A Lorentzian CSS Duality in Causal Diamond Quantum Error-Correcting Codes - Four Codes from One Geometry via Orientation Reversal

Abstract

This paper shows that the discrete Lorentzian causal diamond D — the two-complex built on the twelve lightlike nearest-neighbour vectors of the ternary Minkowski lattice {-1, 0, +1}^4 — generates not two but four CSS quantum error-correcting codes via a geometric duality, and that this duality makes the distance asymmetry dZ = 2 of the original construction algebraically inevitable rather than merely observed.

The central mechanism is that the causal diamond supports two natural orientations of its boundary: the Lorentzian orientation, in which past links are incoming, produces the temporal charge n^0_eff = 12 and motivates an all-ones X-check; the Euclidean orientation, in which all links are outgoing and the boundary sum vanishes, yields a dual code family in which the 21 plaquettes serve as X-type checks and the three spatial-axis groups serve as Z-type checks. These two orientations are related by Wick rotation t → ix, so the duality between the primal and dual code families is a quantum error-correction realisation of Wick rotation.

The four codes derived from this geometry are:

Additional contributions include: a two-stage combined protocol that measures the 21 plaquettes alternately in Z- and X-basis to correct both error types simultaneously (P_log = 0.006 at p = 0.01 with k_eff = 2), a Pigeonhole No-Go theorem proving that dZ ≥ 3 with k ≥ 2 is impossible in the primal CSS family, and an X-Decoration Equivalence theorem extending this bound to weight-≤6 non-CSS codes.

Repository Contents

• Code: Numerical verification script

• Paper: Full PDF pre-print and original LaTeX source files.

License Information

Please note the dual-licensing structure of this repository:

• Software/Source Code: Licensed under the Apache License 2.0.

• PDF Document & LaTeX Source: Licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0).