Quantum computation traditionally operates on tensor product spaces of two-level systems (qubits), with gates represented by unitary matrices acting on these tensor factors. The ratio-based framework introduces a fundamentally different substrate: computation on Bruhat-Tits trees with scaling ratio
q. In this paradigm, quantum gates emerge from tree automorphisms—symmetries of the hierarchical structure—and computation proceeds via scaling transformations along tree paths. This module develops the complete theory of quantum gate implementation on ratio-based trees.
The Bruhat-Tits tree T_p is proposed as a geometric substrate for quantum error-correcting codes. The defining property of ultrametric geometry replaces continuous error accumulation with discrete redundancy steps. This topological reframing connects p-adic mathematical physics, quantum error correction theory, and quantum Darwinism. The central conjecture is that the redundancy-fidelity tradeoff, parameterized on a real quantum processor, may exhibit a staircase structure (ultrametric regime) rather than a smooth curve (Archimedean limit), and that this structure — if detected — would constitute a new QEC design parameter. An experimental protocol is presented for discriminating between these regimes on existing quantum processors. All claims are graded with certainty labels; the paper acknowledges that the experimental program is the path to evidence. [v0.2 — literature search completed 2026-08-06: prior holographic-code work (Marcolli 2018) acknowledged; novelty claim refined to device-level QEC with staircase detection.]
The standard mathematical foundation of quantum computing—Archimedean (Euclidean) geometry—assumes that space is continuous, distances add linearly, and small errors can accumulate into large ones. This assumption creates a fundamental bottleneck: active quantum error correction generates heat that scales faster than cryogenic cooling capacity, producing a
20,000× gap between required and available cooling at millikelvin temperatures. We propose an alternative: encoding quantum states on Bruhat–Tits trees, the geometric realization of
p-adic (ultrametric) spaces, where the strong triangle inequality
d(x,z)≤max{d(x,y),d(y,z)} geometrically confines errors to their local branches. We present a computational validation suite (classical simulation) demonstrating this mechanism. Our results show that tree-encoded circuits produced zero observed logical errors in 500 trials at depths
d≥3 for physical error rates up to
perr=0.40, while equivalent flat (Archimedean) encodings failed with logical error rates up to
0.152. The energy barrier protecting logical states scales as
Ebarrier(d)=2d, confirmed exhaustively for
d=2,3 and computed analytically to
d=10. The strong triangle inequality is verified with zero violations in
15,000 random trials across primes
p=2,3,5. These results provide computational evidence for a geometric threshold theorem in ultrametric quantum error correction and suggest a path toward passive fault tolerance—error suppression as a property of hardware geometry rather than active measurement protocols. The simulation uses classical bits with majority-vote decoding; full quantum validation with superposition and entanglement remains future work.
The laws of thermodynamics impose fundamental limits on all physical computation, determining minimum energy requirements, maximum operating temperatures, and ultimate scalability. In the ratio-based framework, the hierarchical tree structure with scaling ratio
q creates a distinctive energy landscape with barriers scaling as
Ebarrier∝qd, where
d is tree depth. This fundamentally alters thermodynamic constraints compared to conventional quantum architectures, introducing the scaling ratio
q as a key optimization parameter in thermodynamic tradeoffs.
The Bruhat-Tits tree provides an ultrametric substrate for quantum error-correcting codes whose information topology differs from the Archimedean case in ways that may revise the redundancy-fidelity tradeoff established for quantum Darwinism. Two limiting regimes are identified: the large-p limit, in which the ultrametric bound converges to the Archimedean bound, and the deep ultrametric regime of small primes, in which discrete redundancy steps create windows where quantum error correction and classical objectivity may coexist. An experimental protocol is proposed to detect the staircase structure on real quantum processors. All claims are graded with epistemic certainty labels and subjected to an internal red-team audit. Licensed under QNFO Unified License Agreement (QNFO-ULA).