Local Quantum Codes from Subdivided Manifolds
Abstract
For $n \ge 3$, we demonstrate the existence of quantum codes which are local in dimension $n$ with $V$ qubits, distance $V^{\frac{n1}{n}}$, and dimension $V^{\frac{n2}{n}}$, up to a $polylog(V)$ factor. The distance is optimal up to the polylog factor. The dimension is also optimal for this distance up to the polylog factor. The proof combines the existence of asymptotically good quantum codes, a procedure to build a manifold from a code by FreedmanHastings, and a quantitative embedding theorem by GromovGuth.
 Publication:

arXiv eprints
 Pub Date:
 March 2023
 DOI:
 10.48550/arXiv.2303.06755
 arXiv:
 arXiv:2303.06755
 Bibcode:
 2023arXiv230306755P
 Keywords:

 Quantum Physics;
 Mathematics  Differential Geometry