Sample-Based Krylov Quantum Diagonalization for the Schwinger Model on Trapped-Ion and Superconducting Quantum Processors

TL;DR

This work applies Sample-Based Krylov Quantum Diagonalization (SKQD) to lattice gauge theory, using the Schwinger model with a θ-term to benchmark ground-state energies and particle numbers across trapped-ion and superconducting quantum processors. SKQD constructs a Krylov subspace from bitstrings sampled during time evolution and diagonalizes the Hamiltonian within this subspace, achieving noise resilience and reducing the effective Hilbert space. The results demonstrate excellent agreement with exact diagonalization for system sizes up to $N=30$, with subspace dimensions dramatically smaller than the full space (e.g., $\dim\mathcal{K}/\dim\mathcal{H}$ as low as $6.8\times10^{-6}$), enabling accurate capture of the $l_{0,c}$ phase-transition point and CP-symmetry aspects. These findings indicate SKQD’s potential for scalable lattice gauge theory simulations on near-term quantum devices, particularly when leveraging kinetic-only time evolution and mass-based state initialization to extend reach.

Abstract

We apply the recently proposed Sample-based Krylov Quantum Diagonalization (SKQD) method to lattice gauge theories, using the Schwinger model with a $θ$-term as a benchmark. SKQD approximates the ground state of a Hamiltonian, employing a hybrid quantum-classical approach: (i) constructing a Krylov space from bitstrings sampled from time-evolved quantum states, and (ii) classically diagonalizing the Hamiltonian within this subspace. We study the dependence of the ground-state energy and particle number on the value of the $θ$-term, accurately capturing the model's phase structure. The algorithm is implemented on trapped-ion and superconducting quantum processors, demonstrating consistent performance across platforms. We show that SKQD substantially reduces the effective Hilbert space, and although the Krylov space dimension still scales exponentially, the slower growth underscores its promise for simulating lattice gauge theories in larger volumes.

Figures (16)

• Figure 1: Example circuit for preparing $\ket{\psi_k}$ in \ref{['eq:psik']}, with $N=4$ qubits and a single Trotter step ($k=1$). Here, $U(dt)$ is the unitary time-evolution operator for a timestep $dt$, as defined in \ref{['eq:Ugate']}. The label $R$ denotes the $R_{XX+YY}$ gate as defined in Qiskit Qiskit.
• Figure 2: Ground-state energy $E_0$ of the Schwinger model as a function of the constant background field $l_0$, for $N=4$, $N/\sqrt{x}=30$, and $m_{\rm lat}/g=10$. The upper panel shows SKQD results obtained on a trapped-ion quantum processor (blue) compared with exact diagonalization (black), while the lower panel displays their relative difference. Results correspond to a single Trotter step.
• Figure 3: Particle number $\langle P\rangle$ of the Schwinger model as a function of the constant background field $l_0$, for $N=4$, $N/\sqrt{x}=30$, and $m_{\rm lat}/g=10$. Shown are SKQD results obtained on a trapped-ion quantum processor (blue) compared with exact diagonalization (black). Results correspond to a single Trotter step.
• Figure 4: Ground-state energy $E_0$ of the Schwinger model as a function of the constant background field $l_0$, for $N=14$, $N/\sqrt{x}=30$, and $m_{\rm lat}/g=10$. The upper panel shows SKQD results obtained on superconducting quantum processors for different numbers of Trotter steps (colored points) compared with exact diagonalization (black line). The lower panel displays the relative difference between the SKQD result at Trotter step 10 and the exact diagonalization result.
• Figure 5: Particle number $\langle P\rangle$ of the Schwinger model as a function of the constant background field $l_0$, for $N=14$, $N/\sqrt{x}=30$, and $m_{\rm lat}/g=10$. Shown are SKQD results obtained on superconducting quantum processors (blue) compared with exact diagonalization (black). Results correspond to 10 Trotter steps.