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arXiv:2606.30628

Cavity-mediated probabilistic magic TTT-gate injection

Sofia Cocciaretto, Roberto Menta, Vittorio Giovannetti

quant-phcond-mat.quant-gasphysics.atom-phSubmitted Jun 29, 2026• Announced Jun 30, 2026• 0 citations
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Abstract

Non-Clifford gates are a necessary resource for universal quantum computation, yet their fault-tolerant implementation typically relies on magic-state distillation, which incurs significant overhead in qubit count and circuit depth. In this work, we propose a probabilistic cavity-based magic-state injection protocol. Our scheme exploits controlled atom-cavity interactions and conditional measurements to probabilistically prepare an effective magic state encoded in the first two level Fock subspace of a single cavity mode, achieving a success probability of $0.74$ per attempt, independent of the target magic phase. The cavity-encoded magic state is subsequently injected into a computational atom via a teleportation-based protocol mediated by dressed-state transitions, requiring only Clifford operations and a single auxiliary atom for readout. We show that all required operations -- state preparation, two-qubit exchange gates, and projective measurement -- can be implemented with experimentally available techniques in Rydberg atom-cavity platforms. We further discuss how the scheme can in principle be adapted to operate at the logical level, where collective Rydberg interactions and optical nonlinearities provide a route toward cavity-mediated $T$-gate injection directly into code-encoded qubits.

Comments
7 pages, 6 figures. Comments are welcome!

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アヤマリタイセイリョウシケイサンノジツゲンニムケタコウリツテキナマホウジョウタイノセイセイトショウヒ

