Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.01446 · Scientific Reports

Magic state injection on IBM quantum processors above the distillation threshold

Younghun Kim, Martin E. Sevior, Muhammad Usman

Published Feb 26, 2026• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41598-026-40381-1

Related papers

Preparing High-Fidelity Magic States on IBM Quantum Processors

Younghun Kim, Martin E. Sevior, Muhammad Usman

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$\approx 0.37 \%$and$\approx 0.31 \%$, respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states$\vert H_{L}\rangle$and$\vert T_{L}\rangle$achieves fidelities of$0.8806 \pm 0.0002$and$0.8665 \pm 0.0003$, 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.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Aug 30, 2025•0 citations

About Implementation of Magic State Injection in Heavy-Hexagon Structure

Hansol Kim, Wonjae Choi, Younghun Kwon

Implementing fault-tolerant quantum computing necessitates the realization of logical non-Clifford gates, which requires the preparation of specific quantum states known as magic states. However, IBM’s heavy-hexagon structure, which has limited qubit connectivity, presents challenges in adapting quantum error correction codes such as the surface code. Several methods have been proposed to address these challenges by adapting the surface code to the heavy-hexagon architecture. In this study, we implement the magic state injection process within two distinct implementations of surface codes (standard and rotated methods) suitable for the heavy-hexagon structure and compare their logical error rates. Furthermore, we propose initialization methods to enhance the performance of magic state injection in the heavy-hexagon structure, thereby efficiently achieving logical non-Clifford gates with reduced error rates.

Mathematics•Dec 3, 2025•1 citation

Logical Magic State Preparation with Fidelity beyond the Distillation Threshold on a Superconducting Quantum Processor

Yangsen Ye, Tan He, He-Liang Huang +35 more

Fault-tolerant quantum computing based on surface code has emerged as an attractive candidate for practical large-scale quantum computers to achieve robust noise resistance. To achieve universality, magic states preparation is a commonly approach for introducing non-Clifford gates. Here, we present a hardware-efficient and scalable protocol for arbitrary logical state preparation for the rotated surface code, and further experimentally implement it on the Zuchongzhi 2.1 superconducting quantum processor. An average of 0.8983±0.0002 logical fidelity at different logical states with distance three is achieved, taking into account both state preparation and measurement errors. In particular, the logical magic states |A^{π/4}⟩_{L}, |H⟩_{L}, and |T⟩_{L} are prepared nondestructively with logical fidelities of 0.8771±0.0009, 0.9090±0.0009, and 0.8890±0.0010, respectively, which are higher than the state distillation protocol threshold, 0.859 (for H-type magic state) and 0.827 (for T-type magic state). Our work provides a viable and efficient avenue for generating high-fidelity raw logical magic states, which is essential for realizing non-Clifford logical gates in the surface code.

arXiv:2305.15972•Physical Review Letters•Nov 21, 2023•25 citations

Magic state injection on the rotated surface code

Lingling Lao, Ben Criger

Fault-tolerant quantum computing based on the surface code requires magic states to achieve universality. The initially prepared magic states have low fidelity and need to be purified by a costly procedure called magic state distillation. The high spatial-temporal cost for distillation can be reduced by improving the fidelity of initial magic states. Prior works have investigated state injection approaches on the regular surface code. In this work, we propose a magic state injection method for the rotated surface code that requires fewer physical qubits to encode one logical qubit. Analytical results show that our method could achieve lower logical error rates than the state-of-the-art approach for the regular surface code in [1]. When both the two-qubit gate and single-qubit operation error rates are p, the new method results in a logical error rate to 34p/15 compared to 46p/15 in [1]. This fidelity improvement and fewer qubit encoding requirement would help reduce the overhead of fault-tolerant quantum computation, which is crucial for practical implementation in the near term.

ACM International Conference on Computing Frontiers (CF)•May 5, 2022•17 citations

Error-mitigated initialization of surface codes with non-Pauli stabilizers

Zhi-Cheng He, Zheng‐Yuan Xue

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.

