Fault-tolerant, error-corrected quantum computation is commonly acknowledged to be crucial to the realisation of large-scale quantum algorithms that could lead to extremely impactful scientific or commercial results. Achieving a universal set of quantum gate operations in a fault-tolerant error-corrected framework suffers from a `conservation of unpleasantness'. In general, no matter what error correction technique is employed, there is always one element of a universal gate set that carries a significant resource overhead - either in physical qubits, computational time, or both. Specifically, this is due to the application of non-Clifford gates. A common method for realising these gates for stabiliser codes such as the surface code is a combination of three protocols: state injection, distillation and gate teleportation. These protocols contribute to the resource overhead compared to logical operations such as a CNOT gate and contributes to the qubit resources for any error-corrected quantum algorithm. In this paper, we introduce a very simple protocol to potentially reduce this overhead for non-Clifford gates: Transversal Injection. Transversal injection modifies the initial physical states of all data qubits in a stabiliser code before standard encoding and results in the direct preparation of a large class of single qubit states, including resource states for non-Clifford logic gates. Preliminary results hint at high quality fidelities at larger distances and motivate further research on this technique.
Fault-tolerant, error-corrected quantum computation is commonly acknowledged to be crucial to the realization of large-scale quantum algorithms that could lead to extremely impactful scientific or commercial results. Achieving a universal set of quantum gate operations in a fault-tolerant, error-corrected framework suffers from a conservation of unpleasantness. In general, no matter what error-correction technique is employed, there is always one element of a universal gate set that carries a significant resource overhead---either in physical qubits, computational time, or both. Specifically, this is due to the application of non-Clifford gates. A common method for realizing these gates for stabilizer codes such as the surface code is a combination of three protocols: state injection, distillation, and gate teleportation. These protocols contribute to the resource overhead compared with logical operations such as a CNOT gate and contribute to the qubit resources for any error-corrected quantum algorithm. In this paper, we introduce a very simple protocol to potentially reduce this overhead for non-Clifford gates: transversal injection. Transversal injection modifies the initial physical states of all data qubits in a stabilizer code before standard encoding and results in the direct preparation of a large class of single qubit states, including resource states for non-Clifford logic gates. Preliminary results hint at high-quality fidelities at larger distances and motivate further research on this technique.
The development of quantum computing systems for large scale algorithms requires targeted error rates un-achievable through hardware advancements alone. Quantum Error Correction (QEC) allows us to use systems with a large number of physical qubits to form a fault tolerant system with a lower number of logical qubits and a favourable logical error rate. While some gates can be easily implemented in a QEC code transversally, there is no code that has a universal set of transversal gates. Transversal Injection is a new method of preparing logical non-Pauli eigenstates that can be used as resource states for quantum computation. State preparation can be done directly in the surface code and has the potential to prepare higher fidelity injected states. Compared to other techniques, transversal injection can reduce the resource burden for state distillation protocols. In this paper, the authors present the theory behind this new technique as well as an algorithm for calculating the resulting logical states prepared in the surface code
Fault-tolerant logic gates will consume a large proportion of the resources of a two-dimensional quantum computing architecture. Here we show how to perform a fault-tolerant non-Clifford gate with the surface code; a quantum error-correcting code now under intensive development. This alleviates the need for distillation or higher-dimensional components to complete a universal gate set. The operation uses both local transversal gates and code deformations over a time that scales with the size of the qubit array. An important component of the gate is a just-in-time decoder. These decoding algorithms allow us to draw upon the advantages of three-dimensional models using only a two-dimensional array of live qubits. Our gate is completed using parity checks of weight no greater than four. We therefore expect it to be amenable with near-future technology. As the gate circumvents the need for magic-state distillation, it may reduce the resource overhead of surface-code quantum computation considerably.
The Eastin-Knill theorem states that no quantum error-correcting code can have a universal set of transversal gates. For Calderbank-Shor-Steane codes that can implement Clifford gates transversally, it suffices to provide one additional non-Clifford gate, such as the T gate, to achieve universality. Common methods to implement fault-tolerant T gates, e.g., magic state distillation, generate a significant hardware overhead that will likely prevent their practical usage in the near-term future. Recently, methods have been developed to mitigate the effect of noise in shallow quantum circuits that are not protected by error correction. Error mitigation methods require no additional hardware resources but suffer from a bad asymptotic scaling and apply only to a restricted class of quantum algorithms. In this Letter, we combine both approaches and show how to implement encoded Clifford+T circuits where Clifford gates are protected from noise by error correction while errors introduced by noisy encoded T gates are mitigated using the quasiprobability method. As a result, Clifford+T circuits with a number of T gates inversely proportional to the physical noise rate can be implemented on small error-corrected devices without magic state distillation. We argue that such circuits can be out of reach for state-of-the-art classical simulation algorithms.
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.
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.