Realizable Circuit Complexity: Embedding Computation in Space-Time
Benjamin Prada, Ankur Mali
TL;DR
This work introduces Realizable Circuits RC_d, a framework that embeds computation in d-dimensional space under strict causal, geometric, and thermodynamic constraints. By enforcing a volume bound $|C(t)| = O(t^d)$, a surface-bound throughput bound $w(C(t)) = O(t^{d-1})$, and finite gate capacity, RC_d provides universal scaling laws: any algorithm with runtime $ω(n^{d/(d-1)})$ cannot handle maximal-entropy inputs, and d-dimensional parallelism yields at most a polynomial speed-up of degree $(d-1)$. In the limit $d\to\infty$, RC_∞(polylog) equals the classical NC class, recovering standard parallelism as a non-physical idealization. The framework further unifies geometry, causality, and information flow, applying to classical, quantum, and other substrates, and includes a detailed case study mapping transformer attention to flux-bounded RC_d dynamics, yielding a universal $1/d$ exponent for attention scaling. Overall, RC_d extends circuit complexity into the physical domain, offering a principled, dimension-dependent theory of scalable computation with broad implications for hardware design, energy considerations, and architectures such as transformers.
Abstract
Classical circuit complexity characterizes parallel computation in purely combinatorial terms, ignoring the physical constraints that govern real hardware. The standard classes $\mathbf{NC}$, $\mathbf{AC}$, and $\mathbf{TC}$ treat unlimited fan-in, free interconnection, and polynomial gate counts as feasible -- assumptions that conflict with geometric, energetic, and thermodynamic realities. We introduce the family of realizable circuit classes $\mathbf{RC}_d$, which model computation embedded in physical $d$-dimensional space. Each circuit in $\mathbf{RC}_d$ obeys conservative realizability laws: volume scales as $\mathcal{O}(t^d)$, cross-boundary information flux is bounded by $\mathcal{O}(t^{d-1})$ per unit time, and growth occurs through local, physically constructible edits. These bounds apply to all causal systems, classical or quantum. Within this framework, we show that algorithms with runtime $ω(n^{d/(d-1)})$ cannot scale to inputs of maximal entropy, and that any $d$-dimensional parallel implementation offers at most a polynomial speed-up of degree $(d-1)$ over its optimal sequential counterpart. In the limit $d\to\infty$, $\mathbf{RC}_\infty(\mathrm{polylog})=\mathbf{NC}$, recovering classical parallelism as a non-physical idealization. By unifying geometry, causality, and information flow, $\mathbf{RC}_d$ extends circuit complexity into the physical domain, revealing universal scaling laws for computation.