In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates is usually more challenging and resource-intensive than Clifford gates. Methods such as distilling a high-fidelity magic state from noisy copies or transforming to higher-dimensional codes require a significant qubit count overhead. In this work, we propose a new protocol that combines magic state preparation and code transformation to realize logical non-Clifford operations, which has the potential for reducing the qubit count overhead, provided an efficient decoding algorithm exists for the intermediate non-Abelian code involved. Our approach begins with a special logical state in the \({{\mathbb{Z}}}_{4}\) surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian D 4 quantum double model. This process ultimately produces a magic state encoded in the \({{\mathbb{Z}}}_{2}\) surface code. A logical T gate can be implemented in the standard \({{\mathbb{Z}}}_{2}\) surface code by gate teleportation. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding transformations between topological codes.
In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates is usually more challenging and resource-intensive than Clifford gates. Methods such as distilling a high-fidelity magic state from noisy copies or transforming to higher-dimensional codes require a significant qubit count overhead. In this work, we propose a new protocol that combines magic state preparation and code transformation to realize logical non-Clifford operations, which has the potential for reducing the qubit count overhead, provided an efficient decoding algorithm exists for the intermediate non-Abelian code involved. Our approach begins with a special logical state in the ({{\mathbb{Z}}}{4}) surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian D 4 quantum double model. This process ultimately produces a magic state encoded in the ({{\mathbb{Z}}}{2}) surface code. A logical T gate can be implemented in the standard ({{\mathbb{Z}}}_{2}) surface code by gate teleportation. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding transformations between topological codes.
We propose protocols to implement non-Clifford logical gates between stabilizer codes by entangling into a non-Abelian topological order as an intermediate step. Generalizing previous approaches, we provide a framework that generates a large class of non-Clifford and nondiagonal logical gates between qudit surface codes by gauging the topological symmetry of symmetry-enriched topological orders. As our main example, we concretely detail a protocol that utilizes the quantum double of S 3 to generate a controlled-charge-conjugation ( C C ) gate between a qubit and qutrit surface code. Both the preparation of non-Abelian states and logical state injection between the Abelian and non-Abelian codes are executed via finite-depth quantum circuits with measurement and feedforward. We discuss aspects of the fault tolerance of our protocol, presenting insights into how to construct a heralded decoder for the quantum double of S 3 . We also outline how analogous protocols can be used to obtain logical gates between qudit surface codes by entangling into D ( G ) , where G is a semidirect product of Abelian groups. This work serves as a step toward classifying the computational power of non-Abelian quantum phases beyond the paradigm of anyon braiding on near-term quantum devices.
In universal fault-tolerant quantum computing, implementing logical non-Clifford gates often demands substantial spacetime resources for many error-correcting codes, including the high-threshold surface code. A critical mission for realizing large-scale quantum computing is to develop simple and resource-efficient implementations of logical non-Clifford gates. We propose a novel way of implementing non-Clifford operations in the standard surface code based on hybrid lattice surgery. First we generalize the standard lattice surgery to hybrid lattice surgery, where operations of rough merge and rough split happen across different topological codes. Then we apply such procedures between Abelian and non-Abelian codes and show that this can provide non-Clifford operations in the standard surface code, in the form of a magic state or a non-Clifford gate teleportation. Complementing this, we provide a continuum topological field theory description of this hybrid lattice surgery utilizing interfaces between (2+1)d topological orders. From these considerations, we can generalize our protocol to non-Clifford gates and magic states at all finite levels of the Clifford hierarchy, as well as gates beyond the hierarchy. We also discuss protocols extending this framework to qutrits.
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.
We introduce group surface codes, which are a natural generalization of the Z2 surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. We show that group surface codes can be leveraged to perform non-Clifford gates in Z2 surface codes, thus enabling universal computation along with established means of performing logical Clifford gates. Moreover, for suitably chosen groups, we demonstrate that arbitrary reversible classical gates can be implemented transversally in the group surface code. We present the logical operations in terms of a set of elementary logical operations, which include transversal logical gates, a means of transferring encoded information into and out of group surface codes, and preparation and readout. By composing these elementary operations, we implement a wide variety of logical gates and provide a unified perspective on recent constructions in the literature for sliding group surface codes and preparing magic states. We furthermore use tensor networks inspired by ZX-calculus to construct spacetime implementations of the elementary operations. This spacetime perspective also allows us to establish explicit correspondences with topological gauge theories. Our work extends recent efforts in performing universal quantum computation in topological orders without the braiding of anyons, and shows how certain group surface codes allow us to bypass the restrictions set by the Bravyi-König theorem, which limits the computational power of topological Pauli stabilizer models.
A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for n-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of n-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the (n+1)D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted Z23 gauge theory (equivalent to D4 topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in O(d) rounds via code switching. In 3D, we construct a transversal logical T gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted Z24 gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the (n+1)-th level of Clifford hierarchy in n spatial dimension.
The surface code family is a promising approach to implementing fault-tolerant quantum computations. Universal fault-tolerance quantum computing requires error-corrected nonClifford operations, in addition to Clifford gates, and for the former, it is important to experimentally demonstrate additional resources known as magic states. Another challenge is to efficiently embed surface codes into quantum hardware with connectivity constraints. This work simultaneously addresses both challenges by employing a qubit-efficient rotated heavyhexagonal surface code for IBM quantum processors (ibm_fez) and implementing the magic state injection protocol. Our work reports error thresholds for both logical bit- and phase-flip errors, of≈0.37%and≈0.31%, respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states∣HL⟩and∣TL⟩achieves fidelities of0.8806±0.0002and0.8665±0.0003, respectively, which are both above the magic state distillation threshold. Our work demonstrates the potential for realising nonClifford logical gates by producing high-fidelity logical magic states on IBM quantum devices.