Quantum Codes from High-Dimensional Manifolds

Abstract

We construct toric codes on various high-dimensional manifolds. Assuming a conjecture in geometry we find families of quantum CSS stabilizer codes on $N$ qubits with logarithmic weight stabilizers and distance $N^{1-\epsilon }$ for any $\epsilon >0$. The conjecture is that there is a constant $C>0$ such that for any $n$-dimensional torus ${\mathbb{T}}^n={\mathbb{R}}^n/\Lambda$, where $\Lambda$ is a lattice, the least volume unoriented $n/2$-dimensional surface (using the Euclidean metric) representing nontrivial homology has volume at least $C^n$ times the volume of the least volume $n/2$-dimensional hyperplane representing nontrivial homology; in fact, it would suffice to have this result for $\Lambda$ an integral lattice with the surface restricted to faces of a cubulation by unit hypercubes. The main technical result is an estimate of Rankin invariants\cite{rankin} for certain random lattices, showing that in a certain sense they are optimal. Additionally, we construct codes with square-root distance, logarithmic weight stabilizers, and inverse polylogarithmic soundness factor (considered as quantum locally testable codes\cite{qltc}). We also provide an short, alternative proof that the shortest vector in the exterior power of a lattice may be non-split\cite{coulangeon}.

19 pages

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• Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds
• Circuit lower bounds for low-energy states of quantum code Hamiltonians