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ACM International Conference on Computing Frontiers (CF)

Magic state injection on the rotated surface code

Lingling Lao, Ben Criger

Published May 5, 2022• 17 citations
Publisher

Abstract

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.

DOI
10.1145/3528416.3530237

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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.
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.
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.
Zero-level CCZCCZCCZ 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 TTT gate and the CCZCCZCCZ 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 CCZCCZCCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T†T/T^\daggerT/T† operation of the [[8,3,2]][[ 8,3,2 ]][[8,3,2]] code to fault-tolerantly encode the state CCZ‾∣+++⟩\overline{CCZ}|+++\rangleCCZ∣+++⟩, 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×p2p_L \simeq 300 \times p^2pL​≃300×p2 with respect to the physical error rate ppp. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3p = 10^{-3}p=10−3 and p=10−4p = 10^{-4}p=10−4, respectively, compared to conventional seven-TTT-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 CCZCCZCCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional TTT-state generation.
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.
I reduce the cost to prepare magic states with lattice surgery operations on the surface code by using a recursive implementation of 15-to-1 magic state distillation. On a rotated surface code with distance ddd, ∣T⟩|T\rangle∣T⟩ preparation requires a ddd-by-3d3 d3d grid of data qubits for up to 15d15 d15d error correction cycles, and ∣CCZ⟩|CCZ\rangle∣CCZ⟩ preparation requires a 3d3 d3d-by-2d2 d2d grid for up to 10.5d10.5 d10.5d cycles. However, a significantly lower physical error threshold than that of the underlying surface code is required to match the error probability of the output magic state with the logical error rate of the output surface code at large code distances.
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.