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arXiv:2610.14361

Competitive scaling of small-subsystem-optimization heuristics for LABS

Ginger E. Lau, Adil A. Gangat

cond-mat.stat-mechquant-phSubmitted Oct 9, 2026• Announced Oct 13, 2026• 0 citations• Revised Oct 29, 2026 (v3)
v1 · Oct 9, 2026v2 · Oct 14, 2026v3 · Oct 29, 2026
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Abstract

The low autocorrelation binary sequences (LABS) problem, often formulated as a spin-lattice energy-minimization problem, is a combinatorial optimization problem with a very rugged energy landscape. We demonstrate that iterative optimization of very small subsystems of n spins that are decoupled from the rest of the system achieves global optimization of LABS with a time-to-solution scaling that is competitive with heuristics that optimize the global system of N spins directly. Our isolated-subsystem heuristic with n = 4 achieves a time scaling of O(1.33^N ) on the interval N = [27, 60], while the current state-of-the-art classical and quantum scalings are respectively O(1.34^N ) and O(1.21^N ). Our results suggest that specialized optimization hardware with only small numbers of logical spins may be of practical relevance, and that quantum speedups demonstrated in the absence of decomposition may not provide a practical quantum advantage for large-scale problems that require decomposition.

Comments
9 pages, 5 figures

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The quantum approximate optimization algorithm (QAOA) is a leading candidate algorithm for solving optimization problems on quantum computers. However, the potential of QAOA to tackle classically intractable problems remains unclear. Here, we perform an extensive numerical investigation of QAOA on the low autocorrelation binary sequences (LABS) problem, which is classically intractable even for moderately sized instances. We perform noiseless simulations with up to 40 qubits and observe that the runtime of QAOA with fixed parameters scales better than branch-and-bound solvers, which are the state-of-the-art exact solvers for LABS. The combination of QAOA with quantum minimum finding gives the best empirical scaling of any algorithm for the LABS problem. We demonstrate experimental progress in executing QAOA for the LABS problem using an algorithm-specific error detection scheme on Quantinuum trapped-ion processors. Our results provide evidence for the utility of QAOA as an algorithmic component that enables quantum speedups.
Abstract Motivated by near term quantum computing hardware limitations, combinatorial optimization problems that can be addressed by current quantum algorithms and noisy hardware with little or no overhead are used to probe capabilities of quantum algorithms such as the quantum approximate optimization algorithm. In this study, a specific class of near term quantum computing hardware defined combinatorial optimization problems, Ising models on heavy-hex graphs both with and without geometrically local cubic terms, are examined for their classical computational hardness via empirical computation time scaling quantification. Specifically the time-to-solution (TTS) metric using the classical heuristic simulated annealing is measured for finding optimal variable assignments (ground states), as well as the time required for the optimization software Gurobi to find an optimal variable assignment. Because of the sparsity of these Ising models, the classical algorithms are able to find optimal solutions efficiently even for large instances (i.e. 100 000 spin variables). The Ising models both with and without geometrically local cubic terms exhibit average-case linear-time or weakly quadratic scaling when solved exactly using Gurobi, and the Ising models with no cubic terms show evidence of exponential-time TTS scaling when sampled using simulated annealing. These findings point to the necessity of developing and testing more complex, namely more densely connected, optimization problems in order for quantum computing to ever have a practical advantage over classical computing. Our results are another illustration that different classical algorithms can indeed have exponentially different running times, thus making the identification of the best practical classical technique important in any quantum computing vs. classical computing comparison.
Many combinatorial scientific computing problems are NP-hard which in practice requires using heuristics that either decompose a large-scale problem and solve many smaller local subproblems in parallel or iteratively improve the solution by local processing for sufficiently many steps to achieve a good quality/performance trade-off. These subproblems are NP-hard by themselves and the quality of their solution that depends on the subproblem size vitally affects the global solution of the entire problem. As we approach the physical limits predicted by Moore’s law, a variety of specialized combinatorial optimization hardware (such as adiabatic quantum computers, CMOS annealers, and optical parametric oscillators) is emerging to tackle specialized tasks in different domains. Many researchers anticipate that in the near future, these devices will be hybridized with high-performance computing systems to tackle large-scale problems which will open a door for the next breakthrough in performance of the combinatorial scientific computing algorithms. Most of these devices solve the Ising combinatorial optimization model that can represent many fundamental combinatorial scientific computing models. In this paper, we tackle the main challenges of problem size and precision limitation that the Ising hardware model typically suffers from. Although, our demonstration uses one of these novel devices, the models we propose can be generally used in any local search and refinement solvers that are broadly employed in combinatorial scientific computing algorithms.
Quantum approximate optimization is one of the promising candidates for useful quantum computation, particularly in the context of finding approximate solutions to Quadratic Unconstrained Binary Optimization (QUBO) problems. However, the existing quantum processing units (QPUs) are of relatively small size, and canonical mappings of QUBO via the Ising model require one qubit per variable, rendering direct large-scale optimization infeasible. In classical optimization, a general strategy for addressing many large-scale problems is via multilevel/multigrid methods, where the large target problem is iteratively coarsened and the global solution is constructed from multiple small-scale optimization runs. In this work, we experimentally test how existing QPUs perform when used as a sub-solver within such a multilevel strategy. To this aim, we combine and extend (via additional classical processing steps) the recently proposed Noise-Directed Adaptive Remapping (NDAR) and Quantum Relax & Round (QRR) algorithms. We first demonstrate the effectiveness of our heuristic extensions on Rigetti's superconducting transmon device Ankaa-2. We find approximate solutions to 10 instances of fully connected 82-qubit Sherrington-Kirkpatrick graphs with random integer-valued coefficients obtaining normalized approximation ratios (ARs) in the range∼0.98−1.0\sim 0.98-1.0∼0.98−1.0, and the same class with real-valued coefficients (ARs∼0.94−1.0)\sim 0.94-1.0)∼0.94−1.0). Then, we implement the extended NDAR and QRR algorithms as subsolvers in the multilevel algorithm for 6 large-scale graphs with at most∼27,000\sim 27,000∼27,000variables. In practice, the QPU (with classical post-processing steps) is used to find approximate solutions to dozens of at most 82-qubit problems, which are iteratively used to construct the global solution. We observe that quantum optimization results are competitive in terms of the quality of solutions when compared to classical heuristics used as subsolvers within the multilevel approach. Reproducibility: source code and data will be available at [1].
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