Measurement-Guided State Refinement for Shallow Feedback-Based Quantum Optimization Algorithm

TL;DR

Measurement-Guided Initialization (MGI), an iterative strategy that uses measurement outcomes from previous executions to update the initialization of subsequent runs, is introduced, showing that measurement statistics can be exploited to improve shallow quantum optimization protocols compatible with NISQ devices.

Abstract

Limited circuit depth remains a central constraint for quantum optimization in the noisy intermediate-scale quantum (NISQ) regime, where shallow unitary dynamics may fail to sufficiently concentrate probability on low-energy configurations. We introduce Measurement-Guided Initialization (MGI), an iterative strategy that uses measurement outcomes from previous executions to update the initialization of subsequent runs. The method extracts single-qubit marginal probabilities from dominant measurement outcomes and prepares a biased product-state initialization, allowing information obtained during optimization to be reused without introducing classical parameter optimization. We implement this approach in the context of the Feedback-Based Algorithm for Quantum Optimization (FALQON) and evaluate its performance on weighted MaxCut instances. Numerical results show that measurement-guided initialization improves the performance of shallow-depth circuits and enables iterative refinement toward high-quality solutions while preserving the non-variational structure of the algorithm. These results indicate that measurement statistics can be exploited to improve shallow quantum optimization protocols compatible with NISQ devices.

Figures (6)

• Figure 1: Schematic representation of the MaxCut problem on a graph. Vertices are partitioned into two subsets, where purple squares represent vertices encoded in the state $|1\rangle$ and green triangles represent vertices encoded in the state $|0\rangle$. The objective is to maximize the total weight of the edges crossing the partition (highlighted in blue), corresponding to the contribution captured in Eq. \ref{['eq:cut_set']}.
• Figure 2: Conceptual comparison between standard FALQON and MGI-FALQON. In standard FALQON (left), probability concentration toward low-energy configurations is achieved by increasing circuit depth. In MGI-FALQON (right), shallow circuits are repeatedly executed and measurement outcomes are used to refine the initialization through empirical marginals, effectively replacing part of the required circuit depth by measurement-driven iterative refinement.
• Figure 3: The diagram shows the operation of MGI-FALQON in a single iteration for a 4-qubit system. The iteration starts with the initial preparation, where each qubit receives an $R_y$ gate parametrized by the angles $\theta_i$ obtained in the previous iteration. The FALQON circuit with $L$ layers is then executed. The final state is measured $N_{\text{shots}} = 10$ times, producing 10 bitstrings. From these results, the $n = 3$ most frequent bitstrings are selected. Based on this reduced set, the probabilities $c_i$ of observing the value 1 on each qubit are computed. Finally, the angle $\theta_i$ used in the next iteration is determined so that the gate $R_y(\theta_i)$ produces a probability $c_i$ of measuring the state 1.
• Figure 4: The figure presents, in Panel (1), the weighted graph with eight vertices considered in the MaxCut analysis, where edge weights are represented by line thickness and the vertices belonging to the two subsets of the maximum cut partition are highlighted with distinct colors, blue and purple. Panel (2) shows the evolution of the expected value of the cost Hamiltonian energy as a function of the number of FALQON layers. Panel (3) displays heat maps characterizing the performance of the MGI as a function of the number of selected bitstrings, $n$, and the number of FALQON layers. Subfigure (3a) reports the fraction of runs in which the probability of measuring the optimal solution exceeds 0.5, while subfigure (3b) presents the average probability of obtaining the optimal solution. Each point corresponds to the mean over 100 independent MGI executions. Panel (4) shows a specific execution of the MGI with a two-layer FALQON circuit, where subfigure (4a) depicts the evolution of the expected energy and subfigure (4b) the evolution of the probability of measuring the optimal solution across MGI iterations. Panel (5) summarizes the results of 100 MGI executions on the same graph, considering three strategies for $n$. Subfigure (5a) presents the final FALQON energy and subfigure (5b) the final probability of measuring the optimal solution at each MGI iteration, with curves showing the mean values together with their corresponding standard deviations.
• Figure 5: Comparison between MGI-FALQON and standard FALQON for a 12-node weighted MaxCut instance. The left panel shows the evolution of the expected cost Hamiltonian energy $\langle H_p \rangle_k$ as a function of the layer index for MGI-FALQON, where each curve corresponds to a different MGI iteration. The right panel shows the corresponding evolution for standard FALQON. The dashed horizontal line indicates the optimal energy. While standard FALQON requires a large number of layers to approach the optimal solution, MGI-FALQON achieves comparable energy reduction through successive measurement-guided refinements while keeping the circuit depth fixed.