Simulations

The Quantum Family Tree trilogy is supported by two complementary streams of computational evidence: classical Monte Carlo simulations at large tree depths, and an independent verification run on real quantum hardware. Below is a summary of the main results. Full details are in the trilogy papers, and reproducibility code is in the GitHub repository.

The headline result

The quenched von Neumann entanglement entropy along boundary intervals grows linearly with tree distance at a rate that agrees with the analytic prediction (9/10) · log₂(5/2) ≈ 1.1897 bits per generation to 10.34 standard deviations. The measurement combines six independent tree depths (D = 20, 30, 40, 50, 100, 150) with 500+ independent trees per depth.

Quantum hardware verification

An independent experimental run on IBM's 156-qubit Heron quantum processor reproduces the entropy growth signature predicted by the trilogy. The classical simulations and the quantum run agree within error bars, giving two independent physical platforms supporting the same result.

Exact moments and Rényi-2 rate

These rational values drop out of Haar integration on U(4) via Weingarten calculus. They are not fit. The h_S2 = 14/45 identity is derived from the exact (b,c) recursion preserving position-variance cross-correlations; numerical simulation recovers it to within Monte Carlo precision.

First law of entanglement

The first law of entanglement entropy — the thermodynamic identity connecting modular energy to entropy — holds at all tested scales. This was one of the outstanding problems of the earlier (v30) version of the work; it is now an unconditional theorem.

One scalar identity remains open

Historical: v30 depth-12 GPU simulations

The trilogy's exact results subsume an earlier depth-12 GPU simulation run (March 2026, ∼8.4 million leaf pairs), which remains useful for its direct visualization of the emergent geometry. The figures below are from that run.

The information distance d_I(i,j) = −log MI(i,j) satisfies the ultrametric strong triangle inequality for 98.3% of triples. This places the emergent geometry in the same mathematical class as p-adic spaces and Bruhat–Tits trees, the structures used in p-adic holography (Gubser et al. 2017).

Figure: Two-dimensional MDS embedding of information distance d_ij = −log MI_ij between 256 leaves at depth 8 (single tree). Left: leaves colored by top-level subtree cluster. Right: leaves colored by distance from leaf 0. Negative eigenvalues (82 of 256) confirm non-Euclidean geometry. Variance explained by 2D projection: 17% (expected for high-dimensional tree metric).

MDS eigenvalue spectrum. The negative eigenvalues are the signature of non-Euclidean (hyperbolic) geometry — the tree metric cannot be embedded in flat space.