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

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 Rényi 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 Rényi 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 nontrivial questions in the context of classical statistical mechanics.

Thermodynamic state convertibility is determined by qubit cooling and heating

“…where [n] is a shorthand notation for {1, 2, 3, • • • , n}, and p ↓ is the vector with the absolute values of the entries of p arranged in non-increasing order. A technical tool with many applications and generalizations in the quantum regime [45][46][47][48][49][50][51][52][53][54][55][56][57][58] on which we rely is the concept of (relative) majorization: Let p, r ∈ Prob(n) and q, s ∈ Prob(m). We say that (p, r) relatively majorizes (q, s), written as (p, r) (q, s), iff there exists a m × n column stochastic matrix E such that Ep = q, Er = s.…”

Measuring magic on a quantum processor

“…This means that the power of quantum advance requires resources beyond the Clifford group, like the Phase π/8 gate (T gate) or the Toffoli gate and non-Gaussian states for the MCGs 14,15 . The precious resource that makes quantum computers special is colloquially dubbed as 'magic' and a resource theory of magic has been developed in recent years [2][3][4][16][17][18][19][20][21][22][23][24][25][26][27] .…”

Thermodynamic state convertibility is determined by qubit cooling and heating

“…where [n] is a shorthand notation for {1, 2, 3, • • • , n}, and p ↓ is the vector with the absolute values of the entries of p arranged in non-increasing order. A technical tool with many applications and generalization in the quantum regime [38][39][40][41][42][43][44][45][46][47][48][49][50] on which we rely is the concept of (relative) majorization: Let p, r ∈ Prob(n) and q, s ∈ Prob(m). We say that (p, r) relatively majorizes (q, s), written as (p, r) (q, s), iff there exists a m × n column stochastic matrix E such that Ep = q, Er = s.…”