Error Mitigation for Thermodynamic Computing

TL;DR

A method is introduced that reduces the overall error from a linear to a quadratic dependence (from $\epsilon$ to $\epsilon^2$ to $\epsilon^2$) on the imprecision $\epsilon$ for Gaussian sampling and linear algebra applications.

Abstract

While physics-based computing can offer speed and energy efficiency compared to digital computing, it also is subject to errors that must be mitigated. For example, many error mitigation methods have been proposed for quantum computing. However this error mitigation framework has yet to be applied to other physics-based computing paradigms. In this work, we consider thermodynamic computing, which has recently captured attention due to its relevance to artificial intelligence (AI) applications, such as probabilistic AI and generative AI. A key source of errors in this paradigm is the imprecision of the analog hardware components. Here, we introduce a method that reduces the overall error from a linear to a quadratic dependence (from $ε$ to $ε^2$) on the imprecision $ε$, for Gaussian sampling and linear algebra applications. The method involves sampling from an ensemble of imprecise distributions associated with various rounding events and then merging these samples. We numerically demonstrate the scalability of this method for dimensions greater than 1000. Finally, we implement this method on an actual thermodynamic computer and show $20\%$ error reduction for matrix inversion; the first thermodynamic error mitigation experiment.

Figures (8)

• Figure 1: Overview of Error Mitigation. (a) The general framework for error mitigation involves probing the physical computing device with multiple inputs $\{x^{(i)}\}_i$, where such inputs are typically related to the target in some way (e.g., some perturbation of the target input). The collection of outputs $\{y^{(i)}\}_i$ are then post-processed to obtain a high quality output $y^*$. (b) A plethora of error mitigation methods have been developed for quantum computers. Here we show a common one, Zero-Noise Extrapolation, where the noise level of the device is varied and the resulting expectation values $\{\langle O^{(i)}\rangle\}_i$ are fit with a curve that extrapolates to the zero noise limit, leading to the improved expectation value $\langle O^*\rangle$. (c) In thermodynamic computing, the goal is to sample from a target probability distribution $\mathcal{N}(0, \Sigma)$ with an imprecise physical device. Our error mitigation method, Thermies, randomly samples from multiple distributions $\{\mathcal{N}(0, \Sigma^{(i)})\}_i$ that are close to the target and then merges the obtained samples $\{X^{(i)}\}_i$ to obtain a high quality set of samples $X^*$ for the target distribution.
• Figure 2: Univariate Thermies. A target distribution $f_t$ with variance $\sigma^2_t= 1.5\varepsilon$ is approximated by interpolating between $f_1$ and $f_2$ whose variances are respectively $\varepsilon$ and $2\varepsilon$. While $f_t$, $f_1$, and $f_2$ are Gaussian, $f_a$ is a Gaussian mixture.
• Figure 3: Two-Dimensional Example. (a) The hypercube (in this case a cube) representation of the nearest neighbor ensemble is shown for a two-dimensional target covariance matrix $\Sigma^t$. The red dot is $\Sigma^t$ and the blue dots are (imprecise) approximations of the target. The target is inside the convex hull of its nearest neighbors, so it can be expressed as a convex combination of them. (b) The covariance ellipses are shown for the target covariance (black), the $8$ nearest neighbor covariances (blue), and the convex combination of the nearest neighbor covariances (red dashed). The target and approximating covariances exactly coincide due to the covariance matching property of Thermies.
• Figure 4: Imprecision Dependence. The $L_\infty$ distance between the approximate distribution $f_a$ and the target $f_t$ is plotted versus $\varepsilon$ for dimensions $1$ to $4$ with error mitigation (solid lines) and without error mitigation (dashed lines). With error mitigation, the error is proportional to $\varepsilon^2$ near $\varepsilon = 0$, so the first order dependence is zero as $\varepsilon$ goes to zero. Without error mitigation the error is proportional to $\varepsilon$ for small $\varepsilon$, so the error is sensitive to imprecision.
• Figure 5: Sampling Complexity. The Root Mean Square (RMS) error between $\Sigma^t$ and the average of $M$ covariance matrices drawn from the nearest neighbor ensemble is plotted versus $M$. The RMS error shows a dependence of roughly $M^{-1/2}$. This behavior is independent of $d$.

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• proof