A Euclidean Monte-Carlo-informed route to ground-state preparation for quantum simulation of scalar field theory

TL;DR

This work addresses the challenge of preparing non-trivial initial states for real-time quantum-field-theory dynamics by using Euclidean-time Monte-Carlo data to guide the construction of a finite-rank bosonic ansatz and its efficient quantum-circuit implementation. The authors introduce a $(R,Q)$ ansatz that combines a product of single-mode Gaussian squeezers with a polynomial core in the field operators, and they optimize both energy and ground-state moments $\langle \hat{O}\rangle$ against moments computed from PIMC data, yielding states that reproduce non-Gaussian features with only modest energy penalties. The optimized ansatz can be compiled into quantum circuits with gate counts that scale polynomially with system size $N$, and explicit cost estimates are given for core-state preparation and Gaussian-unitsary implementations. This Euclidean-Monte-Carlo-informed approach provides a principled pathway to initialize quantum simulations of scalar-field dynamics and motivates future extensions to fermionic and gauge theories in lattice QCD.

Abstract

Quantum simulators hold great promise for studying real-time (Minkowski) dynamics of quantum field theories. Nonetheless, preparing non-trivial initial states remains a major obstacle. Euclidean-time Monte-Carlo methods yield ground-state spectra and static correlation functions that can, in principle, guide state preparation. In this work, we exploit this classical information to bridge Euclidean and Minkowski descriptions for a (1+1)-dimensional interacting scalar field theory. We propose variational ansatz families which achieve comparable ground-state energies, yet exhibit distinct correlations and local non-Gaussianity. By optimizing selected wavefunction moments with Monte-Carlo data, we obtain ansatzes that can be efficiently translated into quantum circuits. Our algorithmic cost analysis shows these circuits' gate complexity scales polynomially in system size. Our work paves the way for systematically leveraging classically-computed information to prepare initial states in quantum field theories of interest in nature.

Figures (4)

• Figure 1: Schematic overview of classically informed ground-state preparation. Using ground-state correlation functions in an interacting scalar field theory, sourced from Euclidean path-integral Monte Carlo, a joint optimization of the energy and moments of the ansatz wavefunction is performed. Based on a mapping between the field and simulator degrees of freedom, the optimized ansatz is translated into a quantum circuit using a classical algorithm. This classically determined circuit can thereafter be implemented on quantum hardware, and be used as the starting point for performing other tasks, such as simulating real-time dynamics and estimating dynamical correlation functions. The goal of the work presented is to demonstrate a classical determination of the quantum circuit for preparing the ground state.
• Figure 2: Top: Minimum energy of various ansatzes compared with the Monte-Carlo energy estimate for various values of $(m^2,\lambda)$. The two bands indicate the Monte-Carlo energy estimate for the ground state and first excited state. Center: Two-point function for various values of $(m^2,\lambda)$. Bottom: Local $\phi$-moments and moment ratio for the minimum energy $(R,Q)$ and GEP ansatzes for various values of $(m^2,\lambda)$. Black points with error bars in the center and bottom panels show the values from PIMC. While the GEP approximates the two-point function very well, it completely fails to capture the state's non-Gaussianity.
• Figure 3: Optimization of the moment ratio for the $(R,Q)=(2,2)$ ansatz for $(m^2,\lambda)=(0.6,1.5)$. Moment optimization is performed for the target set $\mathcal{T}=\{\hat{\phi}_j^6,\hat{\phi}_j^8,\hat{\phi}_j^{10}\}$ for various values of $w$. The behavior of the moment ratio improves continuously as $w$ is increased. The moment optimization results in a small increase in energy as a function of $w$. The behavior of the two-point function is also only slightly modified as compared to the minimum-energy case. The values associated with different weights and PIMC are slightly offset in the horizontal direction to improve visibility of the error bars.
• Figure 4: Optimization of the two-point function $\hat{\phi}_0\hat{\phi}_4$ for the $(R,Q)=(2,2)$ ansatz for $(m^2,\lambda)=(0.6,1.5)$. Moment optimization is performed for the target set $\mathcal{T}=\{\hat{\phi}_0\hat{\phi}_4\}$ for various values of $w$. The behavior of this two-point function improves continuously as $w$ is increased. The moment optimization results in a small increase in energy as a function of $w$. The behavior of the moment ratio is also only slightly modified as compared to the minimum-energy case. The values associated with different weights and PIMC are slightly offset in the horizontal direction to improve visibility of the error bars.