Fast gradient-free optimization of excitations in variational quantum eigensolvers

TL;DR

ExcitationSolve addresses the challenge of efficiently optimizing excitation-based VQE ansätze by exploiting a known second-order Fourier energy landscape for each parameter, enabling exact classical minimization with minimal quantum resources. The method extends to ADAPT-VQE through a globally-informed operator-selection step and supports multi-parameter optimization with controlled cost via Nyquist-inspired strategies. Across simulated benchmarks and NISQ hardware, ExcitationSolve delivers faster convergence, shallower circuits, and robust performance under noise, often reaching chemical accuracy in a single iteration. By uniting physical insight with efficient optimization, it offers a scalable path toward practical quantum chemistry on near- to mid-term devices, with potential extensions to multi-parameter spaces and dynamics.

Abstract

Finding molecular ground states and energies with variational quantum eigensolvers is central to chemistry applications on quantum computers. Physically motivated ansätze based on excitation operators respect physical symmetries, but existing quantum-aware optimizers, such as Rotosolve, have been limited to simpler operator types. To fill this gap, we introduce ExcitationSolve, a fast quantum-aware optimizer that is globally-informed, gradient-free, and hyperparameter-free. ExcitationSolve extends these optimizers to parameterized unitaries with generators $G$ of the form $G^3=G$ exhibited by excitation operators in approaches such as unitary coupled cluster. ExcitationSolve determines the global optimum along each variational parameter using the same quantum resources that gradient-based optimizers require for one update step. We provide optimization strategies for both fixed and adaptive variational ansätze, along with generalizations for simultaneously selecting and optimizing multiple excitations. On molecular ground state energy benchmarks, ExcitationSolve outperforms state-of-the-art optimizers by converging faster, achieving chemical accuracy for equilibrium geometries in a single parameter sweep, yielding shallower adaptive ansätze and remaining robust to real hardware noise. By uniting physical insight with efficient optimization, ExcitationSolve paves the way for scalable quantum chemistry calculations.

Figures (16)

• Figure 1: Schematic overview of our work.Top: while hardware-efficient ansätze, typically composed of parameterized rotations, allow for fast quantum-aware optimization ostaszewski_StructureOptimizationParameterized_2021nakanishi_SequentialMinimalOptimization_2020parrish_JacobiDiagonalizationAnderson_2019vidal_CalculusParameterizedQuantum_2018, they do not preserve physical properties gard2020efficient, e.g., vary the average particle number $\langle N \rangle$, the opposite is true for physically-motivated ansätze such as those assembled from fermionic excitation operators peruzzo_VariationalEigenvalueSolver_2014. Our new optimizer, ExcitationSolve, fills this gap and combines fast optimization with physical guarantees. Bottom: ExcitationSolve (purple) relies on the same quantum resources, i.e., same number of energy measurements, to jump to the global energy minimum along a single parameter $\theta_j$, as a gradient-based optimizer (red) evaluating and following the (partial) derivative in $\theta_j$. The latter does not consider global information of the energy landscape, thus being limited to a local parameter region. Note that since gradient descent is based on the full gradient evaluated over $N$ parameters, ExcitationSolve in fact performs $N$ update steps while gradient descent updates locally once.
• Figure 2: Flow chart for ExcitationSolve for fixed ansätze. In this iterative algorithm, the $k$-th iteration updates a single parameter $\theta_j$ through repeated sweeps over all $N$ parameters until convergence. To reflect the flexibility in the sweep order, we use separate indices $j$ and $k$. Per iteration, the parameter is shifted to four different positions $\theta^{(k)}_{j,1}, \theta^{(k)}_{j,2}, \theta^{(k)}_{j,3}, \theta^{(k)}_{j,4}$, and the quantum computer (QC) is used to obtain the corresponding energy values. This is the only part requiring the quantum hardware (purple). All remaining steps are efficiently computed classically. The energy associated with the unshifted current parameter value $\theta_j^{(k)}$ is re-used from the previous iteration $k-1$.
• Figure 3: Flow chart for ExcitationSolve for ADAPT-VQE (adpative ansätze). ExcitationSolve is integrated into ADAPT-VQE in two parts. First, for the selection criterion from the operator pool $\mathcal{P}$ by determining the immediate energy improvements via a single ExcitationSolve iteration when appending each of the operator candidates $U_m$ separately. Second, to re-optimize all parameter $\bm{\theta}^{(\ell)}$ at the end of each ADAPT iteration. The usage of the quantum computer (QC, red) solely happens in $\mathrm{ExcitationSolve(Step)}$ (Fig. \ref{['fig:algorithm_diagram_fixed']}) when invoked as sub-routines. Note that ADAPT iteration $\ell$ denotes how many operators have been appended to the ansatz.
• Figure 4: ADAPT-VQE vs. ExcitationSolve. We consider the selection among two excitation operator candidates A and B in the adaptive setting. In the original ADAPT-VQE approach grimsley_AdaptiveVariationalAlgorithm_2019, excitation A is selected based on the gradient criterion, i.e. the steepest gradient at $\theta=0$. This is then converged with a gradient descent towards a (potentially only local) minimum, typically requiring multiple gradient evaluations. ExcitationSolve chooses excitation B (despite the smaller gradient at $\theta=0$) based on the energy criterion, i.e., the attainable global energy minimum, and already initializes $\theta$ in its optimal configuration.
• Figure 5: 1D ExcitationSolve with Coordinate Descent vs. 2D ExcitationSolve. The simultaneous optimization of two parameters can be achieved either by effectively reducing it to a 1D optimization task using coordinate descent or employing a true 2D optimization based on the energy landscape from Eq. \ref{['eq:analytic_form_fourier_d_dim']}. Color indicates energy linearly from low (violet) to high (yellow). In this example, no matter which parameter is tuned first, the 1D coordinate descent approach (white arrows) converges only to a local minimum (black star marker). Also, this convergence takes up multiple iterations. Meanwhile, in the proper 2D case, the global reconstruction of the 2D second-order Fourier series permits an immediate jump (black arrow) to the global minimum (white star marker).

Theorems & Definitions (5)

• proof
• proof : Proof (alternative I)
• proof : Proof (alternative II)
• proof
• proof