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Non-Clifford gates are a necessary resource for universal quantum computation, yet their fault-tolerant implementation typically relies on magic-state distillation, which incurs significant overhead in qubit count and circuit depth. In this work, we propose a probabilistic cavity-based magic-state injection protocol. Our scheme exploits controlled atom-cavity interactions and conditional measurements to probabilistically prepare an effective magic state encoded in the first two level Fock subspace of a single cavity mode, achieving a success probability of 0.740.740.74 per attempt, independent of the target magic phase. The cavity-encoded magic state is subsequently injected into a computational atom via a teleportation-based protocol mediated by dressed-state transitions, requiring only Clifford operations and a single auxiliary atom for readout. We show that all required operations -- state preparation, two-qubit exchange gates, and projective measurement -- can be implemented with experimentally available techniques in Rydberg atom-cavity platforms. We further discuss how the scheme can in principle be adapted to operate at the logical level, where collective Rydberg interactions and optical nonlinearities provide a route toward cavity-mediated TTT-gate injection directly into code-encoded qubits.
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Fault-tolerant ∣T⟩|\sqrt{ \mathrm{T} }\rangle∣T​⟩ state preparation and injection for more efficient fine-grained quantum circuit synthesis
Magic-state injection is a standard route to realize universal fault-tolerant quantum computation. Whereas the set of Clifford gates in combination with the non-Clifford T gate is a widely used universal gate set, extending the available set of non-Clifford primitives can reduce compilation overhead, provided that the additional primitives can be prepared fault-tolerantly with competitive resource costs and at sufficiently low logical noise rates. In this work, we introduce flag fault-tolerant protocols for preparing logical $|\sqrt{\mathrm{T}} \rangle $ magic states on the 3D tetrahedral color code and its smaller morphed variant. Our simulations under circuit-level noise verify fault tolerance, quantify acceptance and logical error rates, and we reconstruct the effective logical channels of the corresponding circuits for gate injection via logical process tomography. We find that access to T\sqrt{\mathrm{T}}T​ reduces the average space-time cost of synthesizing Haar-random single-qubit unitaries by approximately 202020-303030% relative to the Clifford+T gate set across practically relevant approximation regimes. These results demonstrate how expanding the set of fault-tolerant non-Clifford primitives can improve computational efficiency and broaden the design space for universal quantum computation in the early fault-tolerant era.
Quantum gate teleportation is a key technique in fault-tolerant quantum computation that uses resource states to implement logical gates. Here, we develop a theory of magic gate teleportation (MGT), a form of quantum gate teleportation that uses non-stabilizer resource states to implement non-Clifford unitary gates on arbitrary input states without revealing information about those states. This setting is motivated by the canonical approach to implementing logical non-Clifford gates in fault-tolerant quantum computation. We uncover a hidden structure within MGT---after backpropagating the Pauli measurements, MGT protocols can be viewed as encoding the input state into a stabilizer code heralded by the measurement outcomes, followed by a logical non-Clifford gate. Using this structure, we construct MGT protocols for any resource state obtained by applying commuting Pauli rotations to a stabilizer state, and provide an efficient algorithm for synthesizing their circuit implementations. Conversely, we prove that directly useful single-qubit resource states are necessarily Clifford-equivalent to diagonal states, and establish the same constraint for multi-qubit resource states used in a broad class of MGT protocols. In particular, the output state distilled from the [ ⁣[5,1,3] ⁣][\![5, 1, 3]\!][[5,1,3]] protocol is not directly useful for any MGT protocols, though it can be probabilistically converted into directly useful states. Finally, we identify conditions under which the feedforward operators can be implemented by Pauli operators, shedding light on the paradigm of algorithmic fault tolerance and simplifying the feedforward operations needed for quantum computing.
Abstract Magic states have been widely studied in recent years as resource states that help quantum computers achieve fault-tolerant universal quantum computing. The fault-tolerant quantum computing requires fault-tolerant implementation of a set of universal logical gates. Stabilizer code, as a commonly used error correcting code with good properties, can apply the Clifford gates transversally which is fault tolerant. But only Clifford gates cannot realize universal computation. Magic states are introduced to construct non-Clifford gates that combine with Clifford operations to achieve universal quantum computing. Since the preparation of quantum states is inevitably accompanied by noise, preparing the magic state with high fidelity and low overhead is the crucial problem to realizing universal quantum computation. In this paper, we survey the related literature in the past 20 years and introduce the common types of magic states, the protocols to obtain high-fidelity magic states, and overhead analysis for these protocols. Finally, we discuss the future directions of this field.
Magic states, by allowing non-Clifford gates through gate teleportation, are important building blocks of fault-tolerant quantum computation. Magic state distillation protocols aim to create clean copies of magic states from many noisier copies. However, the prevailing protocols require substantial qubit overhead. We present a distillation protocol based on permutation-invariant gnu codes, as small as two qubits. The two-qubit protocol achieves a 0.5 error threshold and 1/2 distillation rate, surpassing prior schemes for comparable codes. Our protocol furthermore distils magic states with arbitrary magic by varying the position of the ideal input states on the Bloch sphere. We achieve this by departing from the usual magic state distillation formalism, allowing the use of non-Clifford gates in the distillation protocol, and allowing the form of the output state to differ from the input state. Our protocol is compatible for use in tandem with existing magic state distillation protocols to enhance their performance.
Quantum computation promises exponential or polynomial speedups over classical computation for a wide range of problems, including integer factoring, quantum chemistry, and linear-system solving.However, current quantum devices remain too noisy to execute complex algorithms reliably.Overcoming this noise and achieving quantum advantage require a fault-tolerant quantum computer (FTQC).The resource overhead associated with fault-tolerant operation is substantial, and reducing this overhead remains one of the central challenges.A major source of this overhead arises from the implementation of non-Clifford gates, which are essential for universal quantum computation.A non-Clifford gate such as the T gate is implemented using a special quantum state known as a magic state, which is prepared via a process called magic state distillation.Magic state distillation is resource-intensive and remains a major bottleneck in FTQC.Hence, reducing the distillation overhead is necessary to lower the overall FTQC overhead.The large overhead of magic state distillation has significantly influenced FTQC architectures.In particular, it has motivated compilation schemes optimized under the constraint of expensive magic state distillation.While reducing the distillation overhead is crucial, it is not sufficient; lowering the overall FTQC overhead also requires revisiting design choices guided by this constraint.Therefore, efficient methods are required not only for preparing magic states but also for consuming them.In this thesis, we reduce the overhead of fault-tolerant quantum computation by focusing on the preparation and consumption of magic states.In contrast to traditional logical-level distillation, physical-level distillation allows direct implementation with physical gates and has the potential to drastically reduce the distillation overhead.However, existing physical-level distillation protocols, such as zero-level distillation, cannot directly yield high-fidelity magic states.We propose two magic state distillation protocols that use physical-level distillation techniques to efficiently prepare high-fidelity magic states.The first protocol is a two-level distillation protocol using physical-level distillation as the first-level stage.The second protocol utilizes lattice surgery to reduce the spacetime overhead of a physical-level distillation protocol.The former targets a high-fidelity regime, whereas the latter targets an intermediate-fidelity regime-lower than the former, but higher than zero-level distillation.Next, we investigate FTQC architectures that can efficiently consume the magic states prepared by these protocols.Because magic state distillation is probabilistic, occasional failures can delay computation.Moreover, physical-level distillation protocols generally exhibit higher failure probabilities than logical-level ones.This delay is amplified by parallel execution, making it an i ii important consideration for FTQC using physical-level distillation protocols.We therefore propose a scheme to mitigate the negative performance impact caused by magic state distillation failures by introducing a pool of magic states for each distillation factory.The proposed distillation protocols allow the simultaneous preparation of many magic states.Leveraging this capability, we propose an FTQC compilation scheme optimized for highly parallel quantum algorithms.By performing magic state distillation locally and consuming magic states in parallel, the scheme enables highly parallel execution and consequently reduces the computational time overhead.Overall, this thesis presents a unified approach to reducing the overhead of fault-tolerant quantum computation.By improving both the preparation and consumption of magic states, it advances the design of resource-efficient FTQC architectures.These results demonstrate that efficient magic state preparation motivates efficient magic state consumption, and vice versa.Together, they bring us closer to the practical realization of fault-tolerant quantum computers.