Quantum error correction with fractal topological codes

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. 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 $D_H\approx 2.966$. 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.

24+13 pages, 10 figures, the simulation code is available at https://github.com/dua-arpit/single_shot_decoding_fractal, minor changes made in v2 and v3, v3 version uses the Quantum journal LaTeX template

References in corpus (5)

• Surface codes: Towards practical large-scale quantum computation
• Topological Quantum Distillation
• Quantum memories based on engineered dissipation
• Error Threshold for Color Codes and Random 3-Body Ising Models
• Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries

Cited by in corpus (6)

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• Engineering 3D Floquet codes by rewinding
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• Entanglement Fractalization
• Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate beyond the $n^{1/3}$ Distance Barrier
• Magic Boundaries of 3D Color Codes