Scaling advantage with quantum-enhanced memetic tabu search for LABS

TL;DR

This work tackles the low-autocorrelation binary sequences (LABS) problem, formulated as minimizing $E(s)=\sum_{k=1}^{N-1} C_k^2$ with $C_k=\sum_i s_i s_{i+k}$, which maps to a long-range Ising/HUBO Hamiltonian with two- and four-body terms. The authors introduce a hybrid quantum–classical solver, QE-MTS, that seeds a memetic tabu search (MTS) with low-energy bitstrings produced by digitized counterdiabatic quantum optimization (DCQO). Across $N=27$ to $37$, QE-MTS achieves a typical time-to-solution scaling of $\mathcal{O}(1.24^N)$, outperforming the classical $\mathcal{O}(1.37^N)$ baseline and the QAOA-based approaches, with a conservative projected crossover near $N\approx 47$. These results suggest quantum sampling can meaningfully enhance the scaling of classical optimization for LABS and point toward practical quantum advantages in near-term hardware through hybrid algorithms.

Abstract

We introduce quantum-enhanced memetic tabu search (QE-MTS), a non-variational hybrid algorithm that achieves state-of-the-art scaling for the low-autocorrelation binary sequence (LABS) problem. By seeding the classical MTS with high-quality initial states from digitized counterdiabatic quantum optimization (DCQO), our method suppresses the empirical time-to-solution scaling to $\mathcal{O}(1.24^N)$ for sequence length $N \in [27,37]$. This scaling surpasses the best-known classical heuristic $\mathcal{O}(1.34^N)$ and improves upon the $\mathcal{O}(1.46^N)$ of the quantum approximate optimization algorithm, achieving superior performance with a $6\times$ reduction in circuit depth. A two-stage bootstrap analysis confirms the scaling advantage and projects a crossover point at $N \gtrsim 47$, beyond which QE-MTS outperforms its classical counterpart. These results provide evidence that quantum enhancement can directly improve the scaling of classical optimization algorithms for the paradigmatic LABS problem.

Figures (7)

• Figure 1: Density of $1$-flip local minima $f_{\mathrm{LO}}$ for different $N$. For the spin-glass case, we report the results over $10$ independent instances. This quantifies the number of sequences in which no single flip reduces the energy, compared to the total number of configurations. Additionally, it supports why LABS is a more challenging problem than a common spin-glass benchmark.
• Figure 2: Per-$N$ distributions of per-replicate medians $\tilde{Y}_{N,m,r}$ for QE-MTS (teal) and MTS (orange) on a log-scaled TTS axis. Each dot is a replicate median; dashed curves are log–linear fits to $Q_{0.50}(N,m)$ (median-of-medians), with fitted bases $\sim1.24^N$ (QE-MTS) and $\sim1.37^N$ (MTS). Lower positions indicate fewer function evaluations (faster).
• Figure 3: Decomposition of the block of two-qubit rotations $\text{R}_{YZ}(\theta)\text{R}_{ZY}(\theta)$, requiring $2$ entangling gates $\text{R}_{ZZ}$ and $4$ single-qubit gates.
• Figure 4: Decomposition of the block of four-qubit rotations $\text{R}_{YZZZ}(\theta)\text{R}_{ZYZZ}(\theta)\text{R}_{ZZYZ}(\theta)\text{R}_{ZZZY}(\theta)$, requiring $10$ entangling gates $\text{R}_{ZZ}$ and $28$ single-qubit gates.
• Figure 5: Per-$N$ distributions of $\log_{10}(\mathrm{TTS}_{\mathrm{QE-MTS}})-\log_{10}(\mathrm{TTS}_{\mathrm{MTS}})$. Negative values indicate QE-MTS is faster (lower TTS) on a log scale. Distributions are obtained via a two-stage bootstrapping (replicates and seeds, $5000$ draws per $N$).