Quantum Simulation of Non-Hermitian Linear Response

Abstract

Linear response theory and Green's functions provide a universal framework for understanding how macroscopic and strongly correlated systems respond to weak external perturbations. While the theoretical foundation for non-Hermitian linear response theory has been recently established to describe open quantum systems, generalizing these predictions onto practical quantum computers remains a formidable algorithmic challenge due to the non-unitary nature of the dynamics. In this work, we present a systematic algorithmic mapping that transforms the non-unitary multi-time correlation functions into a unitary form viable for quantum hardware. By mapping the vectorization of the Lindblad master equation into a unitary Schrödinger-like equation using the continuous-variable Schrödingerization technique, we show that generalized non-Hermitian Green's functions can be systematically extracted. This approach bridges the gap between the established physical theory of non-Hermitian linear response and quantum simulation, achieving optimal state preparation cost.

Figures (4)

• Figure 1: Quantum circuit for the modified Hadamard test used in the Sampler simulation. The auxiliary qubit $A$ controls the preparation of the initial vectorized state $U_{\text{prep\_init}}$ and the Schrödingerization evolution $e^{-i\mathcal{H}_{sch}(\eta)t}$ on the system register $S$ when in the $\ket{0}$ branch. It then controls the preparation of the observable state $U_{\text{prep\_obs}}$ on the $\ket{1}$ branch. Measuring the ancilla in the $X$-basis extracts the real part of the density matrix overlap.
• Figure 2: Numerical simulation of the non-Hermitian linear response function $\chi(\tau)$. The exact solution of the Lindblad master equation (solid black line) is perfectly reproduced by the Schrödingerization simulation (blue circles) with discretized Fourier modes $\eta$ in the high resolution limit.
• Figure 3: Error convergence as a function of the number of Fourier grid points $N$. The log-log plot shows that the precision improves exponentially as $N$ increases, consistent with the theoretical $O(\text{poly}(\log(1/\epsilon)))$ scaling.
• Figure 4: Impact of the warped variable truncation limit $L$ on the simulation accuracy. The tail error $\epsilon_{tail} \sim e^{-L}$ decreases exponentially, confirming the theoretical error bound derived in Appendix \ref{['app:complexity_scaling']}.