This paper is a contribution to the Physical Review Applied collection titled Physics-Inspired Computing.

Many combinatorial optimization problems fall into the nonpolynomial-time NP-hard complexity class, characterized by computational demands that increase exponentially with the size of the problem in the worst case. Solving large-scale combinatorial optimization problems requires hardware solutions beyond the conventional von Neumann architecture. We propose an approach for solving an NP-hard problem based on coupled oscillator networks implemented with charge-density-wave condensate devices. Our prototype hardware, based on the 1T polymorph of ${\mathrm{Ta}\mathrm{S}}_2$, reveals the switching between the charge-density-wave electron-phonon condensate phases, enabling room-temperature operation of the network. The oscillator operation relies on hysteresis in current-voltage characteristics and bistability triggered by applied electrical bias. This work presents a network of injection-locked, coupled oscillators whose phase dynamics follow the Kuramoto model and demonstrates that such coupled quantum oscillators naturally evolve to a ground state capable of solving combinatorial optimization problems. The coupled oscillators based on charge-density-wave condensate phases can efficiently solve NP-hard max-cut benchmark problems, offering advantages over other leading oscillator-based approaches. The nature of the transitions between the charge-density-wave phases, distinctively different from resistive switching, creates the potential for low-power operation and compatibility with conventional $\mathrm{Si}$ technology.

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• Charge density waves
• Coupled oscillators
• Dynamics of networks
• Neuromorphic computing
• Quantum phase transitions
• Nano-oscillators
• Ising model
• Kuramoto model