Utility-Scale Quantum State Preparation: Classical Training using Pauli Path Simulation

TL;DR

The paper presents Pauli Path simulation (PPS), a coefficient-truncation, classical framework to pretrain parametrized quantum circuits for ground-state preparation of large quantum many-body systems at utility scale. By evolving observables in the Heisenberg picture and truncating Pauli terms with threshold $\delta_c$, PPS enables efficient, hardware-free optimization of a Hamiltonian-variational ansatz, with SPSA-ADAM updates and warm-start strategies. PPS achieves high-accuracy ground-state energies and meaningful observables across 1D and 2D Ising models and the Kitaev honeycomb model, including a 48-qubit hardware demonstration that also validates anyon braiding in the topological regime. The work demonstrates a practical, quantum-hardware-friendly approach that blends classical PPS optimization with quantum execution for dynamics-ready starting states and potential quantum advantage in challenging regimes.

Abstract

We use Pauli Path simulation to variationally obtain parametrized circuits for preparing ground states of various quantum many-body Hamiltonians. These include the quantum Ising model in one dimension, in two dimensions on square and heavy-hex lattices, and the Kitaev honeycomb model, all at system sizes of one hundred qubits or more, beyond the reach of exact state-vector simulation, thereby reaching utility scale. We benchmark the Pauli Path simulation results against exact ground-state energies when available, and against density-matrix renormalization group calculations otherwise, finding strong agreement. To further assess the quality of the variational states, we evaluate the magnetization in the x and z directions for the quantum Ising models and compute the topological entanglement entropy for the Kitaev honeycomb model. Finally, we prepare approximate ground states of the Kitaev honeycomb model with 48 qubits, in both the gapped and gapless regimes, on Quantinuum's System Model H2 quantum computer using parametrized circuits obtained from Pauli Path simulation. We achieve a relative energy error of approximately $5\%$ without error mitigation and demonstrate the braiding of Abelian anyons on the quantum device beyond fixed-point models.

Figures (16)

• Figure 1: Flowchart of the variational quantum algorithm framework. In the standard approach, the cost function and its gradient are evaluated on a quantum device. In this work, we instead use the coefficient-based Pauli Path simulation to perform these evaluations, allowing us to optimize the parameters prior to executing tasks on the quantum hardware.
• Figure 2: (a) The relative energy error $\Delta E / |E_0|$ for the 1D quantum Ising model with PBC and $N=100$ at $g_z=0$, where $\Delta E = \langle H \rangle - E_0$. The variational energies $\langle H \rangle$ are re-evaluated at $\delta_c = 10^{-4}$, with circuit repetitions $\ell = 10$ and $\ell = 20$. The relative errors remain below $0.5\%$. (b)(c) Dependence of the relative error on the truncation threshold $\delta_c$ for repetitions $\ell = 10$ and $\ell = 20$, respectively. The results indicate indicate that $\delta_c = 10^{-4}$ already yields accurate energies.
• Figure 3: For the the 1D quantum Ising model with PBC and $N=100$ at $g_z=0$, (a) $M_z$ and (b) $M_x$ for the variational wavefunction with circuit repetitions $\ell =10$ and $\ell =20$, evaluated at $\delta_c=10^{-4}$. The results show good agreement with the DMRG data away from the critical point $g_c =1$. Near the critical point, we expect the results can be improved by increasing the circuit repetition.
• Figure 4: For the 1D quantum Ising model with PBC and $N=100$, the relative energy error $\Delta E /|E_{\text{DMRG}}|$ for (a1) $g_z=0.2$ and (a2) $g_z=0.5$, where $\Delta E = \langle H \rangle - E_{\text{DMRG}}$ and circuit repetition $\ell =5$. (b1)(b2) $M_z$ and (c1)(c2) $M_x$ evaluated for the same parameters at $\delta_c = 10^{-4}$. All results show good agreement with the corresponding DMRG data.
• Figure 5: (a) The square lattice with periodic boundary conditions. The lattice consists of a total of $N = N_x \times N_y$ qubits, where $N_x$ and $N_y$ represent the number of qubits in the $x$ and $y$ directions, respectively. The qubits are labeled according to the order shown in the figure. (b) The heavy-hex lattice with $N=127$ qubits.