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Holographic Quantum Error Correction as AdS/CFT Renormalization-Group Flow on Bruhat–Tits Trees: A Conditional Threshold Analysis at 10⁻⁴

Rowan Brad Quni-Gudzinas

Published Oct 2, 2026• 0 citations
Publisher

Abstract

We analyze a proposal in which quantum error correction (QEC) is organized as renormalization-group (RG) flow on a Bruhat–Tits tree $\mathcal{T}_p$ — the $p$-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 $10^{-4}$ per logical qubit per code cycle. Our method is analytic and conservative: we model the tree code as a $p$-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 $4.4\times10^{-3}$, single-qubit error $1.0\times10^{-3}$, and readout error $1.3\times10^{-2}$ reported for the Tianyan-287 superconducting processor, a cat code with mean photon number $\bar n = 6$ suppressing bit flips to $6.14\times10^{-6}$ per loss event, and a depth-2 tree with branching $p=3$, we obtain a per-cycle logical error of $1.4\times10^{-29}$ under the independence assumption (reported as a mechanism demonstration, not a prediction), and an error-floor estimate of $8.0\times10^{-8}$ once worst-case correlated events are charged. The analytic bit-flip threshold of the tree recursion is $p^\ast = 1/4$ (total depolarizing rate $3/8$), so the advertised threshold $10^{-4}$ holds with a practical margin of at least $19.2\times$ at the tightest (readout-dominated) channel — **conditionally**: the threshold fails if correlated bursts defeat the cat suppression entirely. We state the explicit survivability condition ($p_{\rm corr} \le 7.7\times10^{-6}$ 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/

DOI
10.5281/zenodo.23107745

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•Aug 6, 2026•0 citations
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/
We propose a fault-tolerant quantum computing architecture—the Bruhat-Tits Quantum Processor—that exploits the ultrametric geometry of the Bruhat-Tits tree Tp\mathcal{T}_pTp​ 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 XXX+ZZZ, 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%p_c = 50.0\%pc​=50.0% (4.6×4.6\times4.6× above surface codes at ∼10.9%\sim 10.9\%∼10.9%), a depolarizing threshold of pc=75.0%p_c = 75.0\%pc​=75.0% (75×75\times75× above surface codes at ∼1.0%\sim 1.0\%∼1.0%), and an independent XXX+ZZZ threshold of pc=17.30%p_c = 17.30\%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=15N = 15N=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))d(x,z) = \max(d(x,y), d(y,z))d(x,z)=max(d(x,y),d(y,z)) confines error clusters to disjoint subtrees, enabling parallel, local decoding in O(slog⁡s)O(s \log s)O(slogs) time.
Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.
Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.
Utility-scale quantum computers require quantum error correcting codes with large numbers of physical qubits to achieve sufficiently low logical error rates. The performance of quantum error correction (QEC) is generally predicted through large-scale numerical simulations, used to estimate thresholds, finite-size scaling, and exponential suppression of logical errors below threshold. The connection of QEC to models from statistical mechanics provides an alternative tool for analysing QEC performance. However, predicting the behaviour of these models also requires large-scale numerical simulations, as analytic solutions are not generally known. Here we exploit an exact mapping, from a toric code under bit-flip noise that is post-selected on being syndrome free to the exactly-solvable two-dimensional Ising model on a torus, to derive an analytic solution for the logical failure rate across its full domain of physical error rates. In particular, this mapping provides closed-form expressions for the logical failure rate in four distinct regimes: the path-counting, below-threshold (ordered), near-threshold (critical), and above-threshold (disordered) regimes. Our framework places a number of familiar and long-standing numerical observations on firm theoretical ground. It also motivates explicit ansatze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the domain wall energy cost distributions. By bridging statistical mechanics theory and quantum error correction practice, our results offer a new toolkit for designing, benchmarking, and understanding topological codes beyond current computational limits.
FINDING: Quantum error correction (QEC) is demonstrated below the surface code threshold by Google Quantum AI, and a theoretical link is proposed between QEC and the emergence of spacetime in holography. MATH: - Surface code threshold: ~0.57% error rate per physical qubit (logical error rate decreases exponentially below this). - Logical error rate: PL∝(p/pth)d/2 P_L \propto (p/p_{\text{th}})^{d/2} PL​∝(p/pth​)d/2 for code distance d d d, physical error rate p p p. - Holographic duality: AdS/CFT correspondence maps bulk gravity to boundary quantum field theory; QEC codes (e.g., tensor networks) realize the holographic map via isometric tensors. - Key constants: No explicit golden ratio or base-60 numbers appear in these results. CONNECTION: - Surface code is a topological lattice (square grid) with stabilizer generators on plaquettes and vertices — a Z2 \mathbb{Z}_2 Z2​ gauge theory on a 2D lattice. - Holographic QEC codes (e.g., HaPPY code) use perfect tensors on hyperbolic tilings (e.g., {5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.]