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).
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.
We propose a fault-tolerant quantum computing architecture—the Bruhat-Tits Quantum Processor—that exploits the ultrametric geometry of the Bruhat-Tits tree
Tp to achieve quantum error correction thresholds substantially exceeding those of conventional Euclidean-locality codes. The architecture uses holographic perfect-tensor codes on a truncated Bruhat-Tits tree, tree-automorphism logical gates, and fractal-multiplexed readout. We compute bit-flip, depolarizing, independent
X+
Z, gate-error, and magic-state distillation thresholds via analytical recursion, exact Gottesman-Knill stabilizer simulation, and Monte Carlo methods. The tree code achieves a bit-flip threshold of
pc=50.0% (
4.6× above surface codes at
∼10.9%), a depolarizing threshold of
pc=75.0% (
75× above surface codes at
∼1.0%), and an independent
X+
Z threshold of
pc=17.30%. We benchmark the tree code against the surface code at matched physical qubit counts using Monte Carlo simulation with Minimum-Weight Perfect Matching (MWPM) decoding. A logical-level demonstration of Shor’s algorithm for
N=15 confirms algorithmic completeness and establishes physical resource requirements. All quantitative claims are verified by self-contained Python simulations. We identify open problems—including perfect tensor existence, tree automorphism gate generation, and physical noise modeling—as directions for future investigation. The architecture’s defining innovation is the use of ultrametricity as a quantum error correction resource: the strong triangle inequality
d(x,z)=max(d(x,y),d(y,z)) confines error clusters to disjoint subtrees, enabling parallel, local decoding in
O(slogs) time.