Scattering phase shift in quantum mechanics on quantum computers: non-Hermitian systems and imaginary-time simulations

Abstract

To overcome the fast oscillatory behavior of correlation functions for extracting scattering phase shift in real-time quantum simulations encountered in Ref.\cite{Guo:2026qkx}, we propose and test two solutions in the present work. One is to simulate Hermitian systems in imaginary time, the other is to simulate non-Hermitian systems in real time. We demonstrate that both approaches lead to the problem of non-unitary quantum evolution which can be solved by combining two quantum algorithms: block encoding and Hadamard test. The combined quantum algorithm does not require mid-circuit measurements or adjustment of the input parameters of the Hamiltonian, and can be easily implemented on quantum computers. Both the size and length of quantum circuits grow linearly with evolution time. Numerical tests on quantum simulators show that both approaches agree with exact solutions for a sufficiently long time before the signal is lost in statistical fluctuations. The results bode well for using non-Hermitian and imaginary-time simulations to circumvent oscillations inherent in real-time simulation of other quantum systems.

Figures (8)

• Figure 1: Quantum circuit for computing real part of $\langle \alpha | \frac{A^\dag + A}{2} | \alpha \rangle$, where non-unitary operator $A$ is block encoded in unitary operator $U$.
• Figure 2: Block encoding quantum circuit for unitary evolution of non-Hermitian system in real time, see Eq.\ref{['u1']} to Eq.\ref{['defryangle']}.
• Figure 3: Quantum circuits for computing imaginary time evolution of single-qubit system of Eq.\ref{['traceExpXonequbit']} (upper panel), and the block encoding of non-unitary operator $A$ into $U_A (\delta \tau)$ (lower panel).
• Figure 4: Imaginary time simulation of $\triangle C(\tau)$ for the 1-qubit system in Eq.(\ref{['Honequbit']}). Red curve is the exact solution. The black dots and blue band represent quantum simulator results for the mean value and two-standard-error band generated from 100 trials, each with 100,000 shots. The model parameters are: $m=1$, $a=4$ and $V_0 =2$. The time step is $\delta \tau=0.2$.
• Figure 5: Quantum circuits for computing imaginary time evolution of two-qubit system of Eq.\ref{['2qubitA']} (upper panel), and the block encoding of non-unitary operator $A$ into $U_A (\delta \tau)$ (lower panel).