Theoretical Guarantees of Variational Quantum Algorithm with Guiding States

TL;DR

This work tackles the lack of rigorous guarantees for variational quantum algorithms by introducing a guiding-state VQA and a linearization trick that maps training dynamics to a fixed kernel model. The authors prove convergence and generalization guarantees under a guiding-state assumption, showing that guiding states accelerate convergence and suppress finite-size fluctuations. They establish a concentration of the quantum neural tangent kernel for alternating layered ansätze and demonstrate that the true, nonlinear training dynamics can be well-approximated by its linearized kernel dynamics in the large-system limit. Numerical experiments on 2D random Heisenberg models corroborate the theory, illustrating kernel concentration, lazy training, and improved generalization with increasing data. These results provide a principled framework for warm-started quantum learning and suggest kernel-based design principles for near-term quantum advantage.

Abstract

Variational quantum algorithms (VQAs) are prominent candidates for near-term quantum advantage but lack rigorous guarantees of convergence and generalization. By contrast, quantum phase estimation (QPE) provides provable performance under the guiding state assumption, where access to a state with non-trivial overlap with the ground state enables efficient energy estimation. In this work, we ask whether similar guarantees can be obtained for VQAs. We introduce a variational quantum algorithm with guiding states aiming towards predicting ground-state properties of quantum many-body systems. We then develop a proof technique-the linearization trick-that maps the training dynamics of the algorithm to those of a kernel model. This connection yields the first theoretical guarantees on both convergence and generalization for the VQA under the guiding state assumption. Our analysis shows that guiding states accelerate convergence, suppress finite-size error terms, and ensure stability across system dimensions. Finally, we validate our findings with numerical experiments on 2D random Heisenberg models.

Figures (3)

• Figure 1: Variational quantum algorithm with guiding state: (a) The algorithm starts with a guiding state as a warm start $\rho_0$ and goes through a parameterized quantum circuit $U(\boldsymbol{\theta})$ to learn the representation of the ground state that will be used to generate the properties of the system. The new parameters are updated using gradient descent with learning rate $\eta$ with respect to the loss function $L(\boldsymbol{\theta})$. (b) We present our perspective on warm starts, which is similar to the conventional approach drudis2024variational. However, our focus shifts from the initialization of parameters to the initialization of the quantum state itself. It is worth noting that these two approaches can be mapped onto one another.
• Figure 2: Alternating Layer Ansatz. (a) An illustration of the alternating layered ansatz. Here, each layer is separated by a vertical dashed line. (b) We illustrate the locality property of ALA. The shaded boxes are in the light cone of $O_i$. This means that the actions of $U^{\dagger}(\boldsymbol{\theta})O_iU(\boldsymbol{\theta})$ only depend on the parameters in the shaded boxes and the other will be canceled out.
• Figure 3: Predicting ground state properties in 2D antiferromagnetic random Heisenberg models. (a) The variance of a single entry $K_{\boldsymbol{\theta}(0)}(x, x\prime)$ over $100$ different initialization in a variety of system sizes $n$. The experiment corresponds to the ALA with $L=1$ and $m=2$. (b) The distribution of $\frac{d}{dt} K_{\boldsymbol{\theta}}(x, x\prime)$ across different system size settings $n$ over a range of $t$ from $1$ to $100$. As the system size goes to $20$, the values $K_{\boldsymbol{\theta}}(x, x\prime)$ asymptotically stays constant. (c) Training behavior of the true model corresponding to linearized model with initialized kernel $K_{\boldsymbol{\theta}(0)}$. The models are performed with ALA$(n=20,m=4,L=2)$. (d) The generalization error with three different training dataset sizes of $30,50,80$.

Theorems & Definitions (9)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof