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.
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.
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.
Fault-tolerant preparation of non-Clifford resource states is a major contributor to the overhead of quantum computation, motivating protocols that achieve high output fidelity with minimal qubit and circuit costs. We introduce a low-cost magic-state preparation protocol in which the choice of stabilizer generators is co-designed with the flag gadgets, allowing the syndrome-extraction circuit itself to filter correlated faults across a non-Clifford layer. The protocol prepares a logical plus state in the 15-qubit quantum Reed-Muller code, applies a transversal T gate, and gauge-fixes the same register into the seven-qubit Steane code. By reorganizing equivalent Z-type stabilizer generators into jointly flagged measurement groups, the protocol eliminates all accepted logical-error contributions arising from one or two circuit faults under destructive error detection. Under a uniform circuit-level depolarizing noise model, the postselected infidelity is 210.2p3+O(p4). At p=10−3, exact low-order enumeration combined with stratified sampling bounds the infidelity by 2.2×10−7 at 99.9% joint confidence, while retaining an acceptance probability of 86.9%. The complete circuit requires only 19 qubits and 82 CNOT gates. These results demonstrate that stabilizer-generator design can substantially reduce the cost of postselected magic-state preparation, although corrected operation and the fidelity of an unmeasured output block require separate analysis.
Fault-tolerant quantum computation allows quantum computations to be carried out while resisting unwanted noise. Several error-correcting codes have been developed to achieve this task, but none alone are capable of transversal universal quantum computation. This universality is highly desired and often achieved using techniques such as code concatenation, code switching, magic state distillation, or pieceable fault tolerance. Existing constructions can incur substantial overhead or rely on properties specific to particular codes. This work proposes implementing logical Clifford and T gates through novel, ancilla-mediated, pieceable interactions to construct a universal fault-tolerant quantum gate set. Our construction is deterministic, does not modify the underlying data codes or registers, and is generic over all data stabilizer codes. Thus, any single code becomes capable of universal quantum computation by leveraging helper codes in ancilla registers and mid-circuit measurements. Furthermore, since these logical gates are stabilizer code-generic, these implementations enable communication between heterogeneous stabilizer codes.
Encoding quantum information to protect it from errors is essential for performing large-scale quantum computations. Performing a universal set of quantum gates on encoded states demands a potentially large resource overhead and minimizing this overhead is key for the practical development of large-scale fault-tolerant quantum computers. We propose and experimentally implement a magic-state preparation protocol to fault-tolerantly prepare a pair of logical magic states in a [[6,2,2]] quantum error-detecting code using only eight physical qubits. Implementing this protocol on H1-1, a 20 qubit trapped-ion quantum processor, we prepare magic states with experimental infidelity 7−1+3×10−5 with a 14.8−1+1% discard rate and use these to perform a fault-tolerant non-Clifford gate, the controlled-Hadamard (CH), with logical infidelity ≤2.3−9+9×10−4. Notably, this significantly outperforms the unencoded physical CH infidelity of 10−3. Through circuit-level stabilizer simulations, we show that this protocol can be self-concatenated to produce extremely high-fidelity magic states with low space-time overhead in a [[36,4,4]] quantum error correcting code, with logical error rates of 6×10−10 (5×10−14) at two-qubit error rate of 10−3 (10−4) respectively.
The preparation of high-fidelity logical magic states is a crucial subroutine for universal fault-tolerant quantum computation (FTQC). Predicting the performance of FTQC and developing improved protocols rely on numerical methods to classically simulate logical magic state preparation in the presence of noise. Clifford logic on Pauli-stabilizer codes with circuit-level Pauli errors can be efficiently simulated using Pauli-stabilizer formalism, but the non-Clifford operations required to prepare logical magic states render generic simulation inefficient. We introduce Clifford-stabilizer simulation, an exact and efficient algorithm based on updating a Clifford-stabilizer group to simulate noisy preparation protocols for a broad class of logical magic states used to implement non-Clifford gates in the third level of the Clifford hierarchy under circuit-level Pauli errors. Clifford-stabilizer simulation applies to a range of operations that commonly appear in preparation protocols for such logical magic states, including Pauli-stabilizer measurements, logical Clifford measurements, and transversal non-Clifford gates. Our algorithm for Clifford-stabilizer simulation maps a non-Clifford circuit with sampled circuit-level Pauli errors to a Clifford circuit that exactly reproduces its measurement outcome distribution, achieving time and space complexities polynomial in relevant protocol parameters. We perform exact simulation of magic state cultivation up to fault distance 7 by Clifford-stabilizer simulation. Our method provides a route to perform exact benchmarking of large-scale logical magic state preparation protocols required for useful FTQC.
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.