Charge-Density-Wave Oscillator Networks for Solving Combinatorial Optimization Problems

TL;DR

Solving large NP-hard combinatorial optimization problems remains challenging for digital architectures. The authors propose charge-density-wave quantum oscillator (CDW-QO) networks built from the 1T-TaS2 material, whose phase dynamics follow a Kuramoto-type model and can be injection-locked to realize binary Ising spins. By mapping the Ising Hamiltonian $H(\bm{x}) = -\sum_{i<j} W_{ij} x_i x_j$ to the oscillator couplings $J_{ij}$, the network naturally evolves toward ground states that solve problems such as Max-Cut; simulations on a 6×6 network show rapid convergence, and experiments indicate room-temperature operation with CMOS compatibility. The work demonstrates a promising, low-power hardware paradigm for NP-hard optimization, leveraging the unique quantum CDW transitions in TaS$_2$ to achieve fast, scalable computation with potential integration into conventional silicon technology.

Abstract

Many combinatorial optimization problems fall into the non-polynomial 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 efficiently requires novel hardware solutions beyond the conventional von Neumann architecture. We propose an approach for solving a type of NP-hard problem based on coupled oscillator networks implemented with charge-density-wave condensate devices. Our prototype hardware, based on the 1T polymorph of TaS2, 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 Si technology.

Figures (6)

• Figure 1: $|$ Electrical characteristics and circuit configuration of 1T-TaS2 CDW devices.a, Resistivity versus temperature for a CDW device with a $\sim 100$-nm-thick channel. b, Temperature-dependent I-V characteristics of a 1T-TaS2 CDW device. c, SEM image showing a $\sim 1$ µ m-long 1T-TaS2 device channel in the coupled oscillator circuit. The pseudo colors are used for clarity. d, Circuit schematic of the coupled oscillator device with off-chip coupling resistor RC, and load resistors RSi. The two devices are powered by an applied DC bias voltage VDCi, and all circuit elements are connected to a common ground. The injection locking signal is applied to both devices through an injection capacitor Cinj, and an oscilloscope monitors the output VOi.
• Figure 2: $|$ Oscillatory Characteristics of a CDW Device at room temperature. a, Circuit schematic of a single CDW oscillator device consisting of 1T-TaS2 channel, an off-chip load resistor RS, and a lumped capacitance C from the output node VO to ground. An oscilloscope monitors the output terminal. b, The hysteresis in the I-V characteristics of a 1T-TaS2 device at room temperature. A resistive load line from the circuit described in a intersects the hysteresis. The width of the hysteresis is $\Delta V \sim 0.4$ V. c, Oscillations produced by the circuit shown in a. These oscillations only occur when the voltage across the device intersects the hysteresis, as shown in b. Pink traces represent simulated oscillations, and red traces indicate experimental results obtained at room temperature. d, The frequency of stable oscillations changes as the applied DC bias is adjusted. At $V_{DC} = 4.49$ V, the oscillations begin. Increasing to $V_{DC} = 4.63$ V, $4.67$ V, and $4.71$ V, oscillations stabilize with $f = 219$ kHz, $208$ kHz, and $195$ kHz. At $V_{DC} = 4.81$ V, the oscillations become unstable. The output voltage is normalized.
• Figure 3: $|$ Oscillatory dynamics of coupled and injection-locked CDW-QOs.a, Circuit describing two resistively coupled CDW oscillator devices. b, Circuit describing the injection locking of a single CDW-QO. c, The injection locking signal is applied to two resistively coupled CDW-QOs. d, Coupling scenarios for a pair of coupled CDW-QOs described by the circuit in a. Adjusting the coupling resistance tunes the phase of the frequency-locked coupled oscillators. A larger coupling resistance results in a smaller coupling between oscillators. e Injection locking scenarios for a single oscillator described by the circuit in b. The green dotted line represents the injection signal. Three scenarios are depicted (bottom to top): before injection locking, FHIL, and SHIL after annealing to the stable solution. f Injection locking scenarios of coupled CDW-QOs as described by the circuit in c. Three scenarios are depicted (bottom to top): Strong coupling before injection locking, after FHIL, and after SHIL. Note the output voltage is normalized for d,e, and f.
• Figure 4: Experimental injection-locking scenarios. a, In the absence of injection locking, weakly coupled oscillators synchronize in frequency but not in phase, resulting in a flat energy landscape for the phase space. b, With FHIL, the oscillators synchronize their phases with the injection locking signal. c,d, For SHIL, the oscillator phase is in a bi-stable state and has in-phase c, and out-of-phase d, configurations to the input with equal probability. The green dotted line shows the injection locking signal. Simulated results, shown in light blue and light red, demonstrate the oscillators fit the theoretical framework.
• Figure 5: Phase evolution of three distinct injection-locking scenarios. a,d, In the absence of synchronization, the free-running oscillators exhibit uniform phases across the phase space. b,e, For FHIL, the oscillators are in-phase and locked with the injection signal. Their corresponding phases $\phi_1(t)$ and $\phi_2(t)$ with different random initial values converge to the locked phase value $117^{\circ}$. c,f, For SHIL, the oscillators exhibit in-phase and out-of-phase steady-state waveforms locked to the injection signal, and $\phi_1(t)$ and $\phi_2(t)$ with random initial values converge to the bi-stable phase values $112^{\circ}$ and $292^{\circ}$ with $50\%$ probability respectively.

Theorems & Definitions (2)

• Definition 1
• Definition 2