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arXiv:2610.31115

The limits of magic-state distillation

Llorenç Balada Gaggioli, Jakub Mareček

quant-phSubmitted Oct 22, 2026• Announced Oct 23, 2026• 0 citations
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Abstract

Magic-state factories are a major cost of fault-tolerant quantum computing, yet how few noisy inputs they need is known only for restricted protocols. We show that, for tolerating input faults, any stabilizer protocol, however adaptive, postselected or multistage, is no better than a projection onto one stabilizer code, and that for rational targets a fault pattern is harmful exactly when its complement is. Searches over all codes show that quadratic distillation of one T state needs exactly nine inputs and that the eight-input CCZ factory is optimal. At order $d$, one T state needs $n\geq2d$.

Comments
5 pages, 10 pages of Supplemental Material, 1 figure, 3 tables

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Magic-state factories are a major cost of fault-tolerant quantum computing, yet how few noisy inputs they need is known only for restricted protocols. We show that, for tolerating input faults, any stabilizer protocol, however adaptive, postselected or multistage, is no better than a projection onto one stabilizer code, and that for rational targets a fault pattern is harmful exactly when its complement is. Searches over all codes show that quadratic distillation of one T state needs exactly nine inputs and that the eight-input CCZ factory is optimal. At order ddd, one T state needs n≥2dn\geq2dn≥2d.
Magic-state distillation is a major resource cost in fault-tolerant quantum computing. The cost of a magic-state factory depends strongly on its failure rate, which grows with the number of input magic states. Although symmetry-restricted methods have recently made distance two searches tractable, distance three and above have remained elusive at moderate input counts. We develop symmetry- and AI-assisted methods to search this regime. We present a unified binary-matrix formulation encompassing both triorthogonal-code and direct circuit searches. We show that distance at least three is equivalent to nonzero, pairwise distinct syndromes, separating the choice of syndromes from the search for compatible output gates. We restrict the syndrome search using group symmetry and language-model agents, followed by deterministic solving and independent verification. Our searches yield 699 factory classes, including 564 new ones. These include factories for pure-T states and factories with entangled outputs comprising combinations of T, CS, and CCZ magic states. The pure-T factories [[63, 11, 3]] and [[850, 128, 6]] achieve the lowest overhead exponents we know among protocols with at most 100 and 1000 inputs, respectively, with γ=1.589γ= 1.589γ=1.589 and γ=1.057γ= 1.057γ=1.057. Our [[1715, 287, 6]] factory, with γ=0.998γ= 0.998γ=0.998, is the smallest known pure-T factory with γ<1γ< 1γ<1. We also provide a context directory of search briefs and campaign notes with which readers can train their own agents and tailor the search to their requirements. With these results, we begin constructing an active, open-source repository of magic-state distillation protocols for the quantum community, supplemented by our methods and data, for the practical fault-tolerant quantum computing regime.
Magic-state distillation is one of the leading overheads in fault-tolerant quantum computation. Existing methods for finding distillation factories require a transversal gate to act correctly on the entire codespace, a constraint that limits both generality and search efficiency. We introduce a strictly weaker borrowed-identity condition, requiring only that the distillation circuit act as the identity on a single input state. It applies uniformly across all levels of the Clifford hierarchy and unifies, within a single level, factories that distill different magic states -- for example, the ∣T⟩|T\rangle∣T⟩, ∣CS⟩|CS\rangle∣CS⟩, and ∣CCZ⟩|CCZ\rangle∣CCZ⟩ factories. A brute-force search over borrowed-identity circuits with two-group symmetry recovers, within the search range, all distance-2 factories known from code-construction approaches, including entangled-output and multi-output factories previously outside the scope of any single numerical search. This unification yields parent circuits that encode multiple factories, so the output magic-state type can be chosen at compile time rather than fixed by a hard-coded design. The framework also extends beyond CSS codes, unifying constructions, including synthillation and non-CSS catalytic factories, previously obtained by disparate approaches.
Producing high-fidelity magic states using the smallest possible number of physical qubits and operations stands as a very important challenge to achieve fault-tolerant quantum computation at scale. Besides emerging proposals for alternative methods such as cultivation, magic state distillation remains essential for achieving very low error rates. Known distillation protocols are usually built through quantum codes derived from triorthogonal matrices. Here, exploiting the specific noise structure present in magic state distillation protocols, we show that classical error-correcting codes offer a simpler framework for deriving these protocols. This formulation is particularly well suited to systematic numerical and analytical studies of distillation protocols involving a fixed number of qubits. Specifically, we use a SAT solver to derive a series of no-go theorems that relate key figures of merit, including the number of qubits and the factory distance. In particular, considering the standard implementation as a circuit of Z-diagonal Pauli product rotations followed by measurements, we show that no TTT-to-TTT distillation protocol on fewer than eight qubits can exceed distance 3, and no TTT-to-CCZ\mathrm{CC}ZCCZ protocol distance 2. Our results also include new such distillation protocols with the smallest number of qubits for a given distance in the literature, namely distance 4 and 5 TTT-to-TTT protocols supported on 10 and 11 qubits, as well as distance 3 and 4 TTT-to-CCZ\mathrm{CC}ZCCZ distillation protocols supported on 9 and 10 qubits. Finally, going beyond unitary circuits by recycling qubits through mid-circuit measurement and reinitialization, we obtain an implementation of the distance-5 49T49T49T-to-1T1T1T protocol on only 5 active qubits.
