In the paradigm of thermodynamic computing, instead of behaving deterministically, hardware undergoes a stochastic process in order to sample from a distribution of interest. While it has been hypothesized that thermodynamic computers may achieve better energy efficiency and performance, a theoretical characterization of the resource cost of thermodynamic computations is still lacking. Here, we analyze the fundamental trade-offs between computational accuracy, energy dissipation, and time in thermodynamic computing. Using geometric bounds on entropy production, we derive general limits on the energy-delay-deficiency product (EDDP), a stochastic generalization of the traditional energy-delay product (EDP). While these limits can in principle be saturated, the corresponding optimal driving protocols require full knowledge of the final equilibrium distribution, i.e., the solution itself. To overcome this limitation, we develop quasi-optimal control schemes that require no prior information of the solution and demonstrate their performance for matrix inversion in overdamped quadratic systems. The derived bounds extend beyond this setting to more general potentials, being directly relevant to recent proposals based on non-equilibrium Langevin dynamics.

翻译：在热力学计算范式中，硬件并非确定性地运行，而是通过随机过程从目标分布中进行采样。尽管有假设认为热力学计算机可能实现更高的能效和性能，但热力学计算资源成本的理论刻画仍属空白。本文分析了热力学计算中计算精度、能量耗散与时间之间的基本权衡关系。通过熵产生的几何界限，我们推导出能量-延迟-缺陷积（EDDP）——传统能量-延迟积（EDP）的随机推广——的普适极限。虽然这些极限在理论上可达，但对应的最优驱动协议需要完全掌握最终平衡分布（即解本身）的信息。为突破此限制，我们开发了无需解先验信息的准最优控制方案，并在过阻尼二次系统中展示了其在矩阵求逆任务上的性能。所得界限可推广至更一般的势场情形，与非平衡朗之万动力学的最新研究提案直接相关。