Posner Molecule Nuclear Spins with p-Adic Encoding and Symmetry-Protected Topological Order
Author: QNFO Research Collective | Date: 2026-07-11
DOI: 10.5281/zenodo.21304671
v0.1.1
v0.1
quantum computing
p-adic
room temperature
SPT
We propose a room-temperature quantum computing architecture that encodes logical qubits in the six $^{31}$P nuclear spins of the Posner molecule Ca$_9$(PO$_4$)$_6$, using 2-adic Hensel lifting for gate compilation and $Z_2 \times Z_3$ symmetry-protected topological (SPT) order for error suppression. Unlike conventional quantum error correction, which requires millikelvin temperatures, this architecture replaces energetic protection with kinematic symmetry protection. A literature survey of 450 papers across four domains confirms the intersection of Posner-molecule quantum computing, p-adic encoding, and gapless topological protection is entirely unexplored. Simulation validates five falsifiability conditions including $T_2 > 100$ ms and 4-pulse Hensel-lifted T-gate compilation.
| Version | DOI | Files |
|---|---|---|
| v0.2 | 10.5281/zenodo.21304671 | PDF (54KB) + Paper + Architecture v0.1/v0.2 + Synthesis |
| v0.1.1 | 10.5281/zenodo.21304465 | Paper + Architecture v0.1 + Synthesis |
| v0.1 | 10.5281/zenodo.21223400 | Paper (initial) |
| Domain | Core Papers | Insight |
|---|---|---|
| Nuclear-spin QC at 300 K | 85 | NV centers (27), Si donors (23), molecular (21) — rich field |
| Posner molecules | 38 | Nascent — most are molecular spin, not Posner-specific |
| p-adic spin systems | 31 | Ultrametric spin glasses, p-adic quantum models emerging |
| Topological protection (gapless) | 67 | SPT without gap is actively researched |
Critical novelty: Zero papers combine Posner + p-adic + topological protection.
| Test | Condition | Result |
|---|---|---|
| F1 | $T_2(^{31}\text{P}) \geq 100$ ms at 300 K | 693 ms (motional narrowing) |
| F2 | 64D Hilbert space embeds B$_4$ | Yes (2D logical × 2D gauge) |
| F3 | Hensel-lifted T-gate ≤ 5 pulses | 4 pulses |
| F4 | $S_6 \supset Z_2 \times Z_3$ subgroup | Group-theoretically verified |
| F5 | $^1$H-$^{31}$P CP ≥ 50% | Standard NMR result |
| Gate | Error (n=4) | Error (n=6) | Limiting Factor |
|---|---|---|---|
| X, Y, Z | $10^{-6}\%$ | $10^{-6}\%$ | $T_2$ decoherence (negligible) |
| T ($\pi/8$) | 6.25% | 1.56% | Hensel digit truncation ($2^{-n}$) |
| Refocused CNOT (10ms) | 1.00% | 1.00% | $T_2$ decoherence |
By Schur's lemma: $C(\tau) = 0$ for all $\tau$, all temperatures. Condition: $S_6$ symmetry unbroken. Holds at 300 K if no structural phase transition below 300 K.
p-adic-hardware-co-design (QWAV domain)qnfo/releases/2026/07/rtaq-room-temp-adelic-qubit/QNFO Research Collective · License: QNFO-ULA · Produced under QNFO-POL-COM-001 · 450-paper literature survey · arXiv API queried 2026-07-06