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arXiv:2605.28586

Stabilizer rank bounds for magic-state orbits

Farrokh Labib, Vincent Russo

quant-phSubmitted May 27, 2026• Announced May 28, 2026• 0 citations
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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.

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An explicit ten-term stabilizer decomposition of |T>^{(x)7}: chi(|T>^{(x)7}) <= 10

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Clifford Invariants and Three-Copy Quantum Tests

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Qutrit and ququint magic states

Akalank Jain, Shiroman Prakash

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 $d>3$.

arXiv:2003.07164•Physical Review A•Oct 23, 2020•25 citations
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≤log⁡3(2)/2≈0.316γ_S \le \log_3(2)/2 \approx 0.316γS​≤log3​(2)/2≈0.316 for the Strange state, and γH3,γN≤log⁡3(4)/3≈0.421γ_{H_3}, γ_N \le \log_3(4)/3 \approx 0.421γH3​​,γN​≤log3​(4)/3≈0.421 for the Hadamard-eigenstate and Norrell orbits, all strictly below the prior γT3≤1/2γ_{T_3} \le 1/2γT3​​≤1/2 baseline. We also prove the first nontrivial Ω(m/log⁡m)Ω(m / \log m)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≤log⁡2(3)/4≈0.396γ_T \le \log_2(3)/4 \approx 0.396γT​≤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.
We prove that every pure state of stabilizer rank at most kkk has stabilizer extent at most 2O(klog⁡(k+1))2^{O(\sqrt{k\log(k+1)})}2O(klog(k+1)​), and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-kkk operators. This implies stabilizer fidelity at least 2−O(klog⁡(k+1))2^{-O(\sqrt{k\log(k+1)})}2−O(klog(k+1)​), resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an Ω(n2/log⁡n)Ω(n^2/\log n)Ω(n2/logn) lower bound for the approximate stabilizer rank of tensor powers of any non-stabilizer qubit state. The latter result generalizes the best-known lower bound for tensor powers of TTT-states (Mehraban-Tamasbi, STOC, 2024) to arbitrary non-stabilizer qubit states, including magic states. As a consequence of the Clifford rank--norm inequality, we obtain an Ω(n2/log⁡n)Ω(n^2/\log n)Ω(n2/logn) lower bound for exact representations of nnn-bit AND function by quadratic phases, improving the previous best-known linear bound. Further consequences rule out pseudorandom state and unitary ensembles with approximate stabilizer and Clifford rank O((log⁡n)2/log⁡log⁡n)O((\log n)^2/\log\log n)O((logn)2/loglogn), respectively, a log⁡n\log nlogn improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most kkk, with poly(n)2O(klog⁡(k+1))poly(n)2^{O(\sqrt{k\log(k+1)})}poly(n)2O(klog(k+1)​) time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.
We give an explicit construction proving that the exact stabilizer rank of mmm copies of the qubit face-center magic state is O(23m/8)O(2^{3m/8})O(23m/8). Two symmetry-related fifteen-term identities combine into a twenty-four-term decomposition at twelve copies. The balanced structure of this decomposition yields a substitution that adds eight copies at a cost of eight terms per existing term. The resulting bound γT≤3/8γ_T \leq 3/8γT​≤3/8 improves the previously published exponent log⁡2(3)/4\log_2(3)/4log2​(3)/4.
Short note with supporting code and data on Open Problem Zoo record op_37f7b003ec78a028 (stabilizer rank of tensor powers of the magic state |T> = (|0> + e^{i pi/4}|1>)/sqrt2). The exact stabilizer rank chi(|T>^{(x)m}) is known for m ≤ 4 (2, 2, 3, 4); at m = 5, 6, 7 only the upper bounds 6, 6, 12 were known, the last being the product of the six-copy and one-copy decompositions (no direct search beyond six copies had been reported). This deposit gives an explicit decomposition of |T>^{(x)7} into ten stabilizer states, hence chi(|T>^{(x)7}) ≤ 10, and by sub-multiplicativity chi(|T>^{(x)13}) ≤ 60 (previously 72). The ten states are symmetric under permutations within the qubit blocks {1,2} and {3,...,7}; they are listed in Dehaene-De Moor normal form with exact coefficients in (1/4) Z[e^{i pi/4}]. The decomposition was found by an autonomous AI research agent (the depositor's EVO harness) that closed three seed states under block-symmetric Clifford gates to obtain a dictionary of a few hundred stabilizer states spanning the relevant symmetric subspace and then ran a subset search over it. Verification: exact Gaussian elimination over Q(e^{i pi/4}) (a proof of the identity), a 40-digit least-squares residual of 1.6e-41, and a brute-force recount of the stabilizer group of every state. Companion searches give evidence, not proof, that chi(|T>^{(x)5}) = 6. Claim boundary: an upper bound with an explicit certificate; no optimality, no asymptotic statement, no priority claim. Files: the note (PDF + LaTeX), the decomposition as JSON and as a readable listing, the exact verifier and the independent check with its log, the agent's search code and notes, the task text, SHA-256 sums.
We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored TTT gate magic state. This lowers its asymptotic bound to 2∼0.463t2^{\sim 0.463 t}2∼0.463t for multi-Pauli measurements on ttt magic states, improving over the best previously found bound of 2∼0.468t2^{\sim 0.468 t}2∼0.468t. We numerically demonstrate this reduction. This constructively produces the most efficient strong simulation algorithm of the Clifford+TTT gateset to relative or multiplicative error. We then examine the cost of Pauli measurement in terms of its Gauss sum rank, which is a slight generalization of the stabilizer rank and is a lower bound on its asymptotic scaling. We demonstrate that this lower bound appears to be tight at low ttt-counts, which suggests that the stabilizer rank found at the twelve-qubit state can be lowered further to 2∼0.449t2^{\sim 0.449 t}2∼0.449t and we prove and numerically show that this is the case for single-Pauli measurements. Our construction directly shows how the reduction at 121212 qubits is iteratively based on the reduction obtained at 666, 333, 222, and 111 qubits. This explains why novel reductions are found at tensor factors for these number of qubit primitives, an explanation lacking previously in the literature. Furthermore, in the process we observe an interesting relationship between the T gate magic state's stabilizer rank and decompositions that are Clifford-isomorphic to a computational sub-basis tensored with single-qubit states that produce minimal unique stabilizer state inner products -- the same relationship that allowed for finding minimal numbers of unique Gauss sums in the odd-dimensional qudit Wigner formulation of Pauli measurements.
We give an explicit treatment of the third tensor power of the Clifford representation on nnn qutrits, for n≥2n\geq2n≥2. This representation has five pairwise inequivalent irreducible constituents, with multiplicities 1,1,1,1,21,1,1,1,21,1,1,1,2. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective 333-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.
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>3d>3d>3.