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Bruhat-Tits Quantum Processor: Ultrametric Fault-Tolerant Quantum Computation with Holographic Tree Codes

Rowan Brad Quni-Gudzinas

Published May 10, 2026• 0 citations
Publisher

Abstract

We propose a fault-tolerant quantum computing architecture—the Bruhat-Tits Quantum Processor—that exploits the ultrametric geometry of the Bruhat-Tits tree $\mathcal{T}_p$ 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 $p_c = 50.0\%$ ($4.6\times$ above surface codes at $\sim 10.9\%$), a depolarizing threshold of $p_c = 75.0\%$ ($75\times$ above surface codes at $\sim 1.0\%$), and an independent $X$+$Z$ threshold of $p_c = 17.30\%$. 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 $O(s \log s)$ time.

DOI
10.5281/zenodo.20109835

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We analyze a proposal in which quantum error correction (QEC) is organized as renormalization-group (RG) flow on a Bruhat–Tits tree Tp\mathcal{T}_pTp​ — the ppp-adic analogue of hyperbolic anti-de Sitter (AdS) space — so that the encoding map of a holographic code is literally an RG trajectory from boundary degrees of freedom to an infrared fixed-point subspace. We ask whether such a holographic QEC scheme can meet a threshold of at least 10−410^{-4}10−4 per logical qubit per code cycle. Our method is analytic and conservative: we model the tree code as a ppp-ary majority-vote concatenation layered on a bosonic (cat-state) inner code, derive the exact recursion for logical failure, locate its nontrivial fixed point, and propagate published hardware error rates through the full stack. Using the two-qubit gate error 4.4×10−34.4\times10^{-3}4.4×10−3, single-qubit error 1.0×10−31.0\times10^{-3}1.0×10−3, and readout error 1.3×10−21.3\times10^{-2}1.3×10−2 reported for the Tianyan-287 superconducting processor, a cat code with mean photon number nˉ=6\bar n = 6nˉ=6 suppressing bit flips to 6.14×10−66.14\times10^{-6}6.14×10−6 per loss event, and a depth-2 tree with branching p=3p=3p=3, we obtain a per-cycle logical error of 1.4×10−291.4\times10^{-29}1.4×10−29 under the independence assumption (reported as a mechanism demonstration, not a prediction), and an error-floor estimate of 8.0×10−88.0\times10^{-8}8.0×10−8 once worst-case correlated events are charged. The analytic bit-flip threshold of the tree recursion is p∗=1/4p^\ast = 1/4p∗=1/4 (total depolarizing rate 3/83/83/8), so the advertised threshold 10−410^{-4}10−4 holds with a practical margin of at least 19.2×19.2\times19.2× at the tightest (readout-dominated) channel — **conditionally**: the threshold fails if correlated bursts defeat the cat suppression entirely. We state the explicit survivability condition (pcorr≤7.7×10−6p_{\rm corr} \le 7.7\times10^{-6}pcorr​≤7.7×10−6 per subtree per cycle in the worst channel), compare photon budgets against surface codes, and outline falsifiable tests on trapped-ion and superconducting platforms. ## 1. Introduction Quantum error correc Full text and updates: papers.qnfo.org/papers/holographic-quantum-error-correction-as-ads-cft-renormalization-group-flow-on-br/
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 20,000×20{,}000\times20,000× gap between required and available cooling at millikelvin temperatures. We propose an alternative: encoding quantum states on Bruhat–Tits trees, the geometric realization of ppp-adic (ultrametric) spaces, where the strong triangle inequality d(x,z)≤max⁡{d(x,y),d(y,z)}d(x,z) \leq \max\{d(x,y), d(y,z)\}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≥3d \geq 3d≥3 for physical error rates up to perr=0.40p_{\text{err}} = 0.40perr​=0.40, while equivalent flat (Archimedean) encodings failed with logical error rates up to 0.1520.1520.152. The energy barrier protecting logical states scales as Ebarrier(d)=2dE_{\text{barrier}}(d) = 2^dEbarrier​(d)=2d, confirmed exhaustively for d=2,3d = 2, 3d=2,3 and computed analytically to d=10d = 10d=10. The strong triangle inequality is verified with zero violations in 15,00015{,}00015,000 random trials across primes p=2,3,5p = 2, 3, 5p=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 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 prevailing quantum computing paradigm is fundamentally constrained by the continuous analog fragility inherent to Archimedean spaces, resulting in uncontrolled linear error accumulation that demands massive active correction overheads. To address this structural bottleneck, we propose a transition to a non-Archimedean state space modeled on the p-adic numbers and their graph-theoretic realization, the Bruhat-Tits tree. To validate this non-Archimedean architecture without physical p-adic hardware substrates, we developed a comprehensive software emulation utilizing Bounded Algorithmic Number (BAN) arithmetic logic units to process exact ultrametric valuations. We simulated the compilation of continuous logic gates into discrete tree automorphisms, mapping standard unitary operators to vertex shifts and branch permutations on a p=2 Cayley graph. Furthermore, stochastic Ohmic and burst noise injections were modeled to test the error-filtering properties of the strong triangle inequality dynamically, while van der Put Neural Networks (v-PuNNs) were deployed as read-out trackers to prevent topological distortion. Our empirical simulation results demonstrate that non-Archimedean architectures natively suppress linear error accumulation, with variance saturating flatly at local cluster boundaries, confirming passive geometric fault tolerance. These findings carry implications for the future of post-NISQ hardware, formally bridging holographic tensor network theory with applied digital emulation to outline a scalable computational substrate.
The observation that positional notation carries a native ultrametric tree structure extends into quantum computing through the tensor-product architecture of qubit registers. We formalize three theses: (I) a tree-topology quantum processor confines errors to subtrees via the strong triangle inequality, elevating the error-correction threshold; (II) a 3D-integrated superconducting architecture realizing a 7-ary tree of depth 3 (343 physical qubits) is manufacturable with current fabrication technology; (III) an on-chip bank of prime-frequency resonators enables spectral engineering of the tree's error-confinement properties. We implement a numerical simulation comparing tree-topology and grid-topology codes, deriving threshold estimates and error-propagation statistics. The chapter extends the Ultrametric Foundation thesis (§12) with formal derivations, architecture specifications, simulation results, and a mapping to the broader research program. ---
Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension 2+ϵ , which admits a fault-tolerant non-Clifford CCZ gate \cite{zhu2021topological}. We investigate the performance of such FSCs as fault-tolerant quantum memories. We prove that there exist decoding strategies with non-zero thresholds for bit-flip and phase-flip errors in the FSCs with Hausdorff dimension 2+ϵ . For the bit-flip errors, we adapt the sweep decoder, developed for string-like syndromes in the regular 3D surface code, to the FSCs by designing suitable modifications on the boundaries of the holes in the fractal lattice. Our adaptation of the sweep decoder for the FSCs maintains its self-correcting and single-shot nature. For the phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder for the point-like syndromes. We report a sustainable fault-tolerant threshold ( ∼1.7% ) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by 2.95% ) for the MWPM decoder for a particular FSC with Hausdorff dimension DH≈2.966 . The latter can be mapped to a lower bound of the critical point of a confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.