Magic plays a dual role in quantum computation: it promotes stabilizer dynamics from efficient classical simulability to computational universality, but also challenges fault-tolerant architectures, since non-stabilizer operations are harder to protect against noise. Magic state distillation addresses this issue, yet existing protocols remain largely tailored to specific assumptions about the input or noise model and, more fundamentally, no general information-theoretic theory of optimal magic-state conversion currently exists. Here we develop such a characterization for pure qubit states through universal magic state concentration: a fixed stabilizer protocol that converts a few copies of an unknown pure non-stabilizer qubit state into an exact target magic state. Motivated by the impossibility of exact TTT-state concentration, we build six- and eight-copy protocols producing an exact CCZ\mathrm{CCZ}CCZ state, with six copies being minimal. Their success probabilities are governed by the linearized order-three stabilizer Rényi entropy M3linM^{\mathrm{lin}}_3M3lin​. We show that this connection is structural: for up to nine input copies, M3linM^{\mathrm{lin}}_3M3lin​ fully determines the success probability of every universal Clifford-invariant stabilizer protocol. Remarkably, this characterization persists asymptotically: our constructions achieve optimal rates among universal single-output protocols and remain optimal up to logarithmic factors among arbitrary stabilizer protocols. As a corollary, we show that any unknown pure non-stabilizer state suffices for universal quantum computation via probabilistic CCZ\mathrm{CCZ}CCZ-state injection. Together, these results identify the stabilizer Rényi entropy as a fundamental operational quantity in magic state distillation.
Magic state distillation is a widely considered primitive in fault-tolerant quantum computation for the preparation of high-fidelity non-Clifford resources. The most commonly considered class of such protocols are generalised triorthogonal codes; these distil nnn noisy input T\mathrm{T}T states into purified third-level diagonal magic states. Extensive prior work has searched this space of protocols, often using heuristic methods that do not guarantee optimality. Existing classification work is limited to protocols distilling kkk output T\mathrm{T}T states with n+k≤38n+k\leq 38n+k≤38 (Nezami and Haah, Phys. Rev. A 106, 012437 (2022)). In this work, we significantly expand this classification to all protocols with lengths n≤54n\leq 54n≤54. We restrict to distance d≥3d\geq 3d≥3 to keep the classification to a sensible size, and because efficient searches are well understood at distance 222 (Singh et al., arXiv:2606.28518). Moreover, our results are complementary to synthillation (Campbell and Howard, Phys. Rev. A 95, 022316 (2017)), which can distil third-level states from T\mathrm{T}T states, but only at distance 222. Under optimality in terms of input count, space footprint, and distance for a given output, we find 747474 optimal generalised triorthogonal protocols in our range, 656565 of which are new to the literature. To achieve our classification, we extend the classification of unital triorthogonal spaces of Nezami and Haah from length 383838 to 545454 using a directional derivative method. Using these as the stabiliser spaces, we add logical rows that satisfy the triorthogonality constraints to create full protocols.
Fault-tolerant quantum computation architectures are frequently bottlenecked by the overhead of producing high-fidelity magic states. In this work, we use algebraic geometric techniques to construct codes over binary extension fields F2s\mathbb{F}_{2^s}F2s​, thus discovering new protocols for the distillation of qubit magic states, where our focus is on the regime of practical qubit-based quantum computing architectures. To do this, we show that multi-qubit gates of interest such as CS\text{CS}CS, CCZ\text{CCZ}CCZ, and TOF#=CCZ123CCZ345\text{TOF}\# = \text{CCZ}_{123}\text{CCZ}_{345}TOF#=CCZ123​CCZ345​, can be packaged into simple gates over the larger fields, and we derive simple algebraic conditions in the extension fields allowing the distillation of these gates. Because they are derived from Galois qudits, the corresponding qubits codes naturally handle the correlated errors present on such multi-qubit states. Moreover, the protocols we discover are extremely compact; for example, we show that 4 CS\text{CS}CS states can be distilled to 1 CS\text{CS}CS state at distance 2, using only 4 logical qubits. For a case study, we consider the distillation of CS\text{CS}CS and CCZ\text{CCZ}CCZ states from injected T\text{T}T and CS\text{CS}CS states. When optimized for magic state production per unit time, or logical spacetime volume, we find that our protocols outperform the state-of-the-art in almost every situation, both at input error rates 10−310^{-3}10−3 (direct injection), and 10−610^{-6}10−6 (allowing some cultivation pre-injection).