Quantum error correction with fractal topological codes

Arpit Dua1, Tomas Jochym-O'Connor2,3, and Guanyu Zhu2,3

1Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125 USA
2IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 USA
3IBM Almaden Research Center, San Jose, CA 95120 USA

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Abstract

Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension $2+\epsilon$, which admits a fault-tolerant non-Clifford CCZ gate [1]. We investigate the performance of such FSCs as fault-tolerant quantum memories. We prove that there exist decoding strategies with non-zero thresholds for bit-flip and phase-flip errors in the FSCs with Hausdorff dimension $2+\epsilon$. For the bit-flip errors, we adapt the sweep decoder, developed for string-like syndromes in the regular 3D surface code, to the FSCs by designing suitable modifications on the boundaries of the holes in the fractal lattice. Our adaptation of the sweep decoder for the FSCs maintains its self-correcting and single-shot nature. For the phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder for the point-like syndromes. We report a sustainable fault-tolerant threshold ($\sim 1.7\%$) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by $2.95\%$) for the MWPM decoder for a particular FSC with Hausdorff dimension $D_H\approx2.966$. The latter can be mapped to a lower bound of the critical point of a confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.

Topological codes are a crucial class of error-correcting codes due to local interactions and high error-correcting thresholds. In the past, these codes have been widely studied on $D$-dimensional regular lattices corresponding to tessellations of manifolds. Our work is the first study of error correction protocols and decoders on fractal lattices, which could significantly reduce the space-time overhead for fault-tolerant universal quantum computation. We overcome the challenge of decoding in the presence of the holes at all length scales in the fractal lattice. In particular, we present decoders with provably non-zero error correction thresholds for both point-like and string-like syndromes on the fractal lattice. Remarkably, the desired properties of self-correction and single-shot correction for the string-like syndromes are still maintained in our decoding scheme, even when the fractal dimension approaches two. Such properties were thought to be only possible in three-dimensional (or higher) codes.

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Cited by

[1] Neereja Sundaresan, Theodore J. Yoder, Youngseok Kim, Muyuan Li, Edward H. Chen, Grace Harper, Ted Thorbeck, Andrew W. Cross, Antonio D. Córcoles, and Maika Takita, "Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders", Nature Communications 14 1, 2852 (2023).

[2] Arpit Dua, Nathanan Tantivasadakarn, Joseph Sullivan, and Tyler D. Ellison, "Engineering 3D Floquet Codes by Rewinding", PRX Quantum 5 2, 020305 (2024).

[3] Eric Huang, Arthur Pesah, Christopher T. Chubb, Michael Vasmer, and Arpit Dua, "Tailoring Three-Dimensional Topological Codes for Biased Noise", PRX Quantum 4 3, 030338 (2023).

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