A Lorentzian CSS Duality in Causal Diamond Quantum Error-Correcting Codes

A Lorentzian CSS Duality in Causal Diamond Quantum Error-Correcting Codes presents a comprehensive analysis of discrete Lorentzian causal diamonds, revealing their ability to generate four distinct CSS quantum error-correcting codes through a geometric duality. The basis of this work is the exploration of the twelve lightlike nearest-neighbor vectors within the ternary Minkowski lattice {-1, 0, +1}^4, where a significant structural feature, the distance asymmetry dZ = 2, is shown to be a result of inherent algebraic properties rather than mere observation.

Key Findings

The research posits that the causal diamond framework allows for two definitive orientations of its boundary:

• Lorentzian Orientation: Links are incoming from the past, producing an effective temporal charge of n^0_eff = 12. This setting leads to an all-ones X-check.
• Euclidean Orientation: All links are outgoing, resulting in a vanishing boundary sum. This configuration gives rise to a dual code family where 21 plaquettes function as X-type checks, complemented by three spatial-axis groups serving as Z-type checks. Significantly, these orientations are interconnected through Wick rotation, underpinning a profound duality between the primal and dual code families.

Derived Codes

The research introduces four unique codes derived from this geometric framework:

Validation and Verification

The repository includes a verification script (verification_CSS_Duality_CD_QEC.py) that ensures the accuracy of all theorems and numerical claims presented in the paper. This script adheres to the structure of the paper and outputs the results of each verification with PASS or FAIL. Specific checks encompass a range of topics, including causal diamond geometry, rank gaps, duality, error correction performance, and circuit-level thresholds.

Repository Structure

• /paper: Contains LaTeX source files along with a PDF preprint of the manuscript.
• /script: Includes the verification script for validating code correctness.

Requirements

To run the verification script, the following Python package is necessary:

numpy

Example Usage

The script can be executed in Python as follows:

python verification_CSS_Duality_CD_QEC.py

This script supports both a fast mode for quick verifications and a full mode for comprehensive assessments, including Monte Carlo simulations and circuit-level threshold evaluations.

References

This work fits into a broader series of research studies that dive into the complex relationship between quantum mechanics and geometric structures. Future works will continue to build upon this insightful foundation.