Energy-Time-Accuracy Tradeoffs in Thermodynamic Computing

TL;DR

The paper addresses fundamental energy–time–accuracy limits in thermodynamic computing by defining the energy–delay–deficiency product (EDDP) for stochastic Langevin-based tasks, $EDDP = W_{diss}\,\tau\,(1-\mathcal{Q})$, and proving a universal bound ${\rm EDDP} \ge \mathcal{W}^2(\rho_0,\rho_1)/(\beta\gamma)$. Using overdamped Langevin dynamics and Wasserstein geometry, it shows that geodesic driving saturates this bound in Gaussian/quadratic settings, establishing a minimal-dissipation path between endpoint distributions. Since optimal geodesics require knowledge of the final distribution, the authors develop quasi-optimal protocols—quenches, Bang-Bang, and continuous slow-driving—that operate with limited information yet approach the bound for matrix inversion in quadratic systems. The results extend to general potentials, underdamped dynamics, and thermodynamic neural networks, offering concrete design principles for energy-efficient thermodynamic hardware and clarifying how protocol choice and dimensionality shape achievable energy–time–accuracy tradeoffs.

Abstract

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.

Figures (7)

• Figure 1: Thermodynamic algorithm for computing $f(K)$. In this illustration, the probability density of the state (in blue) is in some arbitrary initial distribution when the potential $V(x) = \frac{1}{2}\vec{x}^TK\vec{x}$ (in black) is applied. The probability density of the state naturally evolves towards thermal equilibrium (\ref{['s1']}), after which the observer can start sampling properties of the system to directly measure $f(K)$ (\ref{['s2']}). For example, the correlation matrix $\Sigma$ of the positions in the thermal distribution corresponds to the inverse of $K$.
• Figure 2: Average EDDP for 500 samples of $K_1$ taken from a Wishart distribution of $10\times10$ matrices with expected value $K_0 = \mathbb{1}$ and $20$ degrees of freedom. (Left) Comparison of the EDDP for geodesic, quench, and Bang-Bang protocols as a function of the interpolation parameter $q$ for a driving time $\tau_D = 10\gamma^{-1}$. Qualitatively, the shape of the Bang-Bang EDDP curve remains the same as one increases the driving time. (Right) Comparison of the EDDP for geodesic, quench, and Bang-Bang with $q=0.5$ protocols as a function of the driving time $\tau_D$. And relative error of the quench protocol ($\sqrt{\Tr[(\Sigma(\tau_D)-K_1^{-1})^2]/\Tr[K_1^{-2}]}$) as a function of the driving time. The relative error of the Bang-Bang protocol is slightly larger (and dependent on $q$) but is of the same order.
• Figure 3: Average Relative error ($\sqrt{\Tr[(\Sigma(\tau_D)-K_1^{-1})^2]/\Tr[K_1^{-2}]}$) for 100 samples of $K_1$ taken from a Wishart distribution of $10\times10$ matrices with expected value $K_0 = \mathbb{1}$ and $20$ degrees of freedom. The relative error is shown for the different protocols as a function of $\tau_D$ ($\tau_D \in [10\gamma^{-1},100\gamma^{-1}]$) for multiple values of $\tau_{\rm wait}$: (Left) no relaxation time, (Middle) fixed waiting time, (Right) adaptive waiting time.
• Figure 4: Examples of continuous driving functions for the Powers and Rationals families.
• Figure 5: Average EDDP for 100 samples of $K_1$ taken from a Wishart distribution of $10\times10$ matrices with expected value $K_0 = \mathbb{1}$ and $20$ degrees of freedom. (Left) EDDP of the Powers, and Rationals protocol families as a function of the free parameter $\alpha$ with $\tau_D = 10\gamma^{-1}$ and $\tau_{\rm wait} = 5\gamma^{-1}$. These are compared to the EDDP of the geodesic, quench, Bang-Bang ($q=0.5$), and linear ramp protocols. (Right) Comparison of the EDDP for a collection of protocols as a function $\tau_D$.