Extracting scattering phase shift in quantum mechanics on quantum computers

TL;DR

The paper tests the feasibility of extracting infinite-volume scattering phase shifts from real-time quantum simulations using the integrated correlation function in a simple 1D trapped system with a contact potential. By formulating both coordinate-space and momentum-space Hamiltonians and deriving the trap-to-infinite-volume relation $C(t)-C_0(t) \to \frac{it}{\pi}\int d\epsilon\,\delta(\epsilon) e^{-i\epsilon t}$, it links spectral information to measurable correlators. It then constructs quantum circuits to time-evolve the Hamiltonian via a $\hat{H}_a+\hat{H}_b+\hat{H}_v$ decomposition and to read out $C(t)$ with a Hadamard test, enabling phase extraction through $\phi(E)$ and Friedel/Krein-type relations. Numerically, the approach reproduces exact results on a single qubit, but hardware experiments on two qubits show significant decoherence, with three-qubit runs failing completely, underscoring current limitations and motivating improvements in circuit design or error mitigation. The work provides a concrete baseline for extending ICF-based scattering studies to more complex field theories and guides future hardware and algorithmic advances required for real-time quantum simulations in scattering physics.

Abstract

We investigate the feasibility of extracting infinite volume scattering phase shift on quantum computers in a simple one-dimensional quantum mechanical model, using the formalism established in Ref.~\cite{Guo:2023ecc} that relates the integrated correlation functions (ICF) for a trapped system to the infinite volume scattering phase shifts through a weighted integral. The system is first discretized in a finite box with periodic boundary conditions, and the formalism in real time is verified by employing a contact interaction potential with exact solutions. Quantum circuits are then designed and constructed to implement the formalism on current quantum computing architectures. To overcome the fast oscillatory behavior of the integrated correlation functions in real-time simulation, different methods of post-data analysis are proposed and discussed. Test results on IBM hardware show that good agreement can be achieved with two qubits, but complete failure ensues with three qubits due to two-qubit gate operation errors and thermal relaxation errors.

Figures (22)

• Figure 1: Convergence of Eq.(\ref{['ICFmainEQanalytic']}) in Euclidean time with $L=4$ (dashed red), $10$ (dashed red) and $L = \infty$ (solid black). Both repulsive potential $V_0 =2$ (upper panel) and attractive potential $V_0 = -0.5$ (lower panel) are demonstrated for parameters $m=1$ and $N=400$. The corresponding lattice spacing is $a = \frac{L}{N-1}=0.01,\, 0.025$.
• Figure 2: Demo plot of real (red) and imaginary (purple) parts of $\triangle C( t ) = C(t) - C_0 (t)$ in real time vs. its infinite volume limit defined in Eq.(\ref{['ICFmainEQanalytic']}) for a repulsive potential $V_0 =2$ with $L=10$ (dashed) and $L = \infty$ (solid), where $m=1$, $N=300$, and $a=0.033$.
• Figure 3: Convergence in the $E+i \varepsilon$ prescription for real (upper panel) and imaginary (lower panel) parts of $\frac{1}{L} \widetilde{C}_0(E + i \varepsilon)$ defined in Eq.(\ref{['C0Lexample']}) vs. its infinite volume limit defined in Eq.(\ref{['C0infexample']}) for $L=20$ (dashed blue), $60$ (dashed red) and $L = \infty$ (solid black), where $\varepsilon = 0.1$ and $m=1$.
• Figure 4: Verification of Eq.\ref{['dlogTdE']} in the $E + i \varepsilon$ prescription: real (top) and imaginary (middle) parts of $(E+ i\varepsilon) \triangle \widetilde{C}(E+ i \varepsilon)$ (red dots) for a periodic trap vs. its infinite volume limit $- (E + i \varepsilon) \frac{d}{d E} \ln T(E+ i \varepsilon)$ (solid black) with $L=100$ and $\varepsilon = 0.1$. Bottom: comparison of $\cot \phi(E + i \varepsilon)$ (red dots) in Eq.(\ref{['phaseshift']}) and Eq.(\ref{['phaseintrap']}) vs. infinite volume limit phase shift $\cot \delta(E)$ (solid black) in Eq.\ref{['invPhase']}. The rest of the parameters are: $V_0 =2$, $m=1$, $N=4000$ and $a \sim 0. 025$.
• Figure 5: Similar to Fig. \ref{['CEplot']}, but in the $L \rightarrow i L$ prescription with $L=100 i$.