Topological quantum error correction in fractal dimensions I: code construction and logical gates

Topological error correcting codes and topological orders in integer spatial dimensions have been widely studied in the fields of quantum information and condensed matter physics. In this work, we consider topological codes defined on a wide class of fractal lattices, which can be considered as a usual d-dimensional lattice with holes at all length scales and correspond to fractal (Hausdorff) dimension $d-\delta$ ($\delta>0$). For simplicity, we call these lattices d-dimensional fractal lattices. We first prove a no-go theorem that topological orders on 2D fractals with Hausdorff dimension $2-\delta$ do not exist in nature. We further construct topological codes on three and higher-dimensional fractals. An important application of these codes is to reduce the space overhead for implementing non-Clifford logical gates. Based on the results of Bravyi and Koenig, there is a trade-off between dimensionality and universality in topological stabilizer codes, i.e., only higher-dimensional codes can implement fault-tolerant logical gates in higher levels of the Clifford hierarchy via local constant depth circuits. By constructing fractal topological codes, we can lower the Hausdorff dimension of the codes and hence reduce the number of qubits needed for a given logical non-Clifford gate.