Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries

Abstract

We investigate topological order on fractal geometries embedded in $n$ dimensions. In particular, we diagnose the existence of the topological order through the lens of quantum information and geometry, i.e., via its equivalence to a quantum error-correcting code with a macroscopic code distance or the presence of macroscopic systoles in systolic geometry. We first prove a no-go theorem that ${\mathbb{Z}}_N$ topological order cannot survive on any fractal embedded in 2D. For fractal lattice models embedded in 3D or higher spatial dimensions, ${\mathbb{Z}}_N$ topological order survives if the boundaries of the interior holes condense only loop or membrane excitations. Moreover, for a class of models containing only loop or membrane excitations, and are hence self-correcting on an $n$-dimensional manifold, we prove that topological order survives on a large class of fractal geometries independent of the type of hole boundaries. We further construct fault-tolerant logical gates using their connection to global and higher-form topological symmetries. In particular, we have discovered a logical CCZ gate corresponding to a global symmetry in a class of fractal codes embedded in 3D with Hausdorff dimension asymptotically approaching $D_H=2+\epsilon$ for arbitrarily small $\epsilon$, which hence only requires a space-overhead $\Omega (d^{2+\epsilon })$ with $d$ being the code distance. This in turn leads to the surprising discovery of certain exotic gapped boundaries that only condense the combination of loop excitations and gapped domain walls. We further obtain logical ${\text{C}}^p\text{Z}$ gates with $p\le n-1$ on fractal codes embedded in $n$D. In particular, for the logical ${\text{C}}^{n-1}\text{Z}$ in the $n^{\text{th}}$ level of Clifford hierarchy, we can reduce the space overhead to $\Omega (d^{n-1+\epsilon })$. Mathematically, our findings correspond to macroscopic relative systoles in fractals.

arXiv comment46+10 pages, fixed typos and the table content, updated funding information

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• Cross-cap defects and fault-tolerant logical gates in the surface code and the honeycomb Floquet code
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• Laughlin topology on fractal lattices without area law entanglement
• Non-Pauli Errors in the Three-Dimensional Surface Code
• Code Properties of the Holographic Sierpinski Triangle
• Entanglement Fractalization
• Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries
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• Approximate Hofstadter- and Kapit-Mueller-like parent Hamiltonians for Laughlin states on fractals
• Magic Boundaries of 3D Color Codes
• Fermionic defects of topological phases and logical gates
• Tessellation codes: encoded quantum gates by geometric rotation
• The geometric phase transition of the three-dimensional ${\mathbb{Z}}_2$ lattice gauge model
• Symmetry-Accelerated Classical Simulation of Clifford-Dominated Circuits
• Color code with a logical control-$S$ gate using transversal $T$ rotations