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

TL;DR

The paper develops a general theory of group-covariant functionals on Hermitian operators and proves that G-invariant pure states are extremal for broad classes of functionals, including mana, stabilizer fidelity, and stabilizer Rényi entropies, when variations are restricted orthogonally to the stabilized subspace. Specializing to Pauli and Clifford groups, Clifford-stabilizer states emerge as universal extremizers, unifying several canonical magic measures and exposing new distillation candidates across qubits and qudits. The authors classify non-degenerate Clifford eigenstates for single qudits (d=2,3,5) and two qubits, and provide explicit examples with new distillation possibilities, including an inefficient protocol that distills a two-qubit magic state with superior stabilizer fidelity. Beyond theory, they introduce the notion of group-stabilizer extent, broaden the resource-theoretic framework, and outline concrete open problems such as complete two-qubit Clifford-stabilizer classifications and more efficient distillation protocols. The work thus links phase-space structure, group symmetry, and operational magic into a cohesive geometric picture with potential impact on fault-tolerant universal quantum computation.

Abstract

Magic states are essential resources enabling universal, fault-tolerant quantum computation within the stabilizer framework. Their non-stabilizerness provides the additional resource required to overcome the constraints of stabilizer codes, as formalized by the Eastin-Knill theorem, while still permitting fault-tolerant distillation. Although numerous measures of magic have been introduced, not every state with nonzero magic has been shown to be distillable by a stabilizer code, and all currently known distillable states arise as special cases of Clifford-stabilizer states, defined as pure states uniquely stabilized by finite subgroups of the Clifford group. In this work, we develop a general framework for group-covariant functionals on the real manifold of Hermitian operators. We formalize the notions of $G$-stabilizer spaces, states, and codes for arbitrary finite subgroups $G \subset \mathrm{U}(\mathcal{H})$, and introduce analytic families of $G$-covariant functionals. Our main theorem shows that any $G$-invariant pure state is an extremal point of a broad class of derived functionals, including symmetric, max-type, and Rényi-type functionals, provided the underlying family is $G$-covariant. This extremality holds for variations restricted to directions orthogonal to the stabilized subspace while preserving purity. Specializing to the Pauli and Clifford groups, our framework unifies the extremality structure of several canonical magic measures, including mana, stabilizer Rényi entropies, and stabilizer fidelity. In particular, Clifford-stabilizer states extremize these measures. We classify such states for qubits, qutrits, ququints, and two-qubit systems, identifying new candidates for magic distillation protocols. We further propose an inefficient distillation protocol for a two-qubit magic state with stabilizer fidelity exceeding that of standard benchmark states.

Figures (4)

• Figure 1: Stabilizer fidelity for all single-qubit states. Each state is plotted in its Bloch sphere representation, but instead of a unit radius, the radial coordinate corresponds to the stabilizer fidelity value. The black points represent the stabilizer states. The red points indicate the $T$-states, while the blue points correspond to the $H$-states. The plot clearly shows that the $T$-states form sharp minima in the stabilizer fidelity function. Similarly, the $H$-states also create sharp minima in all but one direction. In that particular direction, however, the $H$-states give rise to a smooth maximum instead.
• Figure 2: Schematic illustration of the distillation protocol for the state $\ket{\psi_{00}}$. (a) The eight stabilizers of the full ten-qubit system, showing systems $A$ (blue) and $B$ (green) separately, each encoding a logical qubit via the five-qubit perfect code. (b) The same stabilizers represented after reordering the physical qubits according to the convention described in the main text. (c) Preparation stage: each pair $\mathcal{H}_i$ is initialized in the imperfect state $\rho$ defined in Eq. (\ref{['eq:faulty rho']}) or (\ref{['eq:rho_initial']}), representing a dephased noisy version of $\ket{\psi_{00}}$. (d) Conceptual illustration of the distillation step: simultaneous stabilizer measurements project both subsystems onto their respective code spaces whenever all syndromes are trivial (i.e., all stabilizer outcomes are $+1$). Conditioned on this outcome, the resulting logical qubits in $\mathcal{H}$ exhibit a reduced error rate, with the dominant contribution suppressed linearly in the original preparation error.
• Figure 3: Wigner function heatmap visualizations for all nonstabilizer Clifford-inequivalent non-degenerate eigenstates of Clifford operations of single qutrits.
• Figure 4: Wigner function heatmap visualizations for all nonstabilizer Clifford-inequivalent non-degenerate eigenstates of Clifford operations of single ququints.

Theorems & Definitions (19)

• Definition 1: Stabilization
• Definition 2: Stabilized Space of an Operator
• Definition 3: $G$-Invariant Subspace
• Definition 4: $G$-Stabilizer Code
• Definition 5: $G$-Stabilizer State
• Definition 6: Twirling
• Definition 7: Group Covariance
• Definition 8: Extremal Point
• Theorem 1
• Definition 10: Componentwise Map