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Missing Puzzle Pieces in the Performance Landscape of the Quantum Approximate Optimization Algorithm

Elisabeth Wybo, Martin Leib

TL;DR

This work analyzes QAOA performance on random $d$-regular graphs for MaxCut and MIS in the $N\to\infty$ limit by leveraging tree-like RCC contractions (tree QAOA). It unifies both problems as Ising models with a local field, deriving energy-density expressions and combining them with tight upper bounds on optimality to yield refined asymptotic approximation ratios. A key finding is the dichotomy: MaxCut improves with increasing $d$ (consistent with the absence of an OGP barrier), while MIS deteriorates due to OGP, highlighting fundamental locality limits for $p$-local quantum algorithms. The paper also demonstrates practical value by applying precomputed tree angles to small instances, sometimes surpassing classical baselines, and discusses avenues for hybrid non-local or quantum-classical strategies to achieve near-optimal performance in polynomial time.

Abstract

We consider the maximum cut and maximum independent set problems on random regular graphs in the infinite-size limit, and calculate the energy densities achieved by QAOA for high degrees up to $d=100$. Such an analysis is possible because the reverse causal cones of the operators in the Hamiltonian are with high probability associated with tree subgraphs, for which efficient classical contraction schemes can be developed. We combine the QAOA analysis with state-of-the-art upper bounds on optimality for both problems. This yields novel and better bounds on the approximation ratios achieved by QAOA for large problem sizes. We show that the approximation ratios achieved by QAOA improve as the graph degree increases for the maximum cut problem. However, QAOA exhibits the opposite behavior for the maximum independent set problem, i.e. the achieved approximation ratios decrease when the degree of the problem is increased. This phenomenon is explainable by the overlap gap property for large $d$, which restricts local algorithms (like QAOA) from reaching near-optimal solutions with high probability. In addition, we use the QAOA parameters determined on the tree subgraphs for small graph instances, and in that way outperform classical algorithms like Goemans-Williamson for the maximum cut problem and minimal greedy for the maximum independent set problem. In this way we circumvent the parameter optimization problem and are able to compute the expected approximation ratios.

Missing Puzzle Pieces in the Performance Landscape of the Quantum Approximate Optimization Algorithm

TL;DR

This work analyzes QAOA performance on random -regular graphs for MaxCut and MIS in the limit by leveraging tree-like RCC contractions (tree QAOA). It unifies both problems as Ising models with a local field, deriving energy-density expressions and combining them with tight upper bounds on optimality to yield refined asymptotic approximation ratios. A key finding is the dichotomy: MaxCut improves with increasing (consistent with the absence of an OGP barrier), while MIS deteriorates due to OGP, highlighting fundamental locality limits for -local quantum algorithms. The paper also demonstrates practical value by applying precomputed tree angles to small instances, sometimes surpassing classical baselines, and discusses avenues for hybrid non-local or quantum-classical strategies to achieve near-optimal performance in polynomial time.

Abstract

We consider the maximum cut and maximum independent set problems on random regular graphs in the infinite-size limit, and calculate the energy densities achieved by QAOA for high degrees up to . Such an analysis is possible because the reverse causal cones of the operators in the Hamiltonian are with high probability associated with tree subgraphs, for which efficient classical contraction schemes can be developed. We combine the QAOA analysis with state-of-the-art upper bounds on optimality for both problems. This yields novel and better bounds on the approximation ratios achieved by QAOA for large problem sizes. We show that the approximation ratios achieved by QAOA improve as the graph degree increases for the maximum cut problem. However, QAOA exhibits the opposite behavior for the maximum independent set problem, i.e. the achieved approximation ratios decrease when the degree of the problem is increased. This phenomenon is explainable by the overlap gap property for large , which restricts local algorithms (like QAOA) from reaching near-optimal solutions with high probability. In addition, we use the QAOA parameters determined on the tree subgraphs for small graph instances, and in that way outperform classical algorithms like Goemans-Williamson for the maximum cut problem and minimal greedy for the maximum independent set problem. In this way we circumvent the parameter optimization problem and are able to compute the expected approximation ratios.
Paper Structure (20 sections, 1 theorem, 70 equations, 13 figures, 3 tables)

