Abstract
We construct toric codes on various highdimensional 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 ndimensional torus {\mathbb T}^n={\mathbb R}^n/\Lambda, where \Lambda is a lattice, the least volume unoriented n/2dimensional cycle (using the Euclidean metric) representing nontrivial homology has volume at least C^n times the volume of the least volume n/2dimensional hyperplane representing nontrivial homology; in fact, it would suffice to have this result for \Lambda an integral lattice with the cycle restricted to faces of a cubulation by unit hypercubes.
The main technical result is an estimate of Rankin invariants for certain random lattices, showing that in a certain sense they are optimal.
Additionally, we construct codes with squareroot distance, logarithmic weight stabilizers, and inverse polylogarithmic soundness factor (considered as quantum locally testable codes.
We also provide an short, alternative proof that the shortest vector in the exterior power of a lattice may be nonsplit.
BibTeX  Entry
@InProceedings{hastings:LIPIcs:2017:8170,
author = {Matthew B. Hastings},
title = {{Quantum Codes from HighDimensional Manifolds}},
booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
pages = {25:125:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770293},
ISSN = {18688969},
year = {2017},
volume = {67},
editor = {Christos H. Papadimitriou},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8170},
URN = {urn:nbn:de:0030drops81708},
doi = {10.4230/LIPIcs.ITCS.2017.25},
annote = {Keywords: quantum codes, random lattices, Rankin invariants}
}
Keywords: 

quantum codes, random lattices, Rankin invariants 
Collection: 

8th Innovations in Theoretical Computer Science Conference (ITCS 2017) 
Issue Date: 

2017 
Date of publication: 

28.11.2017 