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

Subtitle: Four Codes from One Geometry via Orientation Reversal
Author: Yannick Schmitt
Date: March 2026
Status: Preprint 1.0.0

Overview

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:

Code	Parameters	Highlights
Code I	[[12, 4, (4,2)]]	Rate 1/3; corrects X-errors, detects Z-errors
Code II	[[12, 1, (4,3)]]	Balanced; circuit-level threshold p_c ≈ 3.5%
Dual A	[[12, 2, (2,6)]]	New; corrects all weight-1 and weight-2 Z-errors
Dual II	[[12, 1, (3,4)]]	New; dZ = 4, preferred for dephasing-dominated hardware

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 Structure

• /paper - LaTeX source files and PDF pre-print of the manuscript.
• /script - Verification script

Verification Script

All theorems, propositions, constructions, and numerical claims in the paper are verified by verification_CSS_Duality_CD_QEC.py. The script structure mirrors the paper sections exactly, and every check prints PASS or FAIL with its measured value.

What the script verifies

Section	Content
Sec 2 — Causal Diamond Geometry	Lightlike enumeration; 21 plaquettes; Laplacian spectrum; rank over R and GF(2)
Sec 3 — GF(2) Rank Gap	rank_R(M) = 8, rank_{F2}(M) = 7; all-ones vector as extra flat direction; 12 distinct weight-7 syndromes
Sec 4 — Lorentzian CSS Duality	Duality map Φ; isometric syndrome structure; primal ↔ dual distance exchange
Sec 5 — Code I [[12,4,(4,2)]]	Parameters; disjoint weight-4 X-stabiliser decomposition; code-capacity noise simulation
Sec 6 — Code II [[12,1,(4,3)]]	1248 valid X-check sets; dZ = 3 with 16 weight-3 Z-logicals; role of all-ones vector
Sec 7 — Dual A [[12,2,(2,6)]]	CSS commutativity proof; dZ = 6; 24 weight-6 Z-logicals; code-capacity performance
Sec 8 — Dual II [[12,1,(3,4)]]	Parameters; dZ = 4 correction; P_log(Z) = 0.001 at p = 0.01
Sec 9 — Two-Stage Protocol	Stage 1 X-correction and Stage 2 Z-correction; combined P_log; comparison table
Sec 10 — No-Go Theorems	Pigeonhole No-Go; X-Decoration Equivalence; exhaustive pair-killing counts
Sec 11 — Symmetry Group	Order-96 subgroup; rowspace preservation; duality symmetry
Sec 12 — Circuit-Level Thresholds	Code I and Code II circuit depths; p_c ≈ 3.5% for Code II
Miscellaneous	Additional checks not directly cited in the paper text

Requirements

numpy

Running the script

python verification_CSS_Duality_CD_QEC.py

The script has a fast mode for a quick smoke-test (~2 minutes) and a full mode for complete paper-value verification. The most computationally expensive sections are the Monte Carlo noise simulations (Sec 9) and circuit-level threshold simulations (Sec 12). To enable fast mode, set FAST = True near the top of the script; this reduces trial counts from 20,000 to 2,000 for code-capacity simulations and from 10,000 to 500 for circuit-level simulations.

Key Code Parameters

Code	n	k	dX	dZ	Rate	Max stabiliser weight
Code I	12	4	4	2	1/3	4
Code II	12	1	4	3	1/12	6
Dual A	12	2	2	6	1/6	4
Dual II	12	1	3	4	1/12	6

All four codes share the same 12-qubit physical layout; the qubit interaction graph is K_12 (all-to-all connectivity), making neutral atom arrays and trapped-ion platforms the natural hardware targets.

Series Context

• Paper 1: Exact Discretisation and Boundary Observables in Lorentzian Causal Diamonds — establishes the geometry, Laplacian spectrum, and BF crossover. Yannick Schmitt. (2026). Zenodo. https://doi.org/10.5281/zenodo.19338306
• Paper 4: Algebraic Structure of the D4 Causal Diamond — synthesises the geometry with symmetry group analysis, mass renormalisation, entanglement entropy, and MOND phenomenology. Yannick Schmitt. (2026). Zenodo. https://doi.org/10.5281/zenodo.19343721

Citation

If you use this work, please cite it as:

Yannick Schmitt. (2026). A Lorentzian CSS Duality in Causal Diamond Quantum Error-Correcting Codes. Zenodo. https://doi.org/10.5281/zenodo.19343889

License

• The source code in this repository is licensed under the Apache License 2.0.
• The documentation, LaTeX source files, and PDF papers are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).