Fold-transversal surface code cultivation

The gist: This work presents a new cultivation scheme for magic states in surface codes that achieves the lowest known spacetime overhead by leveraging fold-transversal gates and unitary techniques within the surface code family.

Protocol Overview

The Magic State Cultivation (MSC) protocol involves three main stages: Injection, Cultivation, and Escape The protocol is summarized in Fig. 1 The goal is to prepare a high-fidelity eigenstate of the logical operator HXY = (X + Y)/√2 in a rotated surface code This state is used to implement the logical T = e−iπZ/8 gate via teleportation.

Cultivation Details

The cultivation stage involves several steps, including transformation and measurement 1. Transformation: The protocol transforms the initial code into an unrotated surface code, Reg(3). 2. Logical Clifford Measurement: The fold-transversal logical HXY operator is measured twice using a three-qubit ancilla GHZ state. This operation uses some three-qubit gates, as shown in Appendix A. 3. Postselection: Postselecting on both the stabilizer measurements and the decoded GHZ measurements increases the fault distance of the protocol to f = 3.

Escape Strategy

The escape stage is designed to protect the magic state from future errors by growing the code This three-step escape strategy involves: 1. Reg(3) is grown to Rot(5) via a local unitary circuit. 2. Rot(5) is grown to Rot(7) using a conventional stabilizer measurement approach. 3. Rot(7) is unitarily grown to a final larger-distance code. The stabilizer measurements obtained from the growth to Rot(7) are used for additional postselection via a complementary-gap-based decoder.

Performance and Comparison

The proposed scheme achieves lower logical error rates and significantly lower spacetime overheads compared to previous schemes The performance is summarized by Fig. 2(a) For the Y⟩ state cultivation, our scheme requires a lower number of expected attempts for any target LER compared to Refs. [9, 10] (blue and purple, respectively). Our scheme has a significantly lower overhead, owing to the efficient cultivation and escape steps that use nonlocal operations.

Implementation Advantages

The protocol is most naturally implemented with non-local gates These are native to quantum computing platforms such as neutral atoms [1] and trapped ions [2]. The work demonstrates the dramatic reduction in requirements for scalable quantum computing with surface codes when non-local gates are available. The use of unitary growth is crucial for maintaining a low spacetime overhead compared to using possibly slower and lower-fidelity stabilizer measurements on the larger-distance final code.

Simulation and Benchmarking

The performance of our magic state cultivation scheme is quantified by comparing it against existing protocols We introduce a handoff approach, where outputs from a state vector simulator are passed to a stabilizer simulator to study exact HXY⟩ state cultivation for the first time. The logical error rates for HXY⟩ cultivation are virtually indistinguishable from those for Y⟩ state cultivation in full simulations. The expected spacetime volume comparison shows that our protocol demonstrates significant improvement in spacetime volume compared to Refs. [9, 10]. This is attributed to the use of a more efficient check circuit and the three-stage escape strategy.

Noise Model Analysis

We examine the performance under physically-motivated noise models such as PM1 and PM5 We observe that PM noise models lead to reduced LERs compared to SD6 even when non-local gates are heavily penalized. Our protocol requires less expected attempts for a given LER compared to Ref. [12]. The final success rates are shown in Fig. A14

Conclusion and Future Work

This work presents a fold-transversal cultivation scheme for the surface code Our protocol achieves the lowest known spacetime overhead for MSC. It is conceivable to simulate fault distances higher than f = 5 with lower LERs, perhaps low enough that further distillation is not even required. Future research should extend magic state cultivation to other quantum low-density parity check (LDPC) codes hosting fold-transversal gates. The work aims to inspire new surface code protocols that leverage experimental advances in non-local operations for overhead reduction. The authors thank Ludwig Schmid for modifying the unitary encoding optimizers of Ref. [37]. Related work includes an updated version of Ref. [12] with an investigation of H⟩ state cultivation on the surface code and surface code cultivation using two-qubit gates. The data availability is provided at Ref. [39]. The final results are presented in Section IV.

Appendix Details

Appendix A provides circuits for key stages of our protocol, including the HXY check circuit for Reg(3) Appendix B discusses alternative choices for each stage, including the magic state, injection strategy, GHZ ancilla size, and escape strategy. Appendix C outlines additional steps required for protocols with a higher fault distance f = 5. Appendix D describes our ‘handoff’ simulation that combines both state vector and stabilizer simulators to accurately estimate the LER for H¯XY⟩ magic state cultivation. The results show that the logical error rate of H¯XY⟩ cultivation are virtually indistinguishable from those for Y⟩ state cultivation in full simulations. The comparison between Y¯ ⟩ and H¯XY⟩ state cultivation shows that the relationship between the LER of cultivation and the expected attempts is very similar for the two different states. The paper concludes that magic state preparation can be exactly simulated on a stabilizer simulator if nonClifford gates transform Pauli errors in a simple way. The noise models used are detailed in Table A3. The spacetime volume calculation is defined as V = 1/fM PM i=0 fiVi. The comparison of protocols for f = 3 and f = 5 is summarized in Tables A4 and A5. The survival of shots during cultivation is shown in Fig. A14.

References

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[11] J.

1. textbfProcessing for Ultra-High Fidelity Magic State Generation on Non-Local Architectures: (Improved Quantum Computation Engine). This system can perform magic state cultivation using a new protocol that achieves the lowest known spacetime overhead for magic state cultivation by leveraging non-local gates, making it ideally suited for architectures like neutral atoms or trapped ions.

2. textbfAdaptive Fault Distance Control: (Dynamic QEC Protocol Manager). The system can dynamically adjust the required fault distance based on the target logical error rate, as demonstrated by extending protocols to f = 5 and finding that a simple modification that changes our f = 5 scheme to f = 4 achieves comparable logical error rate with lower overhead.

3. textbfState-of-the-Art Overhead Optimization: (Low Spacetime Overhead Compiler). The AI can automatically select the most efficient cultivation subroutines, as shown in Table A4 and A5, ensuring significantly lower spacetime overheads compared to previous schemes by choosing between different injection and escape strategies.

4. textbf Exact Magic State Characterization: (Handoff Simulation Module). The system can perform exact HXY⟩ magic state cultivation by implementing a state vector to stabilizer handoff simulation, which allows it to compare the performance of the new protocol against previous estimates, as detailed in Section D.

5. textbf Real-Time Noise Model Adaptation: (Physically Motivated Error Simulator). The system can adapt its error handling based on physical hardware constraints by simulating noise models like PM1 and PM5, observing that PM noise models leads to reduced LERs compared to SD6 even when non-local gates are heavily penalized.