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: $γS \le \log_3(2)/2 \approx 0.316$ for the Strange state, and $γ{H_3}, γN \le \log_3(4)/3 \approx 0.421$ 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.

• The paper determines that stabilizer rank varies by orbit, with qubit T-type states showing lower ranks than H-type and qutrit orbits exhibiting distinct bounds.
• It employs explicit algebraic decompositions and rigorous Lean 4 formal verification to certify tight upper and lower bounds at low tensor copies.
• The work introduces concrete magic-state injection protocols for non-Clifford gate conversion, underscoring significant implications for simulation efficiency and fault tolerance.

Stabilizer Rank Bounds for Magic-State Orbits: An Expert Analysis

Motivation and Problem Setting

The paper investigates the stabilizer rank of magic states under tensor powers, explicitly analyzing its dependence on each Clifford orbit for both qubit and qutrit systems (2605.28586). Stabilizer rank is a pivotal quantity governing the efficiency of classical strong simulation algorithms for Clifford-dominated quantum circuits augmented with magic-state ancillae. If $\chi(M^{\otimes t})$—the number of stabilizer states required to span $M^{\otimes t}$—is low, then classical simulation becomes tractable for larger $t$, constraining viable quantum speedup. Conversely, lower bounds on $\chi(M^{\otimes t})$ certify computational hardness. The paper sets out to resolve longstanding ambiguities for qutrit magic states, pinning down upper and lower bounds for all four Clifford-inequivalent orbits: Strange, Norrell, Hadamard-eigenstate, and the qutrit $T$-state.

Main Technical Contributions

Orbit-Dependent Stabilizer Rank Bounds in Qubits

For qubits, the Clifford group yields two inequivalent magic-state orbits ($H$-type and $T$-type). The authors rigorously show that at small tensor powers ($m=4$), the stabilizer rank for $T$-type magic states is strictly below that of $H$-type ($M^{\otimes t}$0 vs $M^{\otimes t}$1), thus demonstrating orbit dependence at finite $M^{\otimes t}$2. Notably, the explicit algebraic decomposition for the $M^{\otimes t}$3-type orbit at $M^{\otimes t}$4 is given, matching the asymptotic exponent $M^{\otimes t}$5, without reliance on entangled cat-state constructions.

Orbit-Dependent Stabilizer Rank Bounds in Qutrits

For qutrits, there are four Clifford orbits of magic states. Prior work established a bound only for the $M^{\otimes t}$6-state orbit ($M^{\otimes t}$7). This paper provides explicit stabilizer decompositions for the remaining three orbits:

• Strange state: $M^{\otimes t}$8 for $M^{\otimes t}$9, the lowest known exponent across all qutrit orbits.
• Hadamard-eigenstate and Norrell orbits: $t$0 for $t$1, both strictly below the $t$2 baseline.

Exhaustive certificate searches confirm tight values at small $t$3, e.g. $t$4 for $t$5.

Asymptotic Lower Bounds

Transferring the subset-sum argument from qubits to qutrits, the paper establishes the first nontrivial lower bounds: $t$6 for the Hadamard-eigenstate and Norrell orbits, given their modulus ratio structure in computational basis amplitudes. The Strange orbit, whose amplitudes have equal modulus and support cardinality two, is shown to be rigid and excluded from this technique.

Magic-State Injection Protocols

The operational utility of these rank reductions is addressed via magic-state injection. Standard single-shot injection is feasible only for phase states, i.e., the $t$7-orbit. For Hadamard-eigenstate and Norrell states, two-copy probabilistic conversion circuits are explicitly constructed, projecting two copies onto phase states with constant success probability ($t$8 for $t$9, $\chi(M^{\otimes t})$0 for Norrell). These enable constant-overhead injection of non-Clifford diagonal gates per orbit. Rigorous exhaustive searches confirm that the Strange orbit remains rigid and non-convertible under low-copy protocols, rendering its low exponent operationally inaccessible.

Software and Formal Verification

All decomposition identities and certificates are formalized in Lean 4, with accompanying code and verification via the open-source library stabrank [stabrank-software]. Machine-checked proofs bolster the credibility of the algebraic decompositions, and exhaustive search results ensure the tightness of bounds at small $\chi(M^{\otimes t})$1.

Strong Numerical Results and Contradictory Claims

• Explicit upper bounds: For qutrits, the Strange state orbit achieves $\chi(M^{\otimes t})$2, strictly below all others. Hadamard-eigenstate and Norrell are $\chi(M^{\otimes t})$3, both below $\chi(M^{\otimes t})$4.
• Lower bounds: $\chi(M^{\otimes t})$5 for Hadamard-eigenstate and Norrell, in contrast to their upper bounds.
• Operational accessibility: Despite the lowest upper-bound exponent for the Strange orbit, exhaustive searches contradict its operational usability—no injection protocol exists for low-copy consumption, even at three copies.

Theoretical Implications

The orbit dependence of stabilizer rank at both small $\chi(M^{\otimes t})$6 and in asymptotic exponents demonstrates that “magic” is not a monolithic resource but highly sensitive to the algebraic structure of Clifford orbits. The separation in exponents underpins the resource theory of quantum computation and contextuality, and also delineates which magic states are practically accessible for quantum algorithms and error correction protocols. The rigidity of the Strange orbit may provide paths to new fault-tolerance schemes, or drive further resource-based analysis in measurement-based quantum computation for qudits.

Practical Implications for Quantum Simulation

Tighter stabilizer rank bounds inform classical simulation algorithms, directly translating to reduced exponential scaling in classical runtime for simulating Clifford-dominated quantum circuits with magic-state ancillae. The explicit injection protocols constructed provide clear recipes for achieving non-Clifford gates in qutrit architectures, with constant overhead. The existence of phase-state conversion protocols for specific orbits expands operational routes for non-Clifford resource injection, crucial for quantum hardware in higher dimensions.

Prospects for Future Developments

• Lower bounds for the Strange orbit: New techniques, potentially from stabilizer nullity or monotone analysis, will be required to establish asymptotic lower bounds for the Strange orbit.
• Approximate stabilizer rank: Extending recent probabilistic lower bounds for approximate stabilizer rank in qubits [mehraban2023quadratic, kalra2025stabilizer] to qutrits.
• Alternate consumption models: The Strange orbit’s rigidity raises the question of whether its low stabilizer rank can be harnessed in non-standard quantum computing paradigms (e.g., non-adaptive MBQC or outcome-deterministic models).
• Clifford conversion protocols: Whether a physical Clifford conversion protocol exists for the Strange orbit at higher copy numbers remains open.
• Resource theory refinement: The explicit orbit classification and tight bounds encourage refined resource monotone analysis and contextuality classification for arbitrary qudit systems.

Conclusion

This work rigorously establishes orbit-dependent stabilizer rank bounds for magic states in both qubits and qutrits, providing explicit algebraic decompositions, certificates, and machine-verified identities. It resolves prior ambiguities for qutrit orbits, gives both upper and lower bounds, and connects these to operational injection protocols. The results sharpen both classical simulation algorithms and resource theory for quantum computation. Future directions include the extension of lower bound techniques, approximate rank analysis, and exploration of non-standard consumption models for rigid orbits.