Quantum Approximate Optimization of Integer Graph Problems and Surpassing Semidefinite Programming for Max-k-Cut

TL;DR

The paper studies qudit-based QAOA for integer graph optimization, focusing on Max-$k$-Cut and encoding each label as a qudit level. It derives an explicit, instance-independent depth-$p$ edge expectation formula on high-girth $d$-regular graphs and shows Hadamard-based acceleration reduces evaluation time to $O(p^2 k^{2p+2}\log k)$, enabling parameter optimization across graphs. Empirically, QAOA at shallow depths surpasses the Frieze-Jerrum SDP in certain $(k,d)$ regimes, while a DSatur-inspired heuristic provides a strong classical baseline; extrapolations suggest QAOA may overtake this heuristic at larger depths. The work broadens quantum advantage prospects by extending QAOA to integer optimization, discusses gate-level implementations on qubit hardware for $k$ a power of two, and outlines future directions to generalize the approach and strengthen classical baselines.

Abstract

Quantum algorithms for binary optimization problems have been the subject of extensive study. However, the application of quantum algorithms to integer optimization problems remains comparatively unexplored. In this paper, we study the Quantum Approximate Optimization Algorithm (QAOA) applied to integer problems on graphs, with each integer variable encoded in a qudit. We derive a general iterative formula for depth-$p$ QAOA expectation on high-girth $d$-regular graphs of arbitrary size. The cost of evaluating the formula is exponential in the QAOA depth $p$ but does not depend on the graph size. Evaluating this formula for Max-$k$-Cut problem for $p\leq 4$, we identify parameter regimes ($k=3$ with degree $d \leq 10$ and $k=4$ with $d \leq 40$) in which QAOA outperforms the Frieze-Jerrum semi-definite programming (SDP) algorithm, which provides the best worst-case guarantee on the approximation ratio. To strengthen the classical baseline we introduce a new heuristic algorithm, based on the degree-of-saturation, that empirically outperforms both the Frieze-Jerrum algorithm and shallow-depth QAOA. Nevertheless, we provide numerical evidence that QAOA may overtake this heuristic at depth $p\leq 20$. Our results show that moving beyond binary to integer optimization problems can open up new avenues for quantum advantage.

Figures (9)

• Figure 1: Qudit Quantum Approximate Optimization Algorithm for Max-$k$-Cut (A) Example input graph for the Max-$k$-Cut problem and its solution for $k=3$. Cuttable edges (between vertices belonging to different sets) are shown as dashed lines, while uncuttable edges (between vertices belonging to the same set) are depicted as solid lines. The dashed box represents the mapping the Max-$k$-Cut problem to an integer labeling by assigning each label to a quantum state $\ket{a}$, where $0 \leq a \leq k-1$. (B) Schematic of the qudit QAOA circuit for Max-$k$-Cut, consisting of $p$ layers of the phaser and mixer. The phaser is constructed from the cost Hamiltonian $H_C$ and mixer allows the exploration of the solution space. In this work we consider three mixers: Transverse Field (TF), BKKT (from Ref. bravyi2022hybrid), and Grover Mixer. The TF mixer is generated by the Hamiltonian $H_M^{\mathrm{TF}}=1/2\sum_{i=1}^{\log_2 k}X_i$, and can only be implemented when $k$ is a power of $2$. For the BKKT mixer, we denote, for $c \in \mathbb{Z}_k$, $\ket{\tilde{c}}=\frac{1}{\sqrt{k}}\sum_{a\in\mathbb{Z}_k} e^{2\pi i a c/k}\ket{a}$. (C) Tree structure for an edge $\{u,v\}$ for QAOA at $p=2$ on a $d$-regular graph. Each vertex branches into $d$ neighbors (excluding the parent edge gives $d-1$ branches at each level).
• Figure 2: Qudit QAOA for Max-$k$-Cut: Mixer comparison. The performance of QAOA for Max-$k$-Cut using the mixers considered in this work is studied for $k=3$(A) and $k=4$(B). For $k=3$, both the Grover and BKKT mixer are studied; notably, after optimization, the latter exhibits behavior indistinguishable from the Grover mixer, a phenomenon already observed for other discrete constraint satisfaction problems peptide_sampling_qaoa. For $k=4$, we additionally include the tensor product mixer, we find that similar to $k=3$, Grover mixer and BKKT mixer yield overlapping performance and both surpass the tensor product mixer. When $k=3$, QAOA outperforms semidefinite programming (SDP) when the graph degree $d\le10$. QAOA outperforms SDP across all graph degrees shown when $k=4$. However, our classical heuristic algorithm consistently outperforms QAOA in this regime. The dashed vertical line gives the bound of the maximum colorable graph degree given in \ref{['tab:degree_threshold_max_k_cut']}.
• Figure 3: QAOA for Max-$k$-Cut at larger circuit depths for the Grover mixer. QAOA performance on Max-$k$-Cut using the Grover mixer is examined as circuit depth $p$ increases, with optimal parameters studied for $k=3$ ($p=7$) in ( A) and $k=4$ ($p=6$) in ( B). The results show that increasing the QAOA depth leads to improved performance across the range of graph degrees considered. To further explore the relationship between QAOA and the classical heuristic algorithm, we fit the QAOA performance to the function $F(p) = m / (p^a + c) + b$, enabling extrapolation to larger depths. The analysis suggests the existence of a finite threshold depth, $p_{\mathrm{th}}$, beyond which QAOA may outperform the classical heuristicalgorithm for both $k=3$ and $k=4$. However, this extrapolation remains speculative, and further validation through direct simulation at greater circuit depths will be required to substantiate these trends.
• Figure 4: Comparison of Heuristic and SDP classical solvers. Cut fractions for the Frieze--Jerrum SDP algorithm and the heuristic algorithm are evaluated for $k=3$ and $k=4$ over varying node counts $n$ and graph degrees. The heuristic algorithm consistently outperforms the SDP solver for all tested values of graph degree. The dashed vertical line indicates the bound for the maximum colorable graph degree, as given in \ref{['tab:degree_threshold_max_k_cut']}.
• Figure 5: Gate-level implementation of the Max-$k$-Cut on qubit hardware.(A) The circuit that implements one of the terms of the QAOA phaser, $e^{-i\gamma P_{v_i,v_j}}$fuchs_2021_maxkcut. This requires in total of $2\times\log_2 k$ CNOT gates and a 2 $C^{\log_2 k}X$ gate, where each $C^{\log_2 k}X$ gate can be decomposed into $\log_2 k-1$ Toffoli gates. (B) The circuit that implements the Grover mixer which requires a total of 2$C^{\log_2 k +1}X$ gate when $k$ is a power of 2 tsvelikhovskiy2026qaoamixer.

Theorems & Definitions (14)

• Proposition 1: Evaluating edge expectations in qudit-QAOA
• Proposition 2: Edge expectations in qudit-QAOA for edge costs translation-invariant in $\mathbb{Z}_k$
• Corollary 1: Evaluating Max-$k$-Cut cost function
• Definition 3: Hadamard transform on qudits
• Theorem 5: Efficient computation of Hadamard transform fwht
• Lemma 6: Matrix-vector multiplication from Hadamard transform
• proof
• Lemma 7: Path integral expansion of QAOA edge expectation for tree problem
• proof
• Proposition 8: Evaluating QAOA edge expectation for tree problem