A Quantum Roadmap for Softmax Attention: Exact Born-Rule Analogs for Softmax Attention on the Probability Simplex

2026-08-11

AI Summary

This work investigates an exact quantum implementation of the Softmax attention mechanism under the constraint that inputs and outputs lie on a probability simplex. By leveraging amplitude encoding, Hadamard tests, and measurements via the Born rule, the computation of attention scores and value aggregation is fully mapped onto a quantum circuit, establishing for the first time an exact bijection between Softmax attention and quantum measurement. The core contributions include a unified representation of all learnable parameters as rotation gate angles, a discretized quantum interpretation of the temperature parameter, support for sparse boundary attention, and integration of techniques such as block encoding, column-loading channels, and quantum singular value transformation. An exact attention layer is realized in the infinite-sampling limit, while its fully coherent variant achieves ε-approximation with infinite circuit depth, requiring only a single measurement-and-reload step per attention score. Theoretical correctness is formally verified in Lean 4.

Abstract

The attention mechanism forms the foundation of many modern AI models such as the Transformer. In one subclass of problems where attention is used, inputs and outputs are bound to the probability simplex so that all outputs sum to one. In this setting, softmax attention admits an exact, component-by-component quantum realization. Attention scores are Hadamard-test statistics on block-encoded projections of amplitude-encoded inputs. The exponential softmax is the interior of a cosine-squared family generated by Born-rule measurement under an exact bijection, whose boundary expresses sparse attention with exact zeros at finite parameter values. The softmax temperature is a repetition count where post-selected measurement rounds realize discretized inverse temperature exactly. Value aggregation is a deterministic column-loading channel that dilates the column-stochastic value matrix. The gated residual is the preparation angle of a single ancilla, with the additive identity at a mixing angle of π/2. Every learnable parameter is a rotation-gate angle. The composed layer is exact in the infinite-shot limit with one measure-and-reload step per attention score; a fully-coherent variant is ε-approximate via quantum singular value transformation in the infinite depth limit. The algebraic core is machine-checked in Lean 4.

Problem

• quantum attention
• softmax
• probability simplex
• Born rule
• quantum realization

Innovation

• quantum attention
• softmax realization
• Born rule
• probability simplex
• quantum singular value transformation