Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code

T. R. Scruby1,2, D. E. Browne2, P. Webster3, and M. Vasmer4,5

1Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan
2Dept. of Physics and Astronomy, University College London, London, WC1E 6BT, UK
3Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia
4Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
5Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada

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We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop and simulate an explicit recipe for a just-in-time decoding scheme in three 3D surface codes, which can be used to implement a transversal (non-Clifford) $\overline{CCZ}$ between three 2D surface codes in time linear in the code distance. We present a fully detailed set of bounded-height lattice slices through the 3D codes which retain the code distance and measurement-error detecting properties of the full 3D code and admit a dimension-jumping process which expands from/collapses to 2D surface codes supported on the boundaries of each slice. At each timestep of the procedure the slices agree on a common set of overlapping qubits on which $CCZ$ should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic $X$ and measurement errors and find evidence for a threshold $p_c \sim 0.1\%$ in all three codes. We expect that this threshold could be improved by optimisation of the decoder.

For an animated illustration of 3D spacetime diagram click here.

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[1] Suhas Vittal, Poulami Das, and Moinuddin Qureshi, Proceedings of the 50th Annual International Symposium on Computer Architecture 1 (2023) ISBN:9798400700958.

[2] Ying-Jie 英杰 Qu 曲, Zhao 钊 Chen 陈, Wei-Jie 伟杰 Wang 王, and Hong-Yang 鸿洋 Ma 马, "Approximate error correction scheme for three-dimensional surface codes based reinforcement learning", Chinese Physics B 32 10, 100307 (2023).

[3] Kaavya Sahay, Junlan Jin, Jahan Claes, Jeff D. Thompson, and Shruti Puri, "High-Threshold Codes for Neutral-Atom Qubits with Biased Erasure Errors", Physical Review X 13 4, 041013 (2023).

[4] Thomas R. Scruby, Michael Vasmer, and Dan E. Browne, "Non-Pauli errors in the three-dimensional surface code", Physical Review Research 4 4, 043052 (2022).

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[6] Michael E. Beverland, Aleksander Kubica, and Krysta M. Svore, "Cost of Universality: A Comparative Study of the Overhead of State Distillation and Code Switching with Color Codes", PRX Quantum 2 2, 020341 (2021).

[7] Paul Webster, Michael Vasmer, Thomas R. Scruby, and Stephen D. Bartlett, "Universal fault-tolerant quantum computing with stabilizer codes", Physical Review Research 4 1, 013092 (2022).

[8] Andreas Bauer, "Low-overhead non-Clifford topological fault-tolerant circuits for all non-chiral abelian topological phases", arXiv:2403.12119, (2024).

[9] Mark Webber, Vincent Elfving, Sebastian Weidt, and Winfried K. Hensinger, "The impact of hardware specifications on reaching quantum advantage in the fault tolerant regime", AVS Quantum Science 4 1, 013801 (2022).

[10] Paul Webster, Michael Vasmer, Thomas R. Scruby, and Stephen D. Bartlett, "Universal Fault-Tolerant Quantum Computing with Stabiliser Codes", arXiv:2012.05260, (2020).

[11] Thomas R. Scruby, Michael Vasmer, and Dan E. Browne, "Non-Pauli Errors in the Three-Dimensional Surface Code", arXiv:2202.05746, (2022).

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