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.
Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.
We propose a design framework for reconstructing gate-based magic state distillation protocols as compact joint-measurement architectures implementable with the surface code. The goal is to reduce the surface-code resource cost of a magic state factory while preserving the logical function and error-detection structure of the distillation protocol. We construct a reduced architecture for implementing an eight-to-three CCZ distillation protocol using smaller surface-code patches. The proposed factory preserves the single-fault-detection property and the leading-order error suppression of the protocol, while producing CCZ magic states with lower spatial cost than the design of Gidney and Fowler. The proposed design perspective can also be applied to T-state factories and other multiqubit non-Clifford resource-state factories. Our approach provides a framework for extending the design space of surface-code magic state factories beyond a single CCZ layout optimization.
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.
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/
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.
Fault-tolerant quantum processors based on topological codes execute programs through lattice surgery, where operations must be mapped, routed, and supplied with magic states across a 2D grid of physical patches. Non-Clifford operations require these magic states, produced either by distillation factories or by cultivation, each trading footprint against preparation latency, and delivering a magic state to the data patches that consume it requires routing through the same shared layout as every other operation. Yet placement, routing, scheduling, and magic-state supply cannot be optimized in isolation: two operations with no circuit-level dependency can still contend for the same ports, routes, or magic-state terminals once placed, so a compiler that decouples instruction scheduling from magic-state generation, or hard-codes a single generation protocol, is forced to trade execution time against layout footprint instead of co-optimizing both across protocols.
We present Harvest, a resource-aware compilation approach for lattice-surgery that co-optimizes magic-state consumption with circuit-aware placement and congestion-aware routing under a protocol-agnostic resource model, then reclaims unused layout footprint after scheduling. Across standard benchmark suites (QAOA, QFT, QASMBench), Harvest achieves an average speedup of
4.83× (up to
17.8×) over sequential execution, improves schedule length by up to
1.35× through circuit-aware placement, and reclaims up to
72.0% of unused magic-state patches and
33.9% of unused routing patches.