arXiv:2411.06407•Physical Review A•Nov 27, 2024•1 citation

Magic State Injection with Erasure Qubits

Shoham Jacoby, Yotam Vaknin, Alex Retzker +1 more

Erasure qubits constitute a promising approach for tackling the daunting resources required for fault-tolerant quantum computing. By heralding of erasure errors, both the error-correction threshold and the sub-threshold scaling of the logical error rate are significantly improved. While previous research has focused primarily on fault-tolerant quantum memories, we extend this investigation to magic state injection—a critical yet resource-intensive component of fault-tolerant quantum computation. We show that, after postselection on erasures, the logical error rate of the injected magic state is set by the residual Pauli error, while the space-time overhead is only marginally increased as compared with the case of non-erasure qubits with a similar noise strength. These conclusions hold both for injection into the surface code and for injection and cultivation on the color code. For the former, we show that most of the gains can be achieved by the use of just three strategically placed erasure qubits in the surface code patch, independent of the patch size. For the latter, in contrast, it is beneficial to have all the qubits in the cultivation patch be erasure qubits. Our results for cultivation suggest that algorithmically relevant logical error rates may be within reach magic state distillation for erasure rates ≲ 4 × 10 − 3 and residual Pauli error rates of ∼ 10 − 4 .

arXiv:2504.02935•PRX Quantum•Sep 4, 2025•3 citations
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.
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.
Implementing fault-tolerant quantum computing necessitates the realization of logical non-Clifford gates, which requires the preparation of specific quantum states known as magic states. However, IBM’s heavy-hexagon structure, which has limited qubit connectivity, presents challenges in adapting quantum error correction codes such as the surface code. Several methods have been proposed to address these challenges by adapting the surface code to the heavy-hexagon architecture. In this study, we implement the magic state injection process within two distinct implementations of surface codes (standard and rotated methods) suitable for the heavy-hexagon structure and compare their logical error rates. Furthermore, we propose initialization methods to enhance the performance of magic state injection in the heavy-hexagon structure, thereby efficiently achieving logical non-Clifford gates with reduced error rates.
Fault-tolerant quantum computing based on surface code has emerged as an attractive candidate for practical large-scale quantum computers to achieve robust noise resistance. To achieve universality, magic states preparation is a commonly approach for introducing non-Clifford gates. Here, we present a hardware-efficient and scalable protocol for arbitrary logical state preparation for the rotated surface code, and further experimentally implement it on the Zuchongzhi 2.1 superconducting quantum processor. An average of 0.8983±0.0002 logical fidelity at different logical states with distance three is achieved, taking into account both state preparation and measurement errors. In particular, the logical magic states |A^{π/4}⟩_{L}, |H⟩_{L}, and |T⟩_{L} are prepared nondestructively with logical fidelities of 0.8771±0.0009, 0.9090±0.0009, and 0.8890±0.0010, respectively, which are higher than the state distillation protocol threshold, 0.859 (for H-type magic state) and 0.827 (for T-type magic state). Our work provides a viable and efficient avenue for generating high-fidelity raw logical magic states, which is essential for realizing non-Clifford logical gates in the surface code.
Fault-tolerant quantum computing based on the surface code requires magic states to achieve universality. The initially prepared magic states have low fidelity and need to be purified by a costly procedure called magic state distillation. The high spatial-temporal cost for distillation can be reduced by improving the fidelity of initial magic states. Prior works have investigated state injection approaches on the regular surface code. In this work, we propose a magic state injection method for the rotated surface code that requires fewer physical qubits to encode one logical qubit. Analytical results show that our method could achieve lower logical error rates than the state-of-the-art approach for the regular surface code in [1]. When both the two-qubit gate and single-qubit operation error rates are p, the new method results in a logical error rate to 34p/15 compared to 46p/15 in [1]. This fidelity improvement and fewer qubit encoding requirement would help reduce the overhead of fault-tolerant quantum computation, which is crucial for practical implementation in the near term.
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.
Erasure qubits constitute a promising approach for tackling the daunting resources required for fault-tolerant quantum computing. By heralding of erasure errors, both the error-correction threshold and the sub-threshold scaling of the logical error rate are significantly improved. While previous research has focused primarily on fault-tolerant quantum memories, we extend this investigation to magic state injection—a critical yet resource-intensive component of fault-tolerant quantum computation. We show that, after postselection on erasures, the logical error rate of the injected magic state is set by the residual Pauli error, while the space-time overhead is only marginally increased as compared with the case of non-erasure qubits with a similar noise strength. These conclusions hold both for injection into the surface code and for injection and cultivation on the color code. For the former, we show that most of the gains can be achieved by the use of just three strategically placed erasure qubits in the surface code patch, independent of the patch size. For the latter, in contrast, it is beneficial to have all the qubits in the cultivation patch be erasure qubits. Our results for cultivation suggest that algorithmically relevant logical error rates may be within reach magic state distillation for erasure rates ≲ 4 × 10 − 3 and residual Pauli error rates of ∼ 10 − 4 .