Accelerating Feedback-based Algorithms for Quantum Optimization Using Gradient Descent

TL;DR

This work proposes a hybrid method that incorporates per-layer gradient estimation to accelerate the convergence of QLC while preserving its low training overhead and stability guarantees, resulting in significantly faster convergence and improved robustness.

Abstract

Feedback-based methods have gained significant attention as an alternative training paradigm for the Quantum Approximate Optimization Algorithm (QAOA) in solving combinatorial optimization problems such as MAX-CUT. In particular, Quantum Lyapunov Control (QLC) employs feedback-driven control laws that guarantee monotonic non-decreasing objective values, can substantially reduce the training overhead of QAOA, and mitigate barren plateaus. However, these methods might require long control sequences, leading to sub-optimal convergence rates. In this work, we propose a hybrid method that incorporates per-layer gradient estimation to accelerate the convergence of QLC while preserving its low training overhead and stability guarantees. By leveraging layer-wise gradient information, the proposed approach selects near-optimal control parameters, resulting in significantly faster convergence and improved robustness. We validate the effectiveness of the method through extensive numerical experiments across a range of problem instances and optimization settings.

Figures (6)

• Figure 1: In our method, for each layer, GD is used to update $\beta_k^{(l)}$. This process is repeated $K$ times to construct all the layers. Our method reduces to FALQON when GD is used for only one iteration (with $\eta=1$) in each layer.
• Figure 2: Comparison of $R_A$ (left) and $p(t)$ (right) for solving weighted MAX-CUT using GD-QLC, FALQON, and SO-FALQON under two timestep settings: $\Delta t = 0.01$ (bottom row) and $\Delta t = 0.1$ (top row). GD-QLC exhibits robust performance across both regimes, outperforming FALQON and achieving performance comparable to SO-FALQON in the large-$\Delta t$ regime. In addition, GD-QLC displays significantly milder variations in the control parameters $\beta_k$ (middle column) compared to the other methods. For visualization purposes, the range of $\beta_k$ is clipped to the interval $[-10, 0]$, and results are shown for up to $200$ layers.
• Figure 3: Performance comparison of GD-QLC, FALQON and SO-FALQON for solving MAX-CUT with $\Delta t = 0.01$.
• Figure 4: Performance comparison of GD-QLC against FALQON and SO-FALQON for MAX-CLIQUE (top row) and MIN-COVER (bottom row) with $\Delta t = 0.005$.
• Figure 5: Results of solving weighted MAX-CUT using GD-QLC, FALQON and SO-FALQON with different timestep choices $\Delta t$. The results illustrate the robustness of GD-QLC for larger timesteps.