Optimal Locality and Parameter Tradeoffs for Subsystem Codes

2025-03-28

AI Summary

This work establishes fundamental locality–parameter trade-offs for quantum subsystem codes embedded in D-dimensional Euclidean space, focusing on lower bounds for the number of interactions (M^*) and their maximum length (ell^*). Using combinatorial topology, geometric embedding theory, and D-dimensional lattice coding techniques, we derive tight (Omega)-bounds for arbitrary D-dimensional embeddings: (M^* = Omega(max(k,d))) and (ell^* = Omegaig(maxig(d/n^{(D-1)/D},, (k d^{1/(D-1)}/n)^{(D-1)/D}ig)ig)). These bounds generalize to commuting projector codes. We further provide explicit constructions achieving both bounds, thereby proving their tightness. Our results expose an intrinsic tension between locality constraints—quantified by interaction number and range—and error-correction capability—governed by code parameters ((n,k,d))—and establish a theoretical benchmark for designing high-dimensional local quantum error-correcting codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We study the tradeoffs between the locality and parameters of subsystem codes. We prove lower bounds on both the number and lengths of interactions in any $D$-dimensional embedding of a subsystem code. Specifically, we show that any embedding of a subsystem code with parameters $[[n,k,d]]$ into $mathbb{R}^D$ must have at least $M^*$ interactions of length at least $ell^*$, where [ M^* = Omega(max(k,d)), quad ext{and}quad ell^* = Omegaigg(maxigg(frac{d}{n^frac{D-1}{D}}, igg(frac{kd^frac{1}{D-1}}{n}igg)^frac{D-1}{D}igg)igg). ] We also give tradeoffs between the locality and parameters of commuting projector codes in $D$-dimensions, generalizing a result of Dai and Li. We provide explicit constructions of embedded codes that show our bounds are optimal in both the interaction count and interaction length.

Problem

• Study tradeoffs between locality and subsystem code parameters
• Prove lower bounds on interaction count and length in D-dimensional embeddings
• Provide optimal constructions for interaction count and length bounds

Innovation

• Proves lower bounds on subsystem code interactions
• Generalizes locality-parameter tradeoffs for codes
• Provides optimal embedded code constructions