Constraints on magic state protocols from the statistical mechanics of
Wigner negativity
Wigner negativity
File(s)
Accepted version
Author(s)
Koukoulekidis, Nikolaos
Jennings, David
Type
Working Paper
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.
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.
Date Issued
2022-03-14
Citation
2022
ISSN
2056-6387
Publisher
Nature Research
Copyright Statement
©2022 The Author(s)
Identifier
http://arxiv.org/abs/2106.15527v1
Subjects
quant-ph
quant-ph
cond-mat.stat-mech
Notes
31 pages, 7 Figures. Comments welcome
Publication Status
Published