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Tradeoffs for reliable quantum information storage in 2D systems
Sergey Bravyi, David Poulin, Barbara Terhal
TL;DR
The paper studies codewords and distances in a two-dimensional lattice setting with local constraints. It derives an exact distance for one lattice family and reports numerical distance scaling for larger lattices.
Problem
The paper examines how code distance behaves for code constructions on a two-dimensional lattice.
Method
The analysis uses codeword locality arguments, combinatorial bounds, explicit codewords, and exhaustive-search optimization for selected lattice sizes.
Results
d(p) = 3p for lattices of height L = 2p, while numerical searches up to L ∼10^4 find d′ ∼L^1.584.
Takeaways & Limitations
The reported lattice family has an exactly characterized distance, and the larger-size numerical results indicate a power-law distance scaling.
Abstract
from arXiv · showhide
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).
APPENDIX A
This appendix studies 2D classical codes built from local constraints and cellular automata, showing that their distance can approach the classical storage tradeoff. The construction encodes information in initial conditions, with distance characterized analytically and numerically.
- Classical-code bound: 2D classical codes obey local constraints on constant-alphabet variables, with distance defined by the minimum separation between distinct codewords.The code encodes k bits when its codespace contains 2^k assignments.
- Classical-code bound: A block-separation argument shows that a codeword is uniquely determined outside blocks containing fewer than d sites, yielding the entropy bound k ≤ |A| ∼ n.The uniform distribution on codewords has zero conditional entropy because the restriction to A uniquely determines the extension to B.
- Cellular-automaton construction: Cellular-automaton codes use the rule x_i^t → x_{i−1}^{t+1} ⊕ x_{i+1}^{t+1}, so every codeword is determined by its first row.The first row supplies the cellular automaton's initial conditions.
- Distance scaling: For height L = 2p, the semi-infinite cellular-automaton code has exact distance d(p) = 3p, attained by a Sierpinski-triangle codeword.The matching upper bound comes from initial conditions with a single active cell.
- Cellular-automaton construction: The construction decomposes into four sublattices that reproduce the original code, and any nonzero codeword occupies at least three of them.This gives the lower bound d(p) ≥ 3p.