Pauli Correlation Encoding for Budget-Contrained Optimization

TL;DR

This work extends Pauli Correlation Encoding (PCE) to budget-constrained optimization by applying it to MinCut with a constraint term, revealing limitations of standard PCE in enforcing global constraints due to continuous relaxation. It introduces Iterative-$\alpha$ PCE, which adaptively increases $\alpha$ to drive binarization while preserving constraint satisfaction, achieving near-100% feasibility and improved cut sizes across graphs up to 300 nodes using a 9-qubit circuit. The authors analyze hyperparameters ($\alpha$, $\beta$) and regularization, propose a principled $\beta(c)$ heuristic, and compare Pauli-encoding schemas, showing substantial gains in constrained performance and scalability. Large-scale experiments with optimized runtimes (QARP) corroborate the approach's practicality on NISQ hardware, motivating further encoding design and efficiency enhancements for quantum-accelerated constrained optimization.

Abstract

Quantum optimization has gained increasing attention as advances in quantum hardware enable the exploration of problem instances approaching real-world scale. Among existing approaches, variational quantum algorithms and quantum annealing dominate current research; however, both typically rely on one-hot encodings that severely limit scalability. Pauli Correlation Encoding (PCE) was recently introduced as an alternative paradigm that reduces qubit requirements by embedding problem variables into Pauli correlations. Despite its promise, PCE has not yet been studied in the context of constrained optimization. In this work, we extend the PCE framework to constrained combinatorial optimization problems and evaluate its performance across multiple problem sizes. Our results show that the standard PCE formulation struggles to reliably enforce constraints, which motivates the introduction of the Iterative-$α$ PCE. This iterative strategy significantly improves solution quality, achieving consistent constraint satisfaction while yielding better cut sizes across a wide range of instances. These findings highlight both the limitations of current PCE formulations for constrained problems and the effectiveness of iterative strategies for advancing quantum optimization in the NISQ era.

Figures (12)

• Figure 1: Pauli Correlation Encoding (PCE) optimzation scheme.
• Figure 2: (a) $\tanh\left(\alpha\left\langle\mathrm{\Pi}\right\rangle\right)$ function. (b) $\tanh\left(\alpha\left\langle\mathrm{\Pi}\right\rangle\right)$ derivative. In both plots a point is marked where $\tanh\left(\alpha\left\langle\mathrm{\Pi}_i\right\rangle\right)=\pm0.99=x_{\pm0.99}$. (c) Width of the plateau region $\mathrm{\Delta x}=x_{+0.99}-x_{-0.99}$ vs $\alpha$ is shown. The points highlighted are for those $\alpha$ in the plots
• Figure 3: (a) Constraint success ratio $\varepsilon_c$ as a function of $\alpha$. (b) Binarization as a function of $\alpha$. (c) CutSize as a function of $\alpha$. Each graph was executed 10 times for all values of the constraint parameter $c \in [2, m/2]$, where $m$ denotes the number of nodes. Reported values correspond to averages over all runs. The classical optimizer used was SLSQP. The penalty parameter was fixed to $\beta = 1000$.
• Figure 4: (a) Constraint success ratio $\varepsilon_c$ as a function of $\alpha$. (b) Binarization as a function of $\alpha$. (c) CutSize as a function of $\alpha$. Each graph was executed 10 times for all values of the constraint parameter $c \in [2, m/2]$, where $m$ denotes the number of nodes. Reported values correspond to averages over all runs. The classical optimizer used was Nelder-Mead. The penalty parameter was fixed to $\beta = 1000$.
• Figure 5: Constraint succeess ratio $\varepsilon_c$ for different configurations of ($\alpha$, $\beta$, $c$) on an 18-node graph. From top to bottom, each row corresponds to $\alpha \in \{4, 8, 16\}$, while each column corresponds to $c \in \{3, 5, 7, 9\}$. At fixed $\alpha$ and $c$, $\varepsilon_c$ is reported for three ranges of $\beta$: $\beta \sim 10^2$, $\beta \sim 10^3$, and $\beta \sim 10^4$. The highlighted bars indicate the highest $\varepsilon_c$ achieved for each $(\alpha, c)$ configuration. Each bar corresponds to the mean ratio over 10 simulations.