Quantum error correction with fractal topological codes

TL;DR

This paper demonstrates that fractal surface codes with Hausdorff dimension greater than 2 can serve as fault-tolerant quantum memories, with proven non-zero thresholds for errors using adapted decoding strategies.

Contribution

It introduces fault-tolerant decoding strategies for fractal topological codes and establishes their error thresholds, extending quantum error correction to fractal lattice structures.

Findings

Achieved a fault-tolerant threshold of ~1.7% for the sweep decoder.

Established a code capacity threshold lower bound of 2.95%.

Mapped thresholds to a confinement-Higgs transition on fractal lattices.

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 $D_H\approx2.966$. 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.