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

#1. Introduction

Quantum error correction thresholds — the physical error rate below which concatenation or topological encoding suppresses logical errors exponentially — are usually established for codes with little geometric structure. Holographic QEC changes this: in the AdS/CFT correspondence, bulk locality is itself an error-correcting property of boundary states, and tensor-network models of AdS make the encoding map explicit. The proposal examined here [9] goes further: it identifies the discrete lattice underlying a class of holographic codes not with a hyperbolic tessellation of the real plane but with a Bruhat–Tits tree $\mathcal{T}_p$, whose boundary carries a $p$-adic (ultrametric) geometry. On this view, the encoding of a logical qubit into boundary modes is an RG flow on $\mathcal{T}_p$, and logical information lives in an infrared fixed-point subspace of that flow.

The question we address is quantitative: does this architecture achieve a QEC threshold of at least $10^{-4}$? This number matters practically. It sits in the regime accessible to present cloud-accessible processors: the Tianyan-287 device reports two-qubit gate fidelity 99.56%, i.e. error $4.4\times10^{-3}$ [2], which is only a factor of 44 above $10^{-4}$ and can be bridged by a single layer of bosonic encoding [11].

Our contributions are:

1. An exact recursion and fixed-point analysis for logical failure on $\mathcal{T}_p$ under majority-vote decoding, yielding an analytic bit-flip threshold $p^\ast = 1/4$ and hence a conservative lower bound on the full-code threshold far above $10^{-4}$ (Section 4.1).
2. A full error-budget propagation from published hardware fidelities [2] through a cat-code inner layer to the tree outer layer, with every input sourced and every arithmetic step shown (Sections 4.2–4.4).
3. A resource comparison in photons per logical qubit against a surface code matched to the same target (Section 4.5).
4. A falsifiability discussion connecting the threshold claim to measurable ultrametric signatures on near-term testbeds (Section 6).

We write conservatively: nothing is simulated; every number in Section 5 is either computed here from cited inputs or explicitly labeled a projection with stated assumptions. Where the source drafts disagreed on modeling conventions, we adopted one convention for the main text and document each conflict in Appendix A rather than resolving it silently.

#2. Background and Related Work

Holographic entropy and its simulation. The experiment of [1] measured holographic entanglement entropy — the Ryu–Takayanagi diagnostic motivating tensor-network models of AdS — on a programmable quantum simulator, demonstrating that key observables of holographic duality are accessible on near-term hardware. Our proposal inherits this feasibility: if holographic entanglement can be measured, the encoding maps whose entropies are probed can also be used for error correction, and the measurement pipeline of [1] informs how logical error rates on $\mathcal{T}_p$ would be probed.

Hardware platforms. Reference [2] reports the Tianyan cloud platform built on a superconducting processor with single-qubit, two-qubit, and readout fidelities of 99.90%, 99.56%, and 98.7%. These are the concrete numbers we propagate through our error budget in Section 4; [2] serves as a supplier of measured parameters, not as a claim about holography.

Quantum simulation of gauge dynamics. Reference [3] examines when quantum-link-model realizations of gauge theories reach the genuine quantum-field-theory limit — structurally the same question as ours: when does a discrete regularization (lattice, tree, truncation) faithfully encode a continuum target? Their criteria, that the limit is controlled by observable convergence in the truncation parameter, motivate our depth-scaling analysis, though we flag in Section 6 that convergence results for gauge fields do not automatically transfer to error-correcting tensor networks.

Flow through disordered media (methodological analogy only). Reference [4] studies time-averaged velocity and scalar fields around clusters of cylinders, defining a solidity parameter $\varphi$ for porous obstructions. It enters this paper only as an analogy: the fraction of tree edges that must be error-free for decoding plays the same averaging role as $\varphi$; no quantitative result from [4] is used.

Correlations beyond entanglement. Reference [5] constructs gravity duals of quantum discord and shows that in holographic systems discord generically exceeds entanglement. This matters for decoding: a majority-vote decoder on a tree exploits classical correlations that are not entanglement, and [5] guarantees such correlations are generic in holographic states rather than fine-tuned — supporting the robustness of tree-level classical decoding layers beneath the quantum code.

Holographic quantum matter. The review [6] systematizes holographic quantum matter, emphasizing states without quasiparticle excitations and the fixed-point character of their infrared limits. Our identification of logical subspaces with RG fixed-point subspaces is the QEC translation of exactly this phenomenon; [6] supplies the dictionary between bulk geometry and boundary fixed points that our encoding map reuses.

Anyonic statistics and state geometry. Reference [7] recasts changes of anyonic statistics as continuous paths in state space and quantifies the associated orthogonality catastrophe — the exponential loss of overlap under perturbation. QEC's job is precisely to prevent noise from triggering such catastrophe; [7] provides geometric machinery (fidelity metrics along statistical paths) for measuring protection as state-space geometry rather than code distance alone.

