Tradeoffs for reliable quantum information storage in 2D systems

TL;DR

This paper establishes fundamental limits on quantum information storage in 2D systems, deriving a tradeoff relation between the number of encoded qubits, code distance, and system size.

Contribution

It introduces a new tradeoff relation for quantum error correcting codes in 2D, linking key parameters and highlighting limitations of reliable quantum storage.

Findings

Derived the relation kd^2=O(n) for quantum codes

Compared quantum and classical storage tradeoffs

Showed limitations depend on locality and particle dimension

Abstract

We ask whether there are fundamental limits on storing quantum information reliably in a bounded volume of space. To investigate this question, we study quantum error correcting codes specified by geometrically local commuting constraints on a 2D lattice of finite-dimensional quantum particles. For these 2D systems, we derive a tradeoff between the number of encoded qubits k, the distance of the code d, and the number of particles n. It is shown that kd^2=O(n) where the coefficient in O(n) depends only on the locality of the constraints and dimension of the Hilbert spaces describing individual particles. We show that the analogous tradeoff for the classical information storage is k\sqrt{d} =O(n).