Nonstabilizer eigenstates of Clifford operators are natural candidates for endpoints of magic-state distillation routines. We provide an explicit bestiary of all inequivalent nonstabilizer Clifford eigenstates for qutrits and ququints. For qutrits, there are four nondegenerate eigenstates and two families of degenerate eigenstates. For ququints, there are eight nondegenerate eigenstates and three families of degenerate eigenstates. Of these states, a simultaneous eigenvector of all Clifford symplectic rotations known as the qutrit strange state is distinguished as both the most magic qutrit state and the most symmetric qutrit state. We show that no analog of the qutrit strange state (i.e., no simultaneous eigenvector of all symplectic rotations) exists for qudits of any odd-prime dimension $d>3$.
Nonstabilizer eigenstates of Clifford operators are natural candidates for endpoints of magic-state distillation routines. We provide an explicit bestiary of all inequivalent nonstabilizer Clifford eigenstates for qutrits and ququints. For qutrits, there are four nondegenerate eigenstates and two families of degenerate eigenstates. For ququints, there are eight nondegenerate eigenstates and three families of degenerate eigenstates. Of these states, a simultaneous eigenvector of all Clifford symplectic rotations known as the qutrit strange state is distinguished as both the most magic qutrit state and the most symmetric qutrit state. We show that no analog of the qutrit strange state (i.e., no simultaneous eigenvector of all symplectic rotations) exists for qudits of any odd-prime dimension d>3.
Distinct Clifford orbits of magic states can exhibit different stabilizer ranks at small tensor powers. We establish this for qutrits, where the single-qutrit Clifford group has four inequivalent orbits of magic states: Strange, Norrell, Hadamard-eigenstate, and the qutrit T-state, but a nontrivial upper bound on the asymptotic exponent had been pinned down for only the qutrit T-state. For the other three orbits we give explicit stabilizer decompositions, yielding upper bounds on the per-copy asymptotic stabilizer-rank exponent: γS≤log3(2)/2≈0.316 for the Strange state, and γH3,γN≤log3(4)/3≈0.421 for the Hadamard-eigenstate and Norrell orbits, all strictly below the prior γT3≤1/2 baseline. We also prove the first nontrivial Ω(m/logm) asymptotic lower bounds for the Hadamard-eigenstate and Norrell orbits, and exhibit two-qutrit Clifford circuits that convert two copies of these states into an injectable phase state with constant success probability, enabling constant-overhead injection of one non-Clifford diagonal gate per orbit. In the case of qubits, we give a closed-form decomposition of the qubit T-type orbit at four copies matching the existing γT≤log2(3)/4≈0.396 exponent via a direct algebraic identity rather than an entangled cat-state construction. An open-source library stabrank accompanies the paper, with Lean 4 proof formalizations of all the decompositions.
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⊂U(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 Rényi-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 Rényi entropies, generalized Rényi 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.
Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all [[n,1]]3 qutrit stabilizer codes with a complete set of transversal Clifford gates for n≤23, and all [[n,1]]3 stabilizer codes with a transversal H2 gate with n≤9 qudits. For n=23, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.
Achieving a genuine quantum advantage relies on two distinct non-classical resources that restrict efficient classical simulation: entanglement and magic (nonstabilizerness). We investigate the interplay between these resources by characterizing the Pareto frontiers of extreme magic at fixed entanglement for systems of two qutrits (d=3) and two ququints (d=5). Unlike the case of two qubits, the Schmidt spectrum for two qutrits features two independent entanglement parameters, resulting in two-dimensional Pareto surfaces. For the lower frontier, we recast the minimal magic as a compact function of concurrence and negativity, with a maximal value of ln2. For the upper frontier, we determine the maximal stabilizer Rényi entropy to be M2=ln(81/17)≈1.561, which tightens the previous theoretical bound of ln5≈1.609 and improves on earlier numerical estimates. The maximum magic is achieved at eighteen distinct maxima categorized into three families of six permutation-equivalent spectra. We provide analytical expressions for the maximal magic in the neighborhood of each maximum and for the corresponding maximally magical states which turn out to be Weyl-Heisenberg-covariant fiducial states for mutually unbiased bases. Finally, numerical analysis of two ququints (d=5) reveals six permutation-inequivalent maxima with a peak magic value of M2=ln(625/49)≈2.546. Based on these findings, we conjecture that the maximal magic for a bipartite system of two qudits with prime dimension d is given by ln[d4/(2d2−1)], which reproduces the previously known value for qubits, as well as the values derived here for qutrits and ququints.
We propose families of protocols for magic-state distillation—important components of fault-tolerance schemes—for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic-state-distillation scheme yet discovered. It excels both in terms of error threshold with respect to depolarizing noise (36.3%) and the efficiency measure known as yield, where, for a large region of parameters, it outperforms its qubit counterpart by many orders of magnitude.
We present applications of quantum quadratic residue codes in magic state distillation. This includes showing that existing codes which are known to distill magic states, like the 5-qubit perfect code, the 7-qubit Steane code, and the 11-qutrit and 23-qubit Golay codes, are equivalent to certain quantum quadratic residue codes. We also present new examples of quantum quadratic residue codes that distill qubit T states and qutrit Strange states with high thresholds, and we show that there are infinitely many quantum quadratic residue codes that distill T states with a non-trivial threshold. All of these codes, including the codes with the highest currently known thresholds for T state and Strange state distillation, are unified under the umbrella of quantum quadratic residue codes.