Harvesting Brownian Motion: Zero Energy Computational Sampling

TL;DR

The paper addresses the energy efficiency of computation by proposing zero-energy, Brownian-motion–driven samplers that generate samples from a target distribution without dissipating energy per step. It introduces two output models, Las Vegas (exact sampling with rejection) and Monte Carlo (approximate sampling), and provides high-level constructions on general machines that realize efficient sampling with inter-sample times of $\Theta(\text{time}(A)^2)$. The authors instantiate these ideas on reversible Turing machines, offering concrete Las Vegas and Monte Carlo implementations with explicit timing bounds and highlighting how adiabatic, zero-energy dynamics can inform the design of energy-efficient randomized algorithms. This work advances the understanding of fundamental thermodynamic costs in computation and points to practical directions for low-energy sampling and computation in future hardware and algorithm design.

Abstract

The key factor currently limiting the advancement of computational power of electronic computation is no longer the manufacturing density and speed of components, but rather their high energy consumption. While it has been widely argued that reversible computation can escape the fundamental Landauer limit of $k_B T\ln(2)$ Joules per irreversible computational step, there is disagreement around whether indefinitely reusable computation can be achieved without energy dissipation. Here we focus on the relatively simpler context of sampling problems, which take no input, so avoids modeling the energy costs of the observer perturbing the machine to change its input. Given an algorithm $A$ for generating samples from a distribution, we desire a device that can perpetually generate samples from that distribution driven entirely by Brownian motion. We show that such a device can efficiently execute algorithm $A$ in the sense that we must wait only $O(\text{time}(A)^2)$ between samples. We consider two output models: Las Vegas, which samples from the exact probability distribution every $4$ tries in expectation, and Monte Carlo, in which every try succeeds but the distribution is only approximated. We base our model on continuous-time random walks over the state space graph of a general computational machine, with a space-bounded Turing machine as one instantiation. The problem of sampling a computationally complex probability distribution with no energy dissipation informs our understanding of the energy requirements of computation, and may lead to more energy efficient randomized algorithms.

Figures (6)

• Figure 1: Las Vegas construction, $M'$, applied to a simple Turing machine $M$ with no transition rules and a single bit of input (i.e. $r=1$). Circles: randomizer with one random bit ($r=1$, for input), one ancillary bit and with metadata $m=0$. Squares: output holding regions labeled with their output bit with metadata $m=-1$ or $m=1$.
• Figure 2: Our general Las Vegas construction $M^\ast$ from a general machine $M$. $R$ is the randomizer which generates random binary input words $b$, and $M_b$ represents the chain of states for $M$ on input $b$. The grey area represents the output holding states.
• Figure 3: Our Monte Carlo construction applied to a general machine with no transition rules and a single bit of input. Each node in the graph is a pair of nodes in the same column of the sub-machines.
• Figure 4: Monte Carlo: Each node in the graph is a pair of nodes from the sub-machines.
• Figure 5: Empirical plots built by simulating our general construction on a reversible Turing machine. Las Vegas: the probability that an observer has found a new sample by each attempt. Monte Carlo: the probability that a sample is independent of the previous one over time.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Corollary 4
• Definition 5
• Definition 6
• Definition 7
• Lemma 8
• Theorem 9