Geo-ADAPT-VQE: Quantum Information Metric-Aware Circuit Optimization for Quantum Chemistry

TL;DR

Geo-ADAPT-VQE is introduced, a geometry-aware adaptive VQE algorithm that selects operators from a pool using the natural gradient rule, thereby improving convergence and reducing the algorithm's susceptibility to shallow local minima and saddle-point regions.

Abstract

Adaptive ansatz construction has emerged as a powerful technique for reducing circuit depth and improving optimization efficiency in variational quantum eigensolvers. However, existing adaptive methods, including ADAPT-VQE, rely solely on first-order gradients and therefore ignore the underlying geometry of the quantum state space, limiting both convergence behavior and operator-selection efficiency. We introduce Geo-ADAPT-VQE, a geometry-aware adaptive VQE algorithm that selects operators from a pool using the natural gradient rule. The geometric operator-selection rule enables the ansatz to grow along directions aligned with the underlying quantum-state geometry, thereby improving convergence and reducing the algorithm's susceptibility to shallow local minima and saddle-point regions. We further provide an asymptotic convergence result. We present numerical simulations involving five molecules, which demonstrate that Geo-ADAPT-VQE achieves faster and more stable convergence compared to existing methods, while producing significantly shorter ansatz. In particular, Geo-ADAPT achieves up to 100-fold reduction in energy error compared to existing methods.

Figures (7)

• Figure 1: Overview of the proposed Geo-ADAPT algorithm. Starting from the Hartree–Fock reference state $|\Psi_{\mathrm{HF}}\rangle$, the method iteratively constructs an ansatz from an operator pool $\mathcal{P}$. At the $k$-th outer iteration, the algorithm evaluates the gradient $\mathbf{g}_{k}$ and the corresponding pool information metric $\mathrm{F}_{k}$ to form the pool natural gradient $\tilde{\mathbf{g}}_{k}$. A geometric operator-selection rule chooses the next operator $j_{k} = \arg\max_j |\tilde{g}_{k,j}|$, and the ansatz is extended by $\exp(-\mathrm{i} \beta O_{j_{k}})$. The resulting parameters are optimized through an inner loop using QNGD, producing the updated ansatz $|\Psi(\boldsymbol{\theta})^{(k)}\rangle$. The procedure repeats until the norm of the natural gradient $\|\tilde{\mathbf{g}}_{k}\|_{\mathrm{F}_{k}}$ falls below a prescribed tolerance, indicating convergence.
• Figure 2: Energy error vs. number of iterations comparing GD, QNG-bd, ADAPT-VQE, and Geo-ADAPT-VQE. Geo-ADAPT consistently requires fewer steps to achieve chemical accuracy and exhibits improved convergence stability, especially for longer bond lengths.
• Figure 3: Energy error vs. number of ansatz parameters for LiH, HF, H5, BeH2, and H2O across several bond lengths. Geo-ADAPT reaches chemical accuracy with substantially fewer operators than ADAPT-VQE.
• Figure 4: Performance comparison for H5 at $1.3\,\text{\AA}$. Pos-Geo-ADAPT achieves faster convergence and lower energy error than ADAPT-VQE and Geo-ADAPT, whereas Pos-ADAPT exhibits performance similar to ADAPT-VQE. The vertical dashed line denotes the number of parameters in the UCCSD ansatz.
• Figure 5: Effect of the number of inner QNGD iterations $\kappa$ on the convergence behavior for the $\ce{H5}$ system at $1.30\,\text{\AA}$. The top panels show ADAPT-VQE and the bottom panels show Geo-ADAPT-VQE. Energy error is plotted as a function of the number of parameters (left) and the number of total optimization iterations (right). Increasing $\kappa$ improves convergence efficiency in Geo-ADAPT-VQE but introduces a trade-off between faster outer-loop convergence and the number of parameters added to the ansatz.

Theorems & Definitions (4)

• Lemma 1
• Lemma 2
• Theorem 1
• Lemma 3