Quantum Simulations of Hadron Dynamics in the Schwinger Model using 112 Qubits

TL;DR

Real-time hadron dynamics in the lattice Schwinger model are demonstrated on 112 qubits using SC-ADAPT-VQE to prepare vacuum and localized hadron wavepackets, combined with a confinement-driven truncation of the electric interaction and second-order Trotter evolution. The approach yields scalable, low-depth circuits and uses extensive error mitigation to reveal hadron propagation on a 56-site lattice, with results qualitatively agreeing with classical cuQuantum MPS simulations. While device errors dominate current measurements, the work highlights a viable pathway to near-term quantum advantage in hadron scattering and outlines extensions to more complex gauge theories. Overall, the framework provides a scalable blueprint for lattice gauge theory simulations on quantum hardware and informs future progress toward inelastic scattering and non-Abelian QCD.

Abstract

Hadron wavepackets are prepared and time evolved in the Schwinger model using 112 qubits of IBM's 133-qubit Heron quantum computer ibm_torino. The initialization of the hadron wavepacket is performed in two steps. First, the vacuum is prepared across the whole lattice using the recently developed SC-ADAPT-VQE algorithm and workflow. SC-ADAPT-VQE is then extended to the preparation of localized states, and used to establish a hadron wavepacket on top of the vacuum. This is done by adaptively constructing low-depth circuits that maximize the overlap with an adiabatically prepared hadron wavepacket. Due to the localized nature of the wavepacket, these circuits can be determined on a sequence of small lattices using classical computers, and then robustly scaled to prepare wavepackets on large lattices for simulations using quantum computers. Time evolution is implemented with a second-order Trotterization. To reduce both the required qubit connectivity and circuit depth, an approximate quasi-local interaction is introduced. This approximation is made possible by the emergence of confinement at long distances, and converges exponentially with increasing distance of the interactions. Using multiple error-mitigation strategies, up to 14 Trotter steps of time evolution are performed, employing 13,858 two-qubit gates (with a CNOT depth of 370). The propagation of hadrons is clearly identified, with results that compare favorably with Matrix Product State simulations. Prospects for a near-term quantum advantage in simulations of hadron scattering are discussed.

Figures (20)

• Figure 1: a) Mapping the $L=56$ lattice onto the qubits of IBM's quantum computer ibm_torino (bottom left). The dynamical re-arrangement of charges in the vacuum screens the interactions between electric charges in the Schwinger model, giving rise to an exponential decay of correlations between spatial-site charges, $\langle \hat{\overline{Q}}_n \hat{\overline{Q}}_{n+d}\rangle$ (top and bottom right). b) The charge screening informs an efficient construction of the quantum circuits used to simulate hadron dynamics. SC-ADAPT-VQE is used to prepare the vacuum and wavepacket, which are time-evolved using Trotterized circuits implementing $e^{-i t \hat{H} }$ with a truncated electric interaction.
• Figure 2: a) The qubit coupling matrix for select electric Hamiltonians with $L=12$: $(i)$ shows the coupling matrix without truncation [Eq. \ref{['eq:Hgf']}], $(ii)$ shows the impact of restricting to the $Q=0$ sector [Eq. \ref{['eq:HelCP']}], and $(iii)$ corresponds to additionally truncating the interaction between charges separated by more than $\overline{\lambda}=1$ spatial sites [Eq. \ref{['eq:spatTrun1']}]. b) The number of $\hat{Z}\hat{Z}$ terms in different versions of the electric Hamiltonian as a function of $L$, showing the quadratic $L^2$ and linear $\overline{\lambda}L$ growth. c) The effects of truncating the electric interaction on the low-lying CP-even and $Q=0$ spectrum as a function of $\overline{\lambda}$ for $L=12$. The transparency of the lines connecting energy levels is proportional to the overlap of their corresponding eigenstates.
• Figure 3: Adiabatic state preparation for $L=12$. Upper panels: the lowest 200 eigenenergies of $\hat{H}_{\text{ad}}(t)$ as a function of adiabatic turn-on time, the energy of the state $|\psi_{\rm WP}(t)\rangle$, and the final overlap between $|\psi_{\rm WP}\rangle$ and the eigenstates of $\hat{H}$, $|E_i\rangle$. Lower panels: evolution of the chiral condensate $\langle\hat{\chi}_j\rangle$, defined in Eq. \ref{['eq:localCC']}, of $|\psi_{\rm WP}(t)\rangle$ for a selection of times, $t_a-t_g$, with the empty markers showing the $\langle \hat{\chi}_j \rangle$ of the vacuum $|\psi_{\rm vac}\rangle$ of $\hat{H}$.
• Figure 4: Infidelity of the wavepacket, defined in Eq. \ref{['eq:inf_wp_scadapt']}, prepared with multiple steps of SC-ADAPT-VQE for a range of $L$.
• Figure 5: Circuits implementing $R^{(XY)}_{\pm}(\theta)={\rm exp}[-i\tfrac{\theta}{2}(\hat{Y}\hat{X}\pm \hat{X}\hat{Y})]$ (left) and $R^{(XX)}_{\pm}(\theta)={\rm exp}[-i\tfrac{\theta}{2}(\hat{X}\hat{X}\pm \hat{Y}\hat{Y})]$ (right).