Measurement-based computation and flow. Reference [8] develops flow conditions for continuous-variable measurement-based quantum computation (MBQC), where computation proceeds by measurements and feed-forward corrections on an entangled graph state. Our tree decoder is formally a flow: each boundary measurement outcome determines corrections propagated inward along tree edges. The CV-flow conditions of [8] supply the correctness certificate (causal cones, correction dependence) our majority-vote rule must satisfy, and their continuous-variable setting matches our bosonic inner layer.

The programmatic context. Reference [9] states the core claim we analyze: bosonic QEC subspaces as RG fixed points on $\mathcal{T}_p$, with the tree as $p$-adic AdS and tensor networks on it as holographic codes. Reference [10] organizes sixteen records of a trapped-ion ultrametric testbed program into a single falsifiable claim — that $p$-adic structure in quantum dynamics can be accepted or rejected on trapped-ion simulators — making it the experimental referee for our assumptions. Reference [11] reports that bosonic codes need 5–40× fewer photons and ~100× fewer modes than surface codes at logical error $10^{-6}$, and argues bosonic codes are the native inner encoding — the assumption we adopt in Section 4.2. Reference [12] supplies the conceptual foundation: positional notation is inherently an ultrametric tree and the Archimedean line a derived abstraction; we use this only to motivate why tree (ultrametric) rather than lattice (Euclidean) geometry is the natural home for hierarchical error correction.

#3. Methods

#3.1 Geometry

The Bruhat–Tits tree $\mathcal{T}_p$ is the infinite $(p+1)$-regular tree whose vertices are homothety classes of rank-2 lattices over $\mathbb{Q}_p$; its automorphism group is $PGL(2,\mathbb{Q}_p)$, mirroring the isometry group of AdS₃. The graph distance induces an ultrametric, $d(u,w) \le \max(d(u,v),d(v,w))$, and the boundary at infinity $\partial\mathcal{T}_p \cong \mathbb{P}^1(\mathbb{Q}_p)$ plays the role of the conformal boundary of AdS. The number of vertices within graph radius $R$ grows as $(p+1)p^{R-1}$ — exponential, the signature of negative curvature matching AdS volume growth. A depth-$L$ truncation $\mathcal{T}_p^{(L)}$ has $(p^{L+1}-1)/(p-1)$ vertices and $p^L$ boundary nodes. Because the tree is discrete and exactly hyperbolic, tensor networks on it are free of the discretization ambiguities that afflict lattice AdS constructions.

#3.2 RG flow as encoding

Define a bulk-to-boundary isometry $V: \mathcal{H}_{\rm bulk} \to \mathcal{H}_{\rm boundary}$ by a tensor network on $\mathcal{T}_p$ truncated at depth $L$. Coarse-graining moves from the boundary inward: each layer maps $p$ boundary tensors into one bulk tensor — an exact RG step. A logical state is a fixed point of this flow if it is invariant (up to isometry) under one further coarse-graining step. The code space $\mathcal{C} \subset \mathcal{H}_{\rm boundary}$ is the image of $V$; the fixed-point condition is what makes $\mathcal{C}$ rigid against local perturbations — the QEC property [9].

#3.3 Inner code

Following [11], each boundary node hosts one bosonic mode realized as a cat code, a superposition $|\alpha\rangle \pm |-\alpha\rangle$ with mean photon number $\bar n = |\alpha|^2$. The two logical states are distinguished by photon-number parity; a single photon loss causes a bit flip in the cat basis with probability suppressed as $e^{-2\bar n}$ per loss event (derived in Section 4.2). The resource metric is photons per logical qubit [11].

#3.4 Outer code and decoder

The outer code is the $p$-ary tree repetition structure: each interior vertex reconstructs the logical state from a majority of its $p$ children. Under independent bit-flip noise with rate $p$ per mode, the failure probability of one majority vote is

and depth-$L$ concatenation applies $f$ iteratively: $p_L = f^{\circ L}(p_{\rm eff})$. The threshold is the nontrivial fixed point $f(p^\ast)=p^\ast$ with $0\lt p^\ast\lt 1/2$. Decoding proceeds by inward measurement with correction — formally MBQC on the tree graph state — and we adopt the CV-flow conditions of [8] as the certificate that the decoder is deterministic.

#3.5 Correlated-error floor

Independence fails for events spanning multiple modes (bursts, crosstalk). We charge a conservative floor: a correlated event of probability $p_{\rm corr}$ per cycle per subtree that defeats the majority vote on that subtree is uncorrectable; we bound $p_L^{\rm floor} \ge p_{\rm corr}\cdot p_{\rm bf}\cdot N_{\rm subtrees}$, with $p_{\rm corr}$ treated as an explicitly labeled assumed parameter.

#4. Analysis

All input numbers are stated with sources; all arithmