Stabilizer rank bounds for magic-state orbits

Abstract

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: $γ_S \le \log_3(2)/2 \approx 0.316$ for the Strange state, and $γ_{H_3}, γ_N \le \log_3(4)/3 \approx 0.421$ for the Hadamard-eigenstate and Norrell orbits, all strictly below the prior $γ_{T_3} \le 1/2$ baseline. We also prove the first nontrivial $Ω(m / \log m)$ 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 $γ_T \le \log_2(3)/4 \approx 0.396$ 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.