Non-unitary Time Evolution via the Chebyshev Expansion Method

TL;DR

The paper addresses computing non-unitary time evolution in quantum systems described by non-Hermitian Hamiltonians. It demonstrates that the Chebyshev expansion for the exponential remains valid across arbitrary complex spectra and derives practical guidelines to control rounding errors and time-step size for stable simulations. Using the Hatano-Nelson model as a test case, it shows accurate non-Hermitian dynamics under both periodic and open boundary conditions and discusses advantages over standard diagonalization approaches. The results provide a robust, scalable framework for simulating non-unitary quantum dynamics and inform parameter choices in non-Hermitian time evolution computations.

Abstract

The Chebyshev expansion method is a well-established technique for computing the time evolution of quantum states, particularly in Hermitian systems with a bounded spectrum. Here, we show that the applicability of the Chebyshev expansion method extends well beyond this constraint: It remains valid across the entire complex plane and is thus suitable for arbitrary non-Hermitian matrices. We identify that numerical rounding errors are the primary source of errors encountered when applying the method outside the conventional spectral bounds, and they are not caused by fundamental limitations. By carefully selecting the spectral radius and the time step, we show how these errors can be effectively suppressed, enabling accurate time evolution calculations in non-Hermitian systems. We derive an analytic upper bound for the rounding error, which serves as a practical guideline for selecting time steps in numerical simulations. As an application, we illustrate the performance of the method by computing the time evolution of wave packets in the Hatano-Nelson model.

Figures (10)

• Figure 1: The absolute difference ($\Delta$) between $\mathrm{e}^{-itz}$ computed using the standard exponential function and obtained using the Chebyshev series expansion Eq. \ref{['eq:expansion']}, truncated by keeping only $m = 250$ terms in the sum, for different values of $z\in\mathbb{C}$. The results are shown for $t=8$.
• Figure 2: Individual terms in the Chebyshev expansion \ref{['eq:expansion']} of $\mathrm{e}^{-itz}$ for various values of $t={8,16}$ and $z=\{3/2,3/2i\}$ as a function of the expansion order $m$.
• Figure 3: The absolute difference ($\Delta$) between $\mathrm{e}^{-itz}$ computed using the standard exponential function and obtained using the Chebyshev series expansion, Eq. \ref{['eq:expansion']}, for different $\rho$ radius values of $z\in\mathbb{C}$. The order of the expansion is $m = 250$. The result is shown for $t=8$. The red curve shows the estimate for the rounding error computed using Eq. \ref{['eq:errorasymp']}.
• Figure 4: Contours in the complex plane where the absolute difference $\Delta$ between $\mathrm{e}^{-itz}$ computed via the standard exponential and the Chebyshev expansion Eq. \ref{['eq:expansion']} satisfies $|\Delta| = 10^{-12}$, for times $t = 3,8$. The order of the expansion is $m = 250$. The dashed lines show the Eq. \ref{['eq:varrhomax']} estimate for the contour and the solid contours show the numerical result.
• Figure 5: The Chebyshev expansion of the complex exponential function $e^{-i z t}$ is shown using integer -- denoted with Int -- and floating-point -- denoted with Float -- arithmetic using $t = 8$. The parameter $\Delta$ represents the difference between the Chebyshev approximation from the exact exponential function, while $m$ denotes the number of terms considered in the expansion.