Extremizing Measures of Magic on Pure States by Clifford-stabilizer States

TL;DR

This paper develops a framework for understanding extremal properties of magic states in quantum computing, showing Clifford-stabilizer states extremize many magic measures and proposing a new distillation protocol.

Contribution

It introduces a general group-covariant framework for magic measures, classifies extremal states, and proposes a new distillation protocol for two-qubit magic states.

Findings

Clifford-stabilizer states extremize key magic measures.

New candidates for magic distillation are identified.

An inefficient distillation protocol surpassing benchmarks is proposed.

Abstract

Magic states enable universal, fault-tolerant quantum computation within the stabilizer framework. Their non-stabilizerness supplies the resource needed to bypass the Eastin-Knill theorem while allowing fault-tolerant distillation. Although many measures of magic exist, not every nonzero-magic state is known to be distillable, and many of currently known distillable states are special cases of Clifford-stabilizer states, defined as pure states uniquely stabilized by finite Clifford subgroups. We develop a general framework for group-covariant functionals on Hermitian operators, introducing the notions of $G$-stabilizer spaces, states, and codes for arbitrary finite subgroups $G \subset \mathrm{U}(\mathcal{H})$. We define analytic families of $G$-covariant functionals and prove that any $G$-invariant pure state is extremal for a broad class of derived functionals, including symmetric, max-type, and R\'enyi-type functionals, whenever the underlying family is $G$-covariant. This extremality holds for purity-preserving variations orthogonal to the stabilized subspace. Specializing to the Pauli and Clifford groups, we recover the extremality structure of canonical magic measures such as mana, stabilizer R\'enyi entropies, generalized R\'enyi entropies, and stabilizer fidelity, and show that Clifford-stabilizer states extremize them. We classify such states for qubits, qutrits, ququints, and two-qubit systems, identifying new candidates for magic distillation. We further propose an inefficient distillation protocol for a two-qubit magic state with stabilizer fidelity exceeding standard benchmarks, and conjecture that SIC-POVM fiducial states are Clifford-stabilizer states.