In order to circumvent that no universal set of gates can be achieved on a given quantum error-correcting codes, additional gates can be performed through magic state injection. This however requires to have access to high fidelity magic state, which can be obtained by the distillation of low quality states to fewer higher quality ones. In this paper, we provide a family of distillation protocols able to produce diagonal states at every level of the Clifford hierarchy. This family is obtained recursively using code doubling techniques and leveraging qubit recycling, the fact that some idle qubits of the protocols can be measured and re-used. At a given level of the Clifford hierarchy, we show that the protocols we derive can be performed at arbitrary high distance on a fix number of logical qubits. This family recovers many known efficient protocols, and uncovers new ones, such as a $111 \to 1$ protocol for $|T\rangle$ state distillation at distance $7$, which are the most compact in term of volume. Finally, we discuss possible extension of this framework for distillation protocols producing more than one output state.
In order to circumvent that no universal set of gates can be achieved on a given quantum error-correcting codes, additional gates can be performed through magic state injection. This however requires to have access to high fidelity magic state, which can be obtained by the distillation of low quality states to fewer higher quality ones. In this paper, we provide a family of distillation protocols able to produce diagonal states at every level of the Clifford hierarchy. This family is obtained recursively using code doubling techniques and leveraging qubit recycling, the fact that some idle qubits of the protocols can be measured and re-used. At a given level of the Clifford hierarchy, we show that the protocols we derive can be performed at arbitrary high distance on a fix number of logical qubits. This family recovers many known efficient protocols, and uncovers new ones, such as a 111→1 protocol for ∣T⟩ state distillation at distance 7, which are the most compact in term of volume. Finally, we discuss possible extension of this framework for distillation protocols producing more than one output state.
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 T-to-T distillation protocol on fewer than eight qubits can exceed distance 3, and no T-to-CCZ 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 T-to-T protocols supported on 10 and 11 qubits, as well as distance 3 and 4 T-to-CCZ 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 49T-to-1T protocol on only 5 active qubits.
Magic state distillation (MSD) is the canonical route to non-Clifford gates in fault-tolerant quantum computation, but conventional nested factories exhibit qubit occupancy that grows multiplicatively with the number of distillation rounds. Recent work on constant-sized-support distillation protocols with qubit recycling (arXiv:2609.17044v1) proposes to hold the active qubit footprint fixed across rounds by recycling output qubits back into input registers. This paper develops a transparent, fully arithmetized resource-budget model of that proposal. We derive (i) the leading-order error suppression of a two-round 15-to-1 distillation ladder, giving output error 1,500,625 p⁹ for input error p (1.500625 × 10⁻²¹ at p = 10⁻³; 1.500625 × 10⁻¹² at p = 10⁻²); (ii) a peak-occupancy comparison showing that a recycled constant-support factory with a 15-state buffer reduces peak qubit count from 240 to 30, a factor-of-8 reduction; (iii) a recycling error budget showing that, under a stated worst-case linear depolarization model with per-reuse error δ = 10⁻⁵ and input error p₀ = 10⁻³, at most 10 reuses preserve input fidelity within 10% of nominal; and (iv) an energy attribution under one stated convention that yields ≈ 0.55 J per distilled magic state (± 0.07 J) when the model is anchored to a publicly reported benchmark energy of 5.5 × 10⁵ J per inference against a classical baseline of 0.0365 J (ratio 1.5 × 10⁷). All numerical results are analytic projections under explicitly stated assumptions, not simulations; divergences among independent draft analyses are documented in Appendix A, and every claim is attributed in Appendix B. We identify the experimental conditions under which the recycling advantage would be falsified. ## 1. Introduction Fault-tolerant quantum computation requires a gate set executed with error rates far below the threshold of the underlying quantum error-correcting code. For most code families, the transversal or lattice-surgery-accessible gate set i Full text and updates: papers.qnfo.org/papers/constant-sized-support-magic-state-distillation-with-qubit-recycling-a-reconcile/
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, 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, CCZ, and TOF#=CCZ123CCZ345, 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 states can be distilled to 1 CS state at distance 2, using only 4 logical qubits. For a case study, we consider the distillation of CS and CCZ states from injected T and 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 10−3 (direct injection), and 10−6 (allowing some cultivation pre-injection).
Magic State Distillation is considered to be one of the promising methods for supplying the non-Clifford resources required to achieve universal fault tolerance. Conventional MSD protocols implemented in surface codes often require multiple code blocks and lattice surgery rounds, resulting in substantial qubit overhead, especially at low target error rates.
In this work, we present practical magic state distillation factories on Bivariate Bicycle (BB) codes that execute Pauli-measurement-based Clifford circuits inside a single BB code block. We formulate distillation circuit design as a joint optimization of logical qubit mapping, gate scheduling, measurement nativization, and protocol compression via qubit recycling. Based on detailed resource analysis and simulations, our BB factories have space-time volume comparable to that of leading distillation factories while delivering lower target error at a smaller qubit footprint, and are particularly compelling as second-round distillers following magic state cultivations.
We present an infinite family of protocols to distill magic states for T -gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T -gate depth of the circuit scale linearly in the logarithm of the target error δ (up to log log 1 / δ ). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.
Zero-level CCZ Distillation
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the T gate and the CCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical CCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T† operation of the [[8,3,2]] code to fault-tolerantly encode the state CCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as pL≃300×p2 with respect to the physical error rate p. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3 and p=10−4, respectively, compared to conventional seven-T-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of CCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional T-state generation.