Warm Starts, Cold States: Exploiting Adiabaticity for Variational Ground-States

TL;DR

The paper addresses the challenge of reliably preparing many-body ground states with variational methods by introducing an iterative warm-start protocol that discretizes an adiabatic path $H(x)=H_0+xH_1$ and solves a sequence of progressively harder problems using the previous step as an initialization. It combines VQE and Meta-VQE with adiabatic principles to track the ground-state manifold, providing a rigorous lower bound on gradient variance that remains polynomial as long as the spectral gap stays open. Numerical simulations—including shot noise—show consistent convergence to the target ground state along the path, while demonstrations near level crossings reveal fundamental limitations when the gap closes. The approach offers a pathway to shorter-depth quantum circuits and potential shortcuts to adiabaticity, with implications for near-term ground-state preparation and fault-tolerant applications, and suggests directions for extending the method to excited-state tracking and optimized Hamiltonian trajectories.

Abstract

Reliable preparation of many-body ground states is an essential task in quantum computing, with applications spanning areas from chemistry and materials modeling to quantum optimization and benchmarking. A variety of approaches have been proposed to tackle this problem, including variational methods. However, variational training often struggle to navigate complex energy landscapes, frequently encountering suboptimal local minima or suffering from barren plateaus. In this work, we introduce an iterative strategy for ground-state preparation based on a stepwise (discretized) Hamiltonian deformation. By complementing the Variational Quantum Eigensolver (VQE) with adiabatic principles, we demonstrate that solving a sequence of intermediate problems facilitates tracking the ground-state manifold toward the target system, even as we scale the system size. We provide a rigorous theoretical foundation for this approach, proving a lower bound on the loss variance that suggests trainability throughout the deformation, provided the system remains away from gap closings. Numerical simulations, including the effects of shot noise, confirm that this path-dependent tracking consistently converges to the target ground state.

Figures (3)

• Figure 1: Schematic overview of the iterative training mechanism and its failure mode at level crossings.(a) Illustrative loss landscape at random initialization: the minimum is narrow and the landscape is otherwise nearly flat, so a random start typically lies in a low-gradient region (barren plateau). (b)Left: The loss landscape $\mathcal{L}(\boldsymbol{\theta})$ as a function of the variational parameters $\boldsymbol{\theta}$ at successive iterations (colors). When the ground state varies smoothly and the spectral gap remains open, the location of the minimum drifts continuously, so initializing each step with the previous optimum keeps the optimizer in a high-gradient region that reliably tracks the moving solution. As the gap closes (yellow dashed box), a competing global minimum can emerge far from the followed basin, causing the iterative procedure to remain trapped in the “wrong” minimum past the crossing. Right: visual representation of the iterative warm-start strategy, where the optimal parameters at iteration $k$ are used to initialize the step $k+1$.
• Figure 2: Variance of the loss function in a hypercube of side $2r$ in parameter space (i.e., side $2r/\pi$ in the rescaled $r/\pi$ units used on the horizontal axis). For each system size $n$ and radius $r$, we sample random directions in the variational parameter space around the optimized point $\theta^\star$ obtained from warm-start training, and estimate the variance of the loss at $x_2 = 0.2$, with the Heisenberg model described in Eq. \ref{['eq:Ising_Ham']}. (a) Variance as a function of $r/\pi$ for different qubit numbers $n$ (log–log scale). (b) Maximum variance as a function of the total number of variational parameters $M$. (c) Radius $r_{\max}$ at which the variance is maximal, plotted versus $M$. In panel (c) we also include a reference line proportional to $M^{-1/2}$, corresponding to regime $r \lesssim 1/\sqrt{M}$ for which Theorem \ref{['thm:main']} guarantees at most polynomially small variance. In contrast, the fitted behavior $r_{\max} \sim M^{-0.274}$ shows a more favorable empirical scaling. All runs use $L = n$ layers, synthetic shot noise with $\texttt{nshots}=10^4$ for the underlying optimization and $10^4$ samples to estimate the variance.
• Figure 3: Energy spectra learned with warm-start VQE and warm-start Meta-VQE. Rows correspond to two target models (top: XY; bottom: transverse-field Ising), while columns compare VQE (left) and Meta-VQE (right). In all panels we use warm starts, i.e., for each value of the field parameter $x$ the optimization is initialized with the parameters obtained at the previous point. Solid lines show the exact ground-state and first-excited energies as functions of $x$, and markers show the corresponding estimates obtained under synthetic shot noise with $\texttt{nshots}=10^4$ (circles: training points; crosses: test points). In the Ising row, the vertical dashed line indicates the critical point at $x=1$. All data are for $n=16$ spins and circuit depth $L=16$.

Theorems & Definitions (9)

• Definition 1: Iterative training strategy
• Theorem 1: Lower-bound on the loss variance for iterative methods, Informal
• Theorem 2: Taylor reminder theorem
• Proposition 1
• proof
• Theorem 3
• proof
• Theorem 4
• proof