Linear-Readout Floors and Threshold Recovery in Computation in Superposition

TL;DR

This paper clarifies the differences between two approaches to computation in superposition, introducing a rank-trace Welch-type bound for linear readouts and analyzing their capacity regimes.

Contribution

It formalizes the distinction between capacity regimes and introduces a Welch-type lower bound for biorthogonal linear readouts, explaining the limits of existing methods.

Findings

Linear readouts have a worst-case off-diagonal cross-talk of (d^{-1/2})

Threshold recovery succeeds at sparsities s=O(d/d) for F=d^2

Linear readouts incur (s/d) error on Bernoulli sparse states

Abstract

Two recent approaches to computation in superposition reach different recursive capacity regimes: H\"anni et al. certify $\tilde{O}(d^{3/2})$ computable features in width $d$ via an approximate-linear recursive template, while Adler and Shavit reach near-quadratic capacity (up to logarithmic factors) using thresholded Boolean recovery. The main contribution of this paper is conceptual: we argue these results are not contradictory because they maintain different interface invariants, and we formalize the distinction. As a tool, we record a rank-trace Welch-type lower bound for biorthogonal linear readouts: for $F \gg d$, the worst-case off-diagonal cross-talk of any unit-diagonal linear readout is $\Omega(d^{-1/2})$, and the bound is tight on average for unit-norm tight frames. At quadratic feature load $F=d^2$, random-support threshold recovery succeeds for sparsities $s=O(d/\log d)$, while linear readouts still incur $\Omega(s/d)$ average per-coordinate squared error on Bernoulli sparse states. Matching the Welch floor against the published tolerance of the H\"anni correction layer explains the $d^{3/2}$ scale as a compatibility threshold for that template, not a universal upper bound. Robust nonlinear reset beyond the H\"anni template is left open.