Quantum simulation of deep inelastic scattering in the Schwinger model

TL;DR

This work demonstrates the feasibility of computing real-time hadronic tensors for deep inelastic scattering in the (1+1)-dimensional Schwinger model using quantum simulation. By employing both quantum-circuit and tensor-network approaches, the authors directly evaluate real-time current–current correlators, extract the hadronic tensor, and obtain the longitudinal structure function $F_L$ in a confining gauge theory. The study provides cross-validation against exact diagonalization for small systems and extends to larger lattices with tensor networks, yielding physically sensible DIS features and showing a path toward applying similar techniques to more realistic gauge theories, including QCD. These results establish a complementary route to traditional Euclidean lattice methods for accessing real-time observables in hadronic structure.

Abstract

Hadronic tensors encode the nonperturbative structure of hadrons probed in deep inelastic scattering (DIS), yet their direct evaluation requires real-time evolution that presents a challenge for traditional Euclidean lattice approaches. In this work, we present the first study of the hadronic tensors in DIS using quantum simulation in the Schwinger model, i.e (1+1)-dimensional QED. Using two complementary quantum-simulation strategies -- quantum-circuit and tensor-network methods -- we compute the real-time current-current correlator directly on the lattice and validate our results against exact diagonalization where applicable. From this correlator, we compute the hadronic tensor and determine the longitudinal structure function, the sole nonvanishing DIS observable in two space-time dimensions. Our study demonstrates that quantum simulation offers a viable complementary pathway towards the evaluation of real-time observables relevant for hadronic structure. It also provides a foundation for extending the calculations from Schwinger model to other gauge theories.

Figures (14)

• Figure 1: Left: Illustration of Deep Inelastic Scattering of an electron and a hadron. The incoming hadron momentum is $P$, and the momentum of the exchanged virtual photon is $q$. Right: The diagram representation of the so-called hadronic tensor, $W^{\mu\nu}(P,q)$.
• Figure 2: Hadron mass evaluated with fixed volume $\mathcal{V}=24$ and 36 with increasing number of qubits up to $N=200$. Extrapolated values at continuum limit $ag=0$ are obtained by a linear fit.
• Figure 3: Quantum circuits for evaluating the components of the two-point correlator $J^\mu(t, x_1) J^\nu(0, x_0)$.
• Figure 4: Benchmark comparison of real and imaginary parts of the matrix elements $\Pi^{00}(t, x)$ and $\Pi^{11}(t, x)$ of the hadron state $\ket{h}$ for $N=12$ qubits calculated using exact diagonalization (ED), quantum circuit (QC), and tensor network (TN). Specifically, $m/g=0.5$ and $ag=0.5$ are used. Qubit locations 5 and 6 are used for the center position $x_0=0$.
• Figure 5: Hadronic tensors $W^{00}(q^0,q^1)$ and $W^{11}(q^0,q^1)$ evaluated at fixed longitudinal momentum transfer $q^1$ with decreasing lattice spacing $a$ at constant volume $\mathcal{V}=24$. Here, $m/g=0.5$.