Constraints on magic state protocols from the statistical mechanics of Wigner negativity

TL;DR

This paper introduces a statistical mechanical framework based on majorization to analyze Wigner negative magic states in quantum computing, providing tighter bounds on magic distillation and new insights into non-classicality measures.

Contribution

It develops a majorization-based approach to quantify Wigner negativity, derive bounds on magic state distillation, and connect these concepts with statistical mechanics and Re9nyi entropies.

Findings

Majorization provides tighter bounds than mana and thauma.

Certain Re9nyi entropies remain meaningful and can be negative.

Mana is linked to divergence of Re9nyi entropies near Shannon entropy.

Abstract

Magic states are key ingredients in schemes to realize universal fault-tolerant quantum computation. Theories of magic states attempt to quantify this computational element via monotones and determine how these states may be efficiently transformed into useful forms. Here, we develop a statistical mechanical framework based on majorization to describe Wigner negative magic states for qudits of odd prime dimension processed under Clifford circuits. We show that majorization allows us to both quantify disorder in the Wigner representation and derive upper bounds for magic distillation. These bounds are shown to be tighter than other bounds, such as from mana and thauma, and can be used to incorporate hardware physics, such as temperature dependence and system Hamiltonians. We also show that a subset of single-shot R\'{e}nyi entropies remain well-defined on quasi-distributions, are fully meaningful in terms of data processing and can acquire negative values that signal magic. We find that the mana of a magic state is the measure of divergence of these R\'{e}nyi entropies as one approaches the Shannon entropy for Wigner distributions, and discuss how distillation lower bounds could be obtained in this setting. This use of majorization for quasi-distributions could find application in other studies of non-classicality, and raises novel questions in the context of classical statistical mechanics.