This paper contains 20 sections, 1 theorem, 70 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. QAOA and tree QAOA
  3. Summary of the recursive iterations
  4. MaxCut and Maximum independent set as Ising models
  5. MaxCut
  6. Maximum independent set
  7. Local algorithms and overlap gap property
  8. Tree QAOA results
  9. Performance of tree QAOA in the asymptotic limit
  10. Changing the local field
  11. Fixing the local field
  12. Tree QAOA angles applied to finite-size problems
  13. Conclusion
  14. Calculation of the QAOA expectation values on trees
  15. A recursive iteration with local field
  16. ...and 5 more sections

Key Result

Theorem 4.1

See Ref. Farhi2020. The MIS problem on $d$-regular graphs has OGP when $d$ is large enough. This means that there exist a $\mu^{\star}$ such that for every $\mu > \mu^{\star}$ there exists $0<\theta_1<\theta_2<\mu$, such that for $N$ large enough with high probability (i.e. a probability converging to 1 exponentially fast in $N$), and with high probability.

Figures (13)

  • Figure 1: (A,B) The 1-tree and 2-tree subgraphs at level $p=3$, for degree $d=3$. (C) The path from outer leaf to root, together with the angle dependencies of each layer. The root variable is shown in black, the other variables are color coded according to their layer: $m=1$ orange, $m=2$ green, $m(=p)=3$ blue.
  • Figure 2: (A) The asymptotic energy densities achieved by $p=1$ QAOA as a function of the local field [see Eq. \ref{['eq:energy_density']}]. The colors indicate the different graph degrees listed in Table \ref{['tb:UBs']}, only a selection is shown in the legend. (B,C) The respective QAOA angles to obtain these energy densities.
  • Figure 3: The approximation ratios for (A) MaxCut and (B) MIS obtained by $p=1$ QAOA optimized as a function of the local field for $N\rightarrow \infty$. The colors indicate the different graph degrees listed in Table \ref{['tb:UBs']}, only a selection is shown in the legend. The dotted vertical lines indicate local fields $h=d-2$. The QAOA angles are chosen such that the energy density given by Eq. \ref{['eq:energy_density']} is minimized [see Fig. \ref{['fig:p1_energy']}]. The approximation ratios $\alpha_{MC}$ and $\alpha_{MIS}$ as discussed in Sec. \ref{['sec:model']}, are evaluated on the resulting QAOA state.
  • Figure 4: The approximation ratios obtained by QAOA for (A) MaxCut, and (B) MIS on 6-regular graphs for various $p$ in the limit $N \rightarrow \infty$. The QAOA ansatz is optimized to minimize the energy given by Eq. \ref{['eq:H_Ising']}. Therefore, there is a sharp transition in solution quality at $h=d$ for both problems. The ground state of Eq. \ref{['eq:H_Ising']} corresponds to the solution of MaxCut when $h\ll d$, and to the solution of MIS when $h\in]d-2,d]$. In these indicated regions the approximation ratio must therefore increase with $p$.
  • Figure 5: The approximation ratios for (A) MaxCut and (B) MIS obtained by depth-$p$ QAOA for $N \rightarrow \infty$. For MaxCut, we compare with the GW guarantee and Random Sampling (RS). For 3-regular graphs there is a specialized version of the GW algorithm, discussed in Ref. Halperin2004, achieving a worst-case approximation ratio of $0.9326$ which is indicated by the star. For MIS, we compare to the performance guarantee of a minimal greedy search Halldrsson1997, and the state-of-the-art linear-time prioritized search algorithm of Ref. Marino2020.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Theorem 4.1