Probabilistic imaginary-time evolution in state-vector-based and shot-based simulations and on quantum devices

TL;DR

The paper develops a state-vector formulation of probabilistic imaginary-time evolution (PITE) to enable efficient ground-state projection on quantum computers, and validates it against shot-based simulations to establish consistency. It derives a unitary-embedded, approximate PITE circuit that uses only a single ancilla in classical simulations and analyzes optimal initial parameters $(\gamma, \Delta\tau)$ to maximize success probabilities. Numerical results on Heisenberg and TFIM spin chains show rapid convergence of energy toward the ground-state values and reveal system-size dependencies linked to energy gaps. The authors demonstrate a TFIM experiment on a trapped-ion device with error mitigation, illustrating practical feasibility and highlighting considerations for near-term digital quantum simulations with larger systems and adaptive parameter tuning.

Abstract

Imaginary-time evolution, an important technique in tensor network and quantum Monte Carlo algorithms on classical computers, has recently been adapted to quantum computing. In this study, we focus on probabilistic imaginary-time evolution (PITE) algorithm and derive its formulation in the context of state-vector-based simulations, where quantum state vectors are directly used to compute observables without statistical errors. We compare the results with those of shot-based simulations, which estimate observables through repeated projective measurements. Applying the PITE algorithm to the Heisenberg chain, we investigate optimal initial conditions for convergence. We further demonstrate the method on the transverse-field Ising model using a state-of-the-art trapped-ion quantum device. Finally, we explore the potential of error mitigation in this framework, highlighting practical considerations for near-term digital quantum simulations.

Figures (9)

• Figure 1: Quantum circuit of the approximate PITE algorithm for a single imaginary-time step PITE. $H$ denotes the Hadamard gate and $R_Z\equiv R_Z(-2\theta_0)$ represents a single-qubit rotation about the $Z$ axis.
• Figure 2: Quantum circuit for the Hadamard test.
• Figure 3: Ground-state energy per site $E_0/L$ vs. the number $n_{\rm step}$ of imaginary-time steps obtained using the PITE algorithm for various values of $\gamma$ in the Heisenberg chain with $L=4$ under PBC. Results are from the noiseless shot-based simulations. The symbol sizes for $\gamma=0.7$ and $0.9$ are proportional to the number of successful outcomes after each imaginary-time step. The dashed line indicates the exact value of $E_0/L$, while the solid lines are guides to the eye.
• Figure 4: (a) $\gamma$ dependence of the success probability $\mathbb{P}_0^{(100)}$ (solid lines) and $|\gamma^2-e^{2\Delta\tau E_0}|$ (dotted lines) for various imaginary-time steps $\Delta\tau$ after 100 imaginary-time steps in the Heisenberg chain with $L=4$ under PBC, computed using state-vector simulations. (b) Estimated ground-state energy for $\Delta\tau=0.2$ and $\gamma=0.83$ using state-vector simulations and shot-based simulations with $N_{\rm shots}^{(0)}=10\,000$ and 50 000. (c) Corresponding success probabilities $\mathbb{P}_0$ for each imaginary-time step in panel (b).
• Figure 5: (a) Ground-state energy per site $E_0/L$ and (b) success probability $\mathbb{P}_0$ as functions of the number $n_{\rm step}$ of imaginary-time steps, obtained using the PITE algorithm with $\gamma=0.45$ (squares) and $0.51$ (circles) for the Heisenberg chain with $L=20$ under PBC. Solid lines represent the results from the state-vector simulations, while the dotted line in panel (a) shows the classical ITE simulation for comparison. Error bars in panel (a) indicate the statistical uncertainty $\sigma_E$ due to sampling, estimated from the standard deviations of the individual energy components $E_{XX}$, $E_{YY}$, and $E_{ZZ}$ as $\sigma_E=\sqrt{\sigma_{XX}^2+\sigma_{YY}^2+\sigma_{ZZ}^2}$ .