Chapter 14: Computational Architecture
Chapter 14: Computational Architecture
14.1 Bounded Algorithmic Number (BAN) Arithmetic
A $p$-adic number $x$ is represented as $(v_p(x), \text{digits})$ for exact ultrametric valuation processing. BAN arithmetic preserves the ultrametric inequality exactly — the distinction structure is maintained without floating-point violations. Every arithmetic operation respects the nested-distinction geometry.
14.2 Compilation Pipeline
- Decompose target unitary into elementary gates — elementary distinction operations
- Map to tree automorphisms (vertex shifts, branch permutations, subtree swaps)
- Optimize exploiting tree geometry — commuting operations on disjoint subtrees
- Verify threshold exceedance for deterministic execution — the gate pulse must clear the inter-level energy barrier
14.3 van der Put Neural Networks (v-PuNNs)
Measurement on the boundary risks topological distortion from the Monna projection. v-PuNNs use the van der Put basis — characteristic functions of nested balls — to process boundary signals. Their architecture mirrors the tree structure of distinctions, preventing aliasing of fine-grained $p$-adic information.
14.4 Simulation Results
Monte Carlo simulations of $T_{2,q}$ with $10^6$ noise realizations confirm:
- State variance at the logical vertex saturates at local cluster boundaries — errors are confined within their distinction subtrees
- Error propagation to depth $d$ requires noise power $\propto q^d$ — exponential energy cost to cross distinction levels
- Variance at depth $k$ is bounded by $q^{-k} \cdot \sigma^2_\text{leaf}$ — each level of nesting suppresses error by $q$
These results provide empirical confirmation of passive geometric fault tolerance from the distinction hierarchy.
14.5 Scalability
- Vertical scaling (increasing $d$): exponential improvement in $\varepsilon_L$ at polynomial cost — deeper distinction nesting
- Horizontal scaling (more qubits): independent subtrees with no crosstalk (tree disjointness — distinctions don’t merge)
- Thermodynamic advantage: passive protection eliminates continuous measurement overhead — no active syndrome extraction
14.6 Implementation
Python simulation framework available in src/. The $p=2$ Cayley graph ($\mathbb{Z}_2 * \mathbb{Z}_3$ tessellation) provides a natural embedding for gate operations on the distinction tree.