Learning parameter curves in feedback-based quantum optimization algorithms

TL;DR

The paper tackles the high sampling cost of feedback-based quantum optimization by predicting full FALQON parameter curves for MaxCut using a teacher–student Graph Neural Network framework in a single classical inference. The model outputs a curve of length $ ext{ell}=1001$ from a weighted graph input, enabling a measurement-free surrogate that can replace or warm-start the iterative FALQON procedure. Empirical results show that the ML-predicted curves closely match reference FALQON dynamics and outperform digitized linear QA schedules, with reasonable generalization to larger, unseen problem sizes within the tested regime. This approach offers a practical path to reducing resource overhead in quantum optimization and suggests avenues for extending ML-based curve design to other problems and QA schedules.

Abstract

Feedback-based quantum algorithms (FQAs) operate by iteratively growing a quantum circuit to optimize a given task. At each step, feedback from qubit measurements is used to inform the next quantum circuit update. In practice, the sampling cost associated with these measurements can be significant. Here, we ask whether FQA parameter sequences can be predicted using classical machine learning, obviating the need for qubit measurements altogether. To this end, we train a teacher-student model to map a MaxCut problem instance to an associated FQA parameter curve in a single classical inference step. Numerical experiments show that this model can accurately predict FQA parameter curves across a range of problem sizes, including problem sizes not seen during model training. To evaluate performance, we compare the predicted parameter curves in simulation against FQA reference curves and linear quantum annealing schedules. We observe similar results to the former and performance improvements over the latter. These results suggest that machine learning can offer a heuristic, practical path to reducing sampling costs and resource overheads in quantum algorithms.

Figures (6)

• Figure 1: FALQON parameter curves found for solving MaxCut on 19 non-isomorphic, unweighted 3-regular graphs with ten nodes, where each curve corresponds to the FALQON solution for one of the 19 graphs. The consistent clustering of these curves around a common profile motivates our work exploring if machine learning could be employed to predict them.
• Figure 2: Schematic of the steps for implementing FALQON. A weighted graph $G$ (left) defines the problem Hamiltonian $H_{\text{p}}$. Each quantum circuit layer $j$ applies $U_p=e^{-i\Delta t\,H_{\text{p}}}$ followed by $U_d(\beta_j)=e^{-i\Delta t\,\beta_j H_{\text{d}}}$. After layer $j$, the expectation value $A_j \equiv \langle i[H_{\text{d}},H_{\text{p}}]\rangle_j$ is estimated from qubit measurements and used to set the next parameter via the feedback law $\beta_{j+1}=-A_j$. Repeating for $\ell$ layers generates the parameter curve $\{\beta_j\}_{j=1}^{\ell}$, presented at right. At the end of the protocol, one may either estimate the final expectation value $\langle H_{\text{p}}\rangle_{\ell}$ or sample the terminal state to obtain a candidate MaxCut bit string solution. In the present work, we examine the prospect of replacing the steps of FALQON, and associated sampling costs, with an ML model trained to predict the full FALQON parameter curve. This figure is adapted from Ref. magann22.
• Figure 3: Teacher–student ML framework for predicting FALQON parameter curves for MaxCut. (a) Teacher: the input graph $G$ and five scalar features $s$ are processed by a graph neural network; pooled graph features feed a small conditioning module that outputs a full parameter curve $\hat{\beta}_{\mathrm{T}}$. (b) Student: a lighter graph network produces an embedding $g$ and an auxiliary estimate $\hat{s}$ (used only during training); a small readout maps $[g,\hat{s}]$ to the predicted curve $\hat{\beta}_{\mathrm{S}}$. At test time, only $\hat{\beta}_{\mathrm{S}}$ is used. Further details are given in Appendix \ref{['app:ml']}.
• Figure 4: Performance is plotted versus layer for the ML model predictions (blue) as well as the unweighted baseline (red) for different figures of merit and different problem sizes. Solid curves show the mean across problem instances, shaded bands indicate associated standard deviation, and dashed curves indicate the maximum error across instances. The three columns of panels correspond to graph sizes $n\in\{8,10,12\}$. The top row of panels shows the parameter error $|\Delta\beta_j|$, the middle row shows the approximation-ratio error $|\Delta r_{A,j}|$, and the bottom row shows the success-probability error $|\Delta\phi_j|$. For a given weighted graph instance, these errors are computed with respect to the true FALQON curve and its performance for solving MaxCut on that instance. The ML model predictions are obtained using the same weighted graph as input to generate an ML-predicted parameter curve, and the unweighted baseline corresponds to using the FALQON parameter curve computed for solving MaxCut on the same graph, but all weights being set equal to 1. The ML model predictions track the FALQON reference quite closely, as indicated by small error and relatively tight associated standard deviation, and they generally show lower errors compared with the unweighted baseline.
• Figure 5: Performance is plotted versus layer for the ML model predictions (blue) as well as the unweighted baseline (red) for different figures of merit and different problem sizes $n\in\{14,16,18,20\}$ not seen by the ML model during training. The plot descriptions match those in Fig. \ref{['fig:ml_falqon_abs_error_bounds']} and are described in the associated figure caption. Here, we observe that for these unseen system sizes, the ML model predictions continue to perform well, as captured by relatively low errors. These findings indicate that the ML model is able to generalize key features of FALQON parameter curves in a manner that can scale to larger sizes. However, this figure also illustrates limitations with scaling. Namely, we observe the ML model performance does begin to deteriorate compared with the unweighted baseline as the problem size increases.