Variational Quantum Optimization with Continuous Bandits

TL;DR

The paper reframes variational quantum optimization as a continuous-bandit best-arm identification problem to circumvent barren plateaus that hinder gradient-based training. It develops an information-theoretic, instance-specific lower bound and presents a simple, near-optimal adaptive algorithm (Reject-and-Refine) with linear-time per-sample complexity and an extension to multi-dimensional parameter spaces. The approach is validated on PQC and QAOA circuits, showing improved sample efficiency over state-of-the-art gradient and finite-difference methods and resilience to flat regions where traditional methods falter. The work offers a principled, scalable framework that leverages global information to train quantum circuits under noise, suggesting a promising direction for robust VQA training beyond gradients.

Abstract

We introduce a novel approach to variational Quantum algorithms (VQA) via continuous bandits. VQA are a class of hybrid Quantum-classical algorithms where the parameters of Quantum circuits are optimized by classical algorithms. Previous work has used zero and first order gradient based methods, however such algorithms suffer from the barren plateau (BP) problem where gradients and loss differences are exponentially small. We introduce an approach using bandits methods which combine global exploration with local exploitation. We show how VQA can be formulated as a best arm identification problem in a continuous space of arms with Lipschitz smoothness. While regret minimization has been addressed in this setting, existing methods for pure exploration only cover discrete spaces. We give the first results for pure exploration in a continuous setting and derive a fixed-confidence, information-theoretic, instance specific lower bound. Under certain assumptions on the expected payoff, we derive a simple algorithm, which is near-optimal with respect to our lower bound. Finally, we apply our continuous bandit algorithm to two VQA schemes: a PQC and a QAOA quantum circuit, showing that we significantly outperform the previously known state of the art methods (which used gradient based methods).

Figures (4)

• Figure 1: Illustration of VQA inspired by La24
• Figure 2: Example with $\epsilon=2^{-5}, \kappa_0=\frac{1}{4},S=3$.
• Figure 3: Toy function (blue) and example run of \ref{['alg:rr']} with $D$ rounds. The red dots indicate the arms sampled from and the gray area the number of samples for each arm. The minimum estimated by the algorithm $\hat{x}$ is depicted by the orange dashed line.
• Figure 4: (1): Distance $|\hat{x}_t -x^*|$ to the true optimum at sample $t$ for runs of \ref{['alg:rr']} and SPSA (left). (2): Median and $(0.25, 0.75)$-quantiles of the sample complexity $N_{\mathrm{total}}$ for optimizing a PQC (middle) and MaxCut-QAOA (right) below the thresholds $C=0.4$ and $C=0.2$, respectively. $N_\mathrm{total}$ refers to the sample complexity and $n$ to the number of qubits. The medians are computed over $20$ and $100$ simulations, respectively. Failure of convergence is indicated by a dashed line.

Theorems & Definitions (20)

• Definition 1: Covering-number
• Definition 2: Zooming-dimension
• Theorem 4.1
• Corollary 1
• Theorem 4.2
• Corollary 2
• Theorem 5.1
• Corollary 3
• proof : Proof of \ref{['thm:exact_lower_bound']}
• proof : Proof of \ref{['cor:trivial_lower_bound']}