Multiple-time Quantum Imaginary Time Evolution

TL;DR

MT-QITE tackles the measurement and fidelity bottlenecks of Quantum Imaginary-Time Evolution by introducing multiple imaginary-time steps per Hamiltonian partition with a shared reference state, enabling parallelization and reduced measurement overhead. The method generalizes QITE to a set of time steps $\{\Delta t_l\}$ and, for quantum chemistry, reformulates the dynamics with a universal anti-Hermitian basis $\hat{T}$ to preserve particle number and spin. Across four benchmark models, MT-QITE achieves 1–2 order-of-magnitude improvements in fidelity and roughly an order of magnitude reduction in Pauli measurements compared to QITE, with partitioning and symmetry exploitation further lowering costs. These results demonstrate a scalable, deterministic, ansatz-free pathway for high-fidelity ground-state preparation on NISQ and fault-tolerant devices, including complex non-local Hamiltonians, and lay groundwork for applying MT-QITE to larger, realistic quantum systems.

Abstract

Quantum Imaginary-Time Evolution (QITE) is a powerful method for preparing ground states on quantum hardware. However, executing QITE has costly measurement budgets for general Hamiltonians. Both fidelity and computational cost are strongly dependent on the definition of suitable local domains and Hamiltonian partitions. In this work, we introduce the Multiple-Time QITE algorithm (MT-QITE). We show how using more than one imaginary time substantially improves the fidelity of the resulting ground state as well as the measurement overhead with respect to the previously published QITE algorithm, while preserving its deterministic character and its independence from ad hoc ansatze. Moreover, unlike QITE and other QITE-based algorithms, MT-QITE is parallelizable, and we show that even in Hamiltonians with non-local interactions, partitioning may entail a computational advantage.

Figures (8)

• Figure 1: Illustration of the MT-QITE algorithm and its relation to the QITE routine. Given a quantum state, a Hamiltonaian term $h[m]$ and a basis ansatz $az[m]$, QITE produces an evolved state through a real-time evolution (RTE) whose parameters are determined by tomography, the desired time step size $\Delta t$, and the classical resolution of a linear system of equations. The idea behind MT-QITE is to use the same reference quantum state for all terms in the Hamiltonian partition (only two are depicted for clarity), and to obtain the RTE parameters for a set of different time step sizes, as unlike QITE, MT-QITE only adds classical processing for a new time step size. The evolved state is the one on which the expectation value of the Hamiltonian is minimum, possibly using a different time step size per partition term.
• Figure 2: Fidelity per extra Pauli measurement per Trotter step in logarithmic scale for a selection of models and parameter regimes, namely: (a) Transverse field for the Ising model h/J, (b) ZZ coupling for the XXZ Heisenberg model J, (c) Electrostatic repulsion for the fermionic Hubbard model U. For the Ising and Heisenberg models simulations become more challenging around the phase transition (h/J = 1 and J= 1 respectively), and for the Hubbard model when strong repulsion is present. Improvement in fidelity per extra measurement is clearly superior for MT-QITE.
• Figure 3: MT-QITE Algorithm
• Figure 4: Ground state fidelity as a function of circuit depth expressed in no. of Pauli string rotations and cost in Pauli measurements (Necessary shots per Pauli not included). (a) XXZ Heisenberg model (8 spins). The MT-QITE algorithm is benchmarked with the previously reported QITE algorithm using a batch of 25 common initial states compatible with system symmetry and 10 Trotter steps using a Hamiltonian partition with two terms. An improvement in fidelity of one order of magnitude is achieved with MT-QITE along with a reduction of the computational cost by a factor 10. MT-QITE outperforms QITE both in average and in the best sample observed. (b) Transverse-field Ising model (6 spins), using a single initial state and varying the partition configuration (two terms in 2P and three in 3P). Only linear system measurements are plotted for comparability, QITE lines display only 3 Trotter steps. In this case MT-QITE improves fidelity in more than two orders of magnitude and the measurement budget is considerably reduced by site inversion symmetry.
• Figure 5: Energy and its absolute error for the $H_4$ chain in function of interatomic distance. Simulations have been run using the pool of anti-hermitian operators given by the UCCGSD ansatz. For stretching geometries, where more correlation is present, QITE and MT-QITE without partitioning (1P) fail, whereas MT-QITE on a three-term partition (3P) converges to chemical accuracy after 6 Trotter steps for the whole span of interatomic distances, and MT-QITE on a two-term partition (2P) follows closely.