Algebraic paradoxes in adaptive quantum computation

2026-07-28

AI Summary

This work addresses a key gap in the algebraic analysis of adaptive measurement-based quantum computation (MBQC), where contextuality is believed to underpin quantum advantage yet suitable tools for adaptive settings have been lacking. By modeling adaptive protocols as ordinary measurements within tree-structured measurement scenarios and integrating ℤ₂ linear algebra with cohomological methods, the authors extend cohomological techniques to adaptive MBQC for the first time. They constructively establish an algebraic criterion for strong contextuality, proving that any adaptive MBQC protocol deterministically computing a non-affine Boolean function necessarily corresponds to an inconsistent system of linear equations. This inconsistency manifests as an algebraic paradox, thereby revealing the protocol’s intrinsic strong contextuality and resolving an open problem posed by Raussendorf.

Abstract

Measurement-based quantum computation (MBQC) is a universal model of quantum computation whose full power requires adaptivity. Contextuality is known to power quantum advantage in MBQC, yet it has resisted algebraic analysis in the adaptive setting. We show that if an adaptive $\mathbb{Z}_2$-linear measurement-based quantum computing protocol deterministically computes a non-affine Boolean function, then the underlying quantum resource satisfies an inconsistent set of linear equations. This witnesses an algebraic form of strong contextuality generalising Mermin's All-versus-Nothing arguments. Such algebraic contextuality can be detected cohomologically, resolving an open question posed by Raussendorf, who had established cohomological witnesses of contextuality for non-adaptive protocols, but left the adaptive case open. We prove this result constructively: we model adaptive measurement protocols as ordinary measurements on a larger scenario of tree-like measurements, and explicitly build the inconsistent equations inductively.

Problem

• adaptive quantum computation
• contextuality
• algebraic paradoxes
• measurement-based quantum computation
• Boolean functions

Innovation

• adaptive MBQC
• algebraic contextuality
• cohomological witness
• non-affine Boolean function
• inconsistent linear equations