convergence of q-Mag is much faster than qHOP, which implies that to achieve the same precision the number of time steps needed in q-Mag is much fewer than that of qHOP.

Theorems & Definitions (28)

• Theorem 1: Local truncation error for general $H(t)$
• proof
• Theorem 2: Quadrature error for general $H(t)$
• proof
• Corollary 3: Quadrature error for interaction picture
• proof
• Theorem 4: long time error for general $H(t)$
• proof
• Lemma 5: The key commutator estimate
• proof