Lower bounding the MaxCut of high girth 3-regular graphs using the QAOA

TL;DR

The paper addresses lower-bounding MaxCut on high-girth, 3-regular graphs by analyzing the Quantum Approximate Optimization Algorithm (QAOA). By pre-computing fixed QAOA parameters for graphs with girth $g\ge 2p+2$, it shows a per-edge contribution $\tilde{c}_{\rm edge}(p)$ such that $M_g\ge \tilde{c}_{\rm edge}(p)$, with numerical results up to $p=17$ yielding a cut fraction of $0.8971$ for $p=17$ and $g=36$, surpassing prior TPM bounds for $g\ge 16$ and approaching asymptotic limits $\ge 0.912$. The work combines tensor-network methods for exact evaluation of $c_{\rm edge}$, fixed-parameter QAOA schedules, and a runtime analysis showing polynomial-time prospects on quantum hardware, along with an MIS extension that translates MaxCut bounds into independent-set guarantees. These results provide a rigorous, algorithmically grounded quantum-classical hybrid bound that implies an exponential speedup over certain classical guarantees for producing large cuts on these graph classes. The study also demonstrates how MIS bounds can be tightened via a more flexible driver, underscoring the broader applicability of QAOA to related combinatorial problems.

Abstract

We study MaxCut on 3-regular graphs of minimum girth $g$ for various $g$'s. We obtain new lower bounds on the maximum cut achievable in such graphs by analyzing the Quantum Approximate Optimization Algorithm (QAOA). For $g \geq 16$, at depth $p \geq 7$, the QAOA improves on previously known lower bounds. Our bounds are established through classical numerical analysis of the QAOA's expected performance. This analysis does not produce the actual cuts but establishes their existence. When implemented on a quantum computer, the QAOA provides an efficient algorithm for finding such cuts, using a constant-depth quantum circuit. To our knowledge, this gives an exponential speedup over the best known classical algorithm guaranteed to achieve cuts of this size on graphs of this girth. We also apply the QAOA to the Maximum Independent Set problem on the same class of graphs.

Figures (7)

• Figure 1: Edge neighborhood of the central edge of a 3-regular graph at $p=3$ and girth $g\geq8$. Every edge in the graph has this neighborhood structure at this minimum girth. We evaluate $\tilde{c}_{\rm edge}(p)$ which is the optimal quantum expected value of the cost function on the central edge.
• Figure 2: Lower bound on $M_g$ for 3-regular graphs given by the present work (blue crosses). As a comparison we show the results of Thompson-Parekh-Marwaha (TPM) thompson2022explicit in orange. The QAOA lower bound exceeds the TPM lower bound at $g\geq 16$ (corresponding to $p \geq 7$). The QAOA lower bound exceeds the TPM $g \to \infty$ bound at $g\geq 32$ ($p \geq 15)$.
• Figure 3: Optimized parameters $\boldsymbol{\tilde{\gamma}}$ and $\boldsymbol{\tilde{\beta}}$ as a function of $(j-1)/(p-1)$ for different values of $p$ up to p=17. These figures suggest that the optimal parameters might approach fixed curves as $p$ increases.
• Figure 4: We plot $\tilde{c}_{\rm edge}$ which is $c_{\rm edge}$ at optimized angles $\boldsymbol{\tilde{\gamma}}$ and $\boldsymbol{\tilde{\beta}}$ as a function of $1/p$. The value 0.912 is the lower bound on $\lim_{g \to \infty} M_g$ given by Refs. gamarnik2018maxcsoka2015invariant. If $\tilde{c}_{\rm edge}$ were to exceed this value at larger $p$, it would provide a new lower bound on $\lim_{g \to \infty} M_g$. The value 0.9351 is the upper bound on the expected cut fraction of large random $3$-regular graphs given in Refs. hladky2006structuralmckay1982maximum, and $\tilde{c}_{\rm edge}$ cannot exceed this value.
• Figure 5: Tensor network for the computation of $\langle {\boldsymbol{\gamma}},{\boldsymbol{\beta}}| Z_i Z_j |{\boldsymbol{\gamma}},{\boldsymbol{\beta}} \rangle$ for $p=3$. On the left (a) we have $d=2$ and on the right (b) arbitrary $d$. In the latter case, the tensor network diagramatic notation is extended to denote extry-wise exponentiation of a tensor. In particular, the result of contracting the tensors in one of the colored boxes is raised to the power $(d-1)$ before proceeding to later contractions, as expressed in Eq. \ref{['eq:tensor_power']}. The time and space complexities of the contraction performed in this way are both $\mathcal{O}(2^{2p})$. The complexity does not depend on $d$. Higher values of $p$ are tackled in a similar fashion.