Landscape-Similarity-Guided Optimization in QAOA

TL;DR

Doubly Optimized QAOA (DO-QAOA), which lowers runtime and quantum measurement overhead while maintaining a competitive approximation ratio gap (ARG) and provides a scalable route to hybrid quantum-classical optimization under realistic hardware constraints, with potential applicability across variational quantum algorithms.

Abstract

Across diverse synthetic and real-world interaction graphs, the variational landscapes of reduced Quantum Approximate Optimization Algorithm (QAOA) instances obtained via variable freezing exhibit a robust universality. Leveraging this structure, we introduce Doubly Optimized QAOA (DO-QAOA), which lowers runtime and quantum measurement overhead while maintaining a competitive approximation ratio gap (ARG). Adapting the replica-overlap framework of spin-glass physics, we define a landscape-overlap order parameter $q$ to quantify geometric correlations between energy landscapes, revealing a sharp landscape-similarity transition as graph connectivity is tuned. Notwithstanding this transition, the dominant convex features of nearly all conditioned sub-instances remain aligned across both phases. Exploiting this persistence, DO-QAOA collapses the nominal $2^m$ reduced instances generated by freezing $m$ qubits into $K = O(1)$ effective landscape classes, eliminating the exponential proliferation in $m$. By leveraging landscape structure, DO-QAOA provides a scalable route to hybrid quantum-classical optimization under realistic hardware constraints, with potential applicability across variational quantum algorithms.

Figures (12)

• Figure 1: Landscape similarity phase transition. Order parameter scaling: landscape overlap $q$ versus connectivity parameter $s$ for varying system sizes. A clear phase transition is marked by the crossing of $q(s)$ curves at $s_c \approx 0.6$, indicating a transition from a fragmented landscape (where sub-problems are unique) to a self-averaging landscape (where sub-problems share a universal basin structure).
• Figure 2: (a) Fragmented Phase ($s < s_c$) vs. (d) Self-Averaging Phase ($s > s_c$). (b--c) Energy Landscape of Fragmented Phase ($s < s_c$): Long-Range Connections Dominate (Minima are instance-dependent). The vertical axis represents the expectation value $E(\bm{\gamma}, \bm{\beta})$ of the cost Hamiltonian. Note that for the depth $p=1$ ansatz used here, the landscape is defined over the single-component parameter space $(\gamma_1, \beta_1)$. Local perturbations spread through the circuit light cone and propagate globally, causing decimated sub-problems to develop sample-specific landscapes with low overlap ($q \ll 1$). Despite this strong decorrelation, the dominant convex features in panels (b) and (c) coincide. (e--f) Energy Landscape Self-Averaging Phase ($s > s_c$): Quasi-1D phase where short-range connections dominate. Information propagation remains localized, resulting in a universal basin structure (Minima are universal) where landscapes become effectively instance-independent ($q \approx 1$).
• Figure 3: Divide-and-conquer approach and energy landscapes. The energy landscapes of $2^3$ sub-problems (overlaid) show that the geometric features (minima/maxima) align closely, despite the disparate linear biases induced by different frozen bitstrings. (a) In the ideal landscape (No Coefficients), the hypothesis holds, so the eight landscapes are mathematically nearly identical. As a result, the eight wireframes overlap perfectly, making it appear as though only a single landscape is plotted. (b) The induced linear-term coefficients distort the landscape, causing the eight wireframes to separate or “fuzz out,” while still remaining similar to the selected representative sub-problem.
• Figure 4: Overview of the DO-QAOA framework. The process begins with partitioning the Input Graph, selecting a Representative Subcircuit for training, and using the Bias-Aware Transfer Rule to efficiently transfer parameters to the remaining $2^m - 1$ sub-problems.
• Figure 5: Reference and distorted QAOA energy landscapes with 1-layer. Panels (a--c) show the ideal (reference) energy landscapes without induced coefficients for $m=1,2,3$. Panels (d–f) show overlaid energy contours of sub-problems generated by freezing $m$ nodes, illustrating how the landscape is distorted. However, basin stability is maintained: the optimization minima (darkest regions) remain localized within the same vicinity of $(\gamma, \beta)$.

Theorems & Definitions (1)

• Theorem 1: Landscape Stability