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arXiv:2502.02897 · Physical Review A

High-fidelity initialization of a logical qubit with multiple injections

Zhicheng He, Zheng‐Yuan Xue

Published May 12, 2025• 1 citation
arXivHTMLPDF

Abstract

Quantum error correction represents a significant advancement in large-scale quantum computing. However, achieving fault-tolerant implementations of non-Clifford logical gates with reduced overhead remains a challenge in the popular surface code strategy. Recent advances have underscored the need for a substantial code distance to attain complete fault tolerance. Here, we introduce a continuous fault-tolerant scheme for non-Clifford logical gates via multiple injections. Unlike existing protocols that focus on a single logical chain, our approach utilizes multiple logical chains, each of which can employ the same or different logical rotation angles to initialize a non-Clifford state. Compared to previous efforts, our protocol significantly alleviates the challenges associated with the requirement for a large code distance and reduces the corresponding resource overheads, making it more feasible to be implemented in current midscale chips via the surface code strategy.

DOI
10.1103/physreva.111.052419

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Quantum error correction represents a significant advancement in large-scale quantum computing. However, achieving fault-tolerant implementations of non-Clifford logical gates with reduced overhead remains a challenge in the popular surface code strategy. Recent advances have underscored the need for a substantial code distance to attain complete fault tolerance. Here, we introduce a continuous fault-tolerant scheme for non-Clifford logical gates via multiple injections. Unlike existing protocols that focus on a single logical chain, our approach utilizes multiple logical chains, each of which can employ the same or different logical rotation angles to initialize a non-Clifford state. Compared to previous efforts, our protocol significantly alleviates the challenges associated with the requirement for a large code distance and reduces the corresponding resource overheads, making it more feasible to be implemented in current midscale chips via the surface code strategy.
Quantum error correction represents a significant milestone in large-scale quantum computing, with the surface code being a prominent strategy due to its high error threshold and experimental feasibility. However, it is challenging to implement non-Clifford logical gates in a fault-tolerant way with low overhead, through the conventional magic-state distillation technique. Here, we enhance the performance of the conventional surface code by incorporating non-Pauli stabilizers and introduce two innovative initialization protocols. Our approach enhances the fidelity of the initialization of the non-Clifford logical state by avoiding unprotected operations before the encoding process. This improved fidelity of the initialization of non-Clifford logical states reduces the necessary number of logical qubits for precise state distillation, ultimately decreasing the overall resource overhead. Furthermore, we demonstrate the ability to entangle logical qubits in non-Pauli and Pauli bases via the lattice surgery technique. This integration enables the use of Pauli-based surface codes for computation while non-Pauli codes are employed for auxiliary qubit initialization, thus making them compatible with the conventional wisdom of logical Clifford operation based on the Pauli-based surface code.
The surface code family is a promising approach to implementing fault-tolerant quantum computations. Universal fault-tolerance requires error-corrected non-Clifford operations, in addition to Clifford gates, and for the former, it is imperative 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 heavy-hexagonal 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 [Formula: see text] and [Formula: see text], respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states [Formula: see text] and [Formula: see text] achieve fidelities of [Formula: see text] and [Formula: see text], respectively, which are both above the magic state distillation threshold. The post-selection process yields an average success rate of [Formula: see text]. Additionally, we report the minimum fidelity among injected arbitrary single logical qubit states as [Formula: see text]. Our work demonstrates the potential for realising non-Clifford logical gates by producing high-fidelity logical magic states on IBM quantum devices.
One of the most promising routes towards fault-tolerant quantum computation utilizes topological quantum error correcting codes, such as the Z2{\mathbb{Z}}_{2}Z2​ surface code. Logical qubits can be encoded in a variety of ways in the surface code, based on either boundary defects, holes, or bulk twist defects. However, proposed fault-tolerant implementations of the Clifford group in these schemes are limited and often require unnecessary overhead. For example, the Clifford phase gate in certain planar and hole encodings has been proposed to be implemented using costly state injection and distillation protocols. In this paper, we show that within any encoding scheme for the logical qubits, we can fault tolerantly implement the full Clifford group by using joint measurements involving a single appropriately encoded logical ancilla. This allows us to provide low overhead implementations of the full Clifford group in surface and color codes. It also provides the first proposed implementations of the full Clifford group in hyperbolic codes. We further use our methods to propose state-of-the art encoding schemes for small numbers of logical qubits; for example, for code distances d=3,5,7d=3,5,7d=3,5,7, we propose a scheme using 60,160,30860,160,30860,160,308 (respectively) physical data qubits, which allow for the full logical Clifford group to be implemented on two logical qubits. To our knowledge, this is the optimal proposal to date, and thus may be useful for demonstration of fault-tolerant logical gates in small near-term quantum computers.
Fault-tolerant logical operations are fundamental for scalable quantum computation. Here, we report the experimental realization of lattice-surgery operations between a pair of distance-three surface-code logical qubits on a planar superconducting processor. During repeated syndrome extraction cycles, the logical qubits exhibit per-cycle error rates of 0.0365(2)0.0365(2)0.0365(2) and 0.0282(1)0.0282(1)0.0282(1), respectively, after leakage events are rejected. By leveraging joint initialization and lattice splitting, we deterministically prepare a logical Bell state, confirming genuine bipartite entanglement via the error-corrected logical state fidelity. We further execute a two-qubit Deutsch-Jozsa algorithm at the logical level to demonstrate algorithmic utility in a fault-tolerant framework. Finally, to achieve universal control, we implement magic-state injection and gate teleportation to realize continuous non-Clifford rotations about the logical XXX axis. For the logical RX(π/4)R_{X}(π/4)RX​(π/4) gate, we achieve a logical gate fidelity of 0.943−9+100.943_{-9}^{+10}0.943−9+10​ conditioned on the absence of detected errors. These results establish lattice surgery as a practical and versatile paradigm for logical computation in near-term surface-code architectures, representing a critical milestone toward scalable fault-tolerant quantum advantage in superconducting circuits.
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 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%\approx 0.37 \%≈0.37%and≈0.31%\approx 0.31 \%≈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⟩\vert H_{L}\rangle∣HL​⟩and∣TL⟩\vert T_{L}\rangle∣TL​⟩achieves fidelities of0.8806±0.00020.8806 \pm 0.00020.8806±0.0002and0.8665±0.00030.8665 \pm 0.00030.8665±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.