Symmetry-Aware Trotterization for Simulating the Heisenberg Model on IBM Quantum Devices

Abstract: Simulating the time-dependent Schr\"odinger equation requires finding the unitary operator that efficiently describes the time evolution. One of the fundamental tools to efficiently simulate quantum dynamics is the Trotter decomposition of the time-evolution operator, and various quantum-classical hybrid algorithms have been proposed to implement Trotterization. Given that some quantum hardware is publicly accessible, it is important to assess the practical performance of Trotterization on them. However, a straightforward Trotter decomposition of the Hamiltonian often leads to quantum circuits with large depth, which hinders accurate simulation on devices with limited coherence time. In this work, we propose a hardware-efficient Trotterization scheme for the time evolution of a 3-site, $J=1$ XXX Heisenberg model. By exploiting the symmetry of the Hamiltonian, we derive an effective Hamiltonian that acts on a reduced subspace, significantly lowering the circuit depth after optimization for specific evolution times. This approach can be interpreted as a change of basis in the standard Trotterization scheme. We also test our method on the IBM Quantum device ibmq_jakarta. Combining with the readout error mitigation and the zero-noise extrapolation, we obtain the fidelity $0.9928 \pm 0.0013$ for the simulation of the time evolution of the Heisenberg model from the time $t=0$ to $t=\pi$.