Out of the Loop: Structural Approximation of Optimisation Landscapes and non-Iterative Quantum Optimisation

TL;DR

This paper develops a rigorous framework linking the structure of combinatorial solution spaces to QAOA optimisation landscapes. By proving a two-level-Hamiltonian–based approximation theorem, it shows that the expected one-layer QAOA landscape $E(F_1)$ can be efficiently approximated by a problem-global, instance-averaged quantity $\\tilde{E}(F_1)$, with a provable error bound. It leverages Hamming-distance counts and target-space statistics to derive a non-iterative, unit-depth QAOA variant that uses problem-wide parameters for all instances, matching or exceeding standard QAOA performance in several canonical problems. The approach provides a principled foundation for parameter clustering and landscape similarity while offering practical routes to reduced quantum resources, demonstrated through five representative NP-related problems including SAT, k-clique, and qr-Factoring.

Abstract

The Quantum Approximate Optimisation Algorithm (QAOA) is a widely studied quantum-classical iterative heuristic for combinatorial optimisation. While QAOA targets problems in complexity class NP, the classical optimisation procedure required in every iteration is itself known to be \NP-hard. Still, advantage over classical approaches is suspected for certain scenarios, but nature and origin of its computational power are not yet satisfactorily understood. By introducing means of efficiently and accurately approximating the QAOA optimisation landscape from solution space structures, we derive a new algorithmic variant of unit-depth QAOA for two-level Hamiltonians (including all problems in NP): Instead of performing an iterative quantum-classical computation for each input instance, our non-iterative method is based on a quantum circuit that is instance-independent, but problem-specific. It matches or outperforms unit-depth QAOA for key combinatorial problems, despite reduced computational effort. Our approach is based on proving a long-standing conjecture regarding instance-independent structures in QAOA. By ensuring generality, we link existing empirical observations on QAOA parameter clustering to established approaches in theoretical computer science, and provide a sound foundation for understanding the link between structural properties of solution spaces and quantum optimisation.

Figures (14)

• Figure 1: Overview of our main foundational contributions. Left: Previously conjectured parameter clustering in qaoa optimisation landscapes $F_{1}^{(i)}(\beta, \gamma)$ (derived from a combinatorial optimisation objective function $c^{(i)}(\vec{z})$ of a decision problem instance) suggests that shared macroscopic similarities exist between instances $i$. Averaging over these reveals that the expected landscape $E(F_{1}(\beta, \gamma))$ is a common structure at the problem-global level. Right: Shared macroscopic features exist across all instances, based on structural properties manifest in the solution spaces. Aggregation (which may be possible analytically, but can always be performed using empirical sampling) leads to a macroscopic description of a problem-global solution space, from which the expected optimisation landscape can be efficiently approximated as $\tilde{E}(F_{1}(\beta, \gamma))$. Most importantly, we prove that the approximation has a bounded difference to the underlying exact quantity. The approximation approach (ingredients indicated by a blue background) and its consequences are subject of this paper.
• Figure 2: Overview of our main practical contributions (blue background indicates problem-global, grey background instance-specific components; yellow boxes mark classical computation. Dashed lines indicate transfer of classical data, and double lines denote querying a resource). Left: Standard qaoa that iteratively (symbolised by $\textcolor{lfd4}{\circlearrowright}$) determines optimal parameters ($\beta, \gamma$) for every instance by repeatedly sampling a structured quantum circuit. Right: Two-phase qaoa approach introduced in this paper that first determines optimal parameters ($\beta, \gamma$) by sampling $\tilde{E}(F_{1}(\beta, \gamma))$ from the problem-global target space, and then uses the derived instance-independent optimal parameters ($\beta, \gamma$) to obtain instance-specific solutions by sampling from a quantum circuit that is constant for each instance $\hat{C}$. As we show, $\tilde{E}(F_{1})$ has a strictly bounded difference to the expectation value of $F_{1}$, the core quantity of interest in qaoa.
• Figure 3: A central part of our approximation theorem is the idea of grouping equal Hamming distances. How this captures structural information of a problem solution space is visualized in (a). Mathematically this concept is expressed by a counting function $\#_d$, its derivation is depicted in (b).
• Figure 4: Three landscape components $\ab|c_k\ab(\beta,\gamma)|^2$ for $k \in \ab\{10, 13, 14\} = T$ and a state space of dimension $n = 5$ are depicted in this order from left to right. The landscapes are represented by an array of vertical cross sections along $\beta$.
• Figure 5: Cross section of the optimisation landscape components $\ab|c_k\ab(\beta,\gamma_{\text{c}})|^2$ depicted in \ref{['fig:3d_overview']} at $\gamma_{\text{c}}=1.2$ for $k \in \ab\{10, 13, 14\} \subset T$ (note matching colours). The dashed line shows $E\ab(|c_k\ab(\beta,\gamma_{\text{c}})|^2)$ for these three components, and illustrates how the mean value captures the globally relevant features of instance-specific information.

Theorems & Definitions (19)

• Definition 1
• Definition 2: qaoa circuit as in Ref. farhi2014quantum
• Definition 3: qaoa
• Lemma 1: Projector version of diez2024connection
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2