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 $|T\rangle$, $|CS\rangle$, and $|CCZ\rangle$ 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.
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⟩, ∣CS⟩, and ∣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.
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=1.589 and γ=1.057. Our [[1715, 287, 6]] factory, with γ=0.998, is the smallest known pure-T factory with γ<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 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.
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.
Practical quantum computation requires high-fidelity instruction executions on qubits. Among them, Clifford instructions are relatively easy to perform, while non-Clifford instructions require the use of magic states. This makes magic state distillation a central procedure in fault-tolerant quantum computing. A magic state distillation factory consumes many low-fidelity input magic states and produces fewer, higher-fidelity states. To reach high fidelities, multiple distillation factories are typically chained together into a multi-level pipeline, consuming significant quantum computational resources. Our work optimizes the resource usage of distillation pipelines by introducing a novel dynamic pipeline architecture. Observing that distillation pipelines consume magic states in a burst-then-steady pattern, we develop dynamic factory scheduling and resource allocation techniques that go beyond existing static pipeline organizations. Dynamic pipelines reduce the qubit cost by 16%-70% for large-scale quantum applications and achieve average reductions of 26%-37% in qubit-time volume on generated distillation benchmarks compared to state-of-the-art static architectures. By significantly reducing the resource overhead of this building block, our work accelerates progress towards the practical realization of fault-tolerant quantum computers.
The standard approach to fault-tolerant quantum computation is to store information in a quantum error correction code, such as the surface code, and process information using a strategy that can be summarized as distill then synthesize. In the distill step, one performs several rounds of distillation to create high-fidelity logical qubits in a magic state. Each such magic state provides one good T gate. In the synthesize step, one seeks the optimal decomposition of an algorithm into a sequence of many T gates interleaved with Clifford gates. This gate-synthesis problem is well understood for multiqubit gates that do not use any Hadamards. We present an in-depth analysis of a unified framework that realizes one round of distillation and multiqubit gate synthesis in a single step. We call these synthillation protocols, and show they lead to a large reduction in resource overheads. This is because synthillation can implement a general class of circuits using the same number of T states as gate synthesis, yet with the benefit of quadratic error suppression. This general class includes all circuits primarily dominated by control-control-Z gates, such as adders and modular exponentiation routines used in Shor's algorithm. Therefore, synthillation removes the need for a costly round of magic state distillation. We also present several additional results on the multiqubit gate-synthesis problem. We provide an efficient algorithm for synthesizing unitaries with the same worst-case resource scaling as optimal solutions. For the special case of synthesizing controlled unitaries, our techniques are not just efficient but exactly optimal. We observe that the gate-synthesis cost, measured by T count, is often strictly subadditive. Numerous explicit applications of our techniques are also presented.