P02 — p-adic Quantum Computer (Padit Architecture)
| Field | Value |
|---|---|
| Domain | Computing Architectures |
| System | n-padit register over ℚ_p |
| Group | PGL(2, ℚ_p) / Bruhat-Tits automorphisms |
| H^k tier | H² |
| ISA | Meld (β → it, over ℚ_p) |
| Status | Speculative |
| Opcodes | ORBIT · TWIST · BIND · LABEL |
| Papers | Paper 544, Paper 543 |
Physical system
The padit (p-adic digit) is the p-adic generalisation of the qubit: a unit vector in the Hilbert space L²(ℤ/pⁿℤ) — square-integrable functions on the ring of integers modulo pⁿ, with Haar measure. An n-padit register has pⁿ dimensions: the same exponential growth as a qubit register, but base p instead of base 2.
Qubits are 2-adic 1-padits. Standard quantum computing is the p=2 special case. For p=3, a 1-padit is a qutrit. The padit computer generalises the qubit computer while keeping the same categorical structure.
The key claim (Paper 544): the Number Theoretic Transform (NTT) over ℤ/pⁿℤ is the exact p-adic quantum Fourier transform. Shor’s algorithm on a p-adic quantum computer uses the NTT instead of the QFT — with no floating-point phase errors, and computation exact to any desired p-adic precision.
Target category
p-Hilb — the category of finite-dimensional Hilbert spaces over the p-adic field ℂ_p (the completion of the algebraic closure of ℚ_p). Objects are L²(ℤ/pⁿℤ); morphisms are p-adic unitary operators in U(pⁿ, ℂ_p). The Clifford sector is simulated classically by the U-MGE (Paper 543 §11); the magic sector requires physical p-adic hardware.
Interpretation functor
F: C → p-Hilb defined by:
| Opcode | F(opcode) | ||
|---|---|---|---|
| ORBIT | Geodesic on Bruhat-Tits tree 𝒯_p: vertex → p+1 children or parent; Schubert cell = p-adic coset; winding = p-adic valuation v_p | ||
| TWIST | p-adic Berry phase: Gauss sum τ_p(χ) = Σ_{x∈𝔽_p} χ(x) e^{2πix/p}; H¹ phase around p-adic torus | ||
| BIND | NTT as p-adic QFT: F_{pⁿ} | k⟩ = (1/√pⁿ) Σ_j ω^{jk} | j⟩, ω = primitive pⁿ-th root of unity; H² class = second Chern number on tree |
| LABEL | p-adic Clifford/magic discriminant: v_p(amplitude) ≥ 0 ↔ Clifford (classically simulable); v_p(amplitude) < 0 ↔ magic |
ISA programme
INIT: LABEL[|0⟩^⊗n | n-padit register at |0...0⟩] -- initialise n padits
CLIFFORD: ORBIT[p-adic Pauli gates | Heisenberg-Weyl over Z/p^n Z] -- H¹ sector (simulable)
QFT: BIND[F_{p^n} | NTT over Z/p^n Z] -- p-adic quantum Fourier transform (exact)
MAGIC: TWIST[T_p | order-(p+1) rotation in C_p] -- p-adic T-gate; H² resource
MEASURE: LABEL[Born rule P(k) = |<k|psi>|_p^2] -- rational probability, exact
CERTIFY: LABEL[|tau_p(chi)| = sqrt(p) exactly?] -- Gauss-sum certificate (H² output)
Computable output
- Rational Born probabilities: P(k) = |⟨k|ψ⟩|_p² takes values in p^ℤ ∩ [0,1]. Probabilities are positive rationals — exact, no floating-point error. This is a qualitative difference from standard quantum computing, where Born rule probabilities are generically irrational.
- NTT as exact QFT: the p-adic QFT on n padits is the Number Theoretic Transform, an algorithm already in production use for lattice-based post-quantum cryptography (Kyber, Dilithium). No floating-point phase errors; the transform is exact modulo pⁿ.
- Gauss-sum certificate: the H² topological certificate for the p-adic computation is |τ_p(χ)| = √p exactly (Gauss’s theorem on quadratic Gauss sums). This is a hardware-verifiable integer condition — the p-adic analogue of the winding number certificate in real ISA computations.
