Time evolution of controlled many-body quantum systems with matrix product operators

TL;DR

This work introduces an MPO-based framework to solve the time-dependent Schrödinger equation for many-body controlled quantum systems by combining the Magnus expansion with Chebyshev polynomials to approximate the unitary evolution $U(T)=\exp(\Omega^{(\infty)}(T))$. By representing $H_0$ and $H_c$ as matrix product operators, the method achieves scalable, linearly growing storage with system size for fixed truncation orders, enabling efficient time evolution and its use in quantum optimal control. The authors demonstrate on a controlled Ising model that the approach attains competitive accuracy with favorable scaling and apply it to QCPOP, a global polynomial-optimization framework, showing improved performance over GRAPE/CRAB for larger systems. This MPO-based strategy thus offers a practical route to large-scale gate synthesis and control of quantum many-body systems, with accuracy governed by truncation orders and a clear path to symbolic tensor-network control problems.

Abstract

We present a method for describing the time evolution of many-body controlled quantum systems using matrix product operators (MPOs). Existing techniques for solving the time-dependent Schrödinger equation (TDSE) with an MPO Hamiltonian often rely on time discretization. In contrast, our approach uses the Magnus expansion and Chebyshev polynomials to model the time evolution, and the MPO representation to efficiently encode the system's dynamics. This results in a scalable method that can be used efficiently for many-body controlled quantum systems. We apply this technique to quantum optimal control, specifically for a gate synthesis problem, demonstrating that it can be used for large-scale optimization problems that are otherwise impractical to formulate in a dense matrix representation.

Figures (6)

• Figure 1: MPO decomposition of an $N$-qubit Hamiltonian.
• Figure 2: Infidelity between the dense and the MPO method. We fix a Magnus expansion of order 1 and vary the Chebyshev order. We compare different time evolutions.
• Figure 3: Time and memory required to compute the dense and MPO unitary evolutions using a first order Magnus expansion, a Chebyshev approximation of third order, and an evolution time $T=\frac{1}{2N}$. We do 5 repetitions for each $N$ to look at the mean and standard deviation.
• Figure 4: Comparison of accuracy of the Magnus and Chebyshev method with Trotterization of $K$ time steps for two different controls $u(t)$.
• Figure 5: Time and memory required to compute the polynomial to minimize for the dense and MPO representations. We use a second order Magnus expansion, an evolution time $T=\frac{\pi}{4}$, and a control function with 3 unknown parameters. We do 5 repetitions per number of qubits to look at the mean and standard deviation.