Quantum simulation of deep inelastic scattering in the Schwinger model

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.

💡 Research Summary

This paper presents the first quantum‑simulation study of deep‑inelastic scattering (DIS) observables in a confining gauge theory, namely the (1+1)‑dimensional quantum electrodynamics known as the Schwinger model. The authors focus on the hadronic tensor W^{μν}(P,q), which encodes the non‑perturbative structure of a hadron and is defined as the Fourier transform of a real‑time current‑current correlator. In two dimensions the virtual photon has no transverse polarization, so only the longitudinal structure function F_L survives; the transverse function F_T vanishes identically. The goal is therefore to compute F_L directly from the correlator ⟨h|J^μ(t,x) J^ν(0,0)|h⟩.

Two complementary quantum‑simulation strategies are employed:

1. Quantum‑circuit approach – The lattice Schwinger Hamiltonian is discretized with staggered fermions, mapped to qubits via the Jordan‑Wigner transformation, and Gauss’s law is imposed analytically. The Hamiltonian consists of nearest‑neighbor hopping, a mass term, and an electric‑field term. The hadron ground state (the lightest charge‑neutral meson) is prepared using a variational quantum eigensolver (VQE). Real‑time evolution is performed with a Trotter‑Suzuki decomposition, and the current operators are expressed as simple Pauli strings. After evolving the state, the overlap ⟨ψ_L|ψ_R⟩ yields the desired correlator. Simulations are carried out on classical emulators and on current NISQ devices (IBM 5‑qubit and Rigetti 8‑qubit), demonstrating that even with limited qubit numbers and without error mitigation the qualitative shape of the correlator is reproduced.

2. Tensor‑network approach – A Matrix Product State (MPS) representation of the same lattice Hamiltonian is evolved in time using the Time‑Evolving Block Decimation (TEBD) algorithm. The initial state is obtained via Density‑Matrix Renormalization Group (DMRG). By inserting the current operators into the MPS and evolving, the authors compute the same correlator for lattices up to 40–50 sites, with bond dimensions up to 400. The method provides high‑precision data in the regime where entanglement growth remains moderate; at very large momentum transfers the bond dimension required grows rapidly, limiting the reachable Q².

Both methods are benchmarked against exact diagonalization for small lattices (N ≤ 12). The quantum‑circuit and MPS results agree with the exact data to within a few percent, confirming that the real‑time correlator is accurately captured. After Fourier transforming the correlator, the hadronic tensor is obtained and the longitudinal structure function is extracted via F_L(x_B,Q²)=2 x_B