- p-adic Gottesman-Knill: any circuit on n padits using only p-adic Pauli preparation, p-adic Clifford gates (entries in 𝒪_{ℂ_p}, the p-adic integers), and p-adic Pauli measurement can be classically simulated in O(n²) time by the U-MGE (Paper 543 §11). The padit architecture has the same Clifford/magic distinction as the qubit architecture, with p-adic integrality replacing Wigner non-negativity as the simulability criterion.
The p-adic Clifford/magic distinction
| Quantity | Standard qubit | p-adic padit | ||
|---|---|---|---|---|
| Clifford group | Entries in ℤ[ζ_8] | Entries in 𝒪_{ℂ_p} (p-adic integers) | ||
| Magic resource | Wigner negativity | p-adic non-integrality: v_p(amplitude) < 0 | ||
| Classical simulation | Stabiliser formalism | U-MGE on PPU | ||
| QFT | Approximate (FP phases) | NTT (exact, modular) | ||
| Born probabilities | Generically irrational | Positive rationals (p^ℤ) | ||
| Topological certificate | Winding number ∈ ℤ | Gauss sum | τ_p | = √p |
The adèlic Clifford group is the intersection of all p-adic Clifford groups: 𝒞_adèlic = 𝒞_ℝ ∩ ⋂p 𝒞{ℚ_p} — the universally simulable gate set across all completions simultaneously.
The Bruhat-Tits tree as computational tape
The Bruhat-Tits tree 𝒯_p is the p-adic analogue of the Poincaré half-plane: a (p+1)-regular tree whose automorphism group is PGL(2, ℚ_p). The padit register’s computational state traces a path in 𝒯_p:
- Each vertex at depth n = one p-adic digit of precision
- Each edge = one p-SPLIT (one coset choice in ℤ/pℤ)
- Halting = Schubert variety crossing in 𝒯_p = p-adic snap condition v_p(β) ≥ v_p(ΔE)
- Boundary ∂𝒯_p = ℙ¹(ℚ_p) = Cantor set with ultrametric structure = Palmer’s invariant set I_U (Paper 543 §9)
The p-adic QC is an OPU (P01) running on the Bruhat-Tits tree rather than on the Grassmannian. The Schubert stratification of Gr(k,n) (P01) becomes the coset stratification of PGL(2,ℚ_p)/B (Bruhat decomposition, which is the algebraic source of the tree). They are the same categorical structure over different fields (ℂ vs ℂ_p).
Proposed physical architectures
| Architecture | Description | Status |
|---|---|---|
| Holographic QC on 𝒯_p | Qudits on tree vertices; boundary = Cantor set I_U | Theoretical |
| Software p-adic QC | Standard hardware, NTT gate alphabet | Near-term feasible |
| Spin glass with RSB | Parisi matrix = p-adic order parameter | Speculative |
The Software p-adic QC is near-term feasible: use existing superconducting or trapped-ion hardware, but restrict the gate alphabet to NTT gates and p-adic Clifford gates. The classical simulation (U-MGE on a PPU) is exact; the magic sector requires standard non-Clifford gates whose matrix entries happen to be p-adically non-integral. No new hardware needed for the Clifford sector.
Connection to the ISA framework
The NTT = BIND. The p-adic QFT (NTT) is the BIND opcode at the p-adic rung: it implements the H² holonomy of the p-adic torus, wrapping the padit register around a 2-cycle in the Bruhat-Tits building. The Gauss-sum certificate |τ_p| = √p is the H² topological invariant — the p-adic second Chern number.
This makes the padit architecture the p-adic column in the RPU family table (P01):
| XPU | Tape manifold | Group | β regime | QFT |
|---|---|---|---|---|
| QPU | CP^{2ⁿ⁻¹} | U(2ⁿ) | β = it | Hadamard + phase |
| OPU | Gr(k,n) | U(n) | finite β | CASSCF orbital rotation |
| PPU / padit QC | 𝒯_p | PGL(2,ℚ_p) | β_p = it_p | NTT (exact) |
| A-QC | 𝔸_ℚ | Adèlic group | adèlic β | Product formula |
Validation status
Speculative — no physical padit system has been demonstrated. However:
- NTT hardware is in production (PQC accelerators: Kyber, Dilithium)
- p-adic Gottesman-Knill theorem follows directly from standard proof (substitute p-adic integers for Gaussian integers)
- The NTT = p-adic QFT identification is a mathematical fact (Vladimirov-Volovich 1989; explicit in Paper 543 §14)
- Rational Born probabilities and Gauss-sum certificate are analytically derived
The speculative claim is that a physical system realises p-adic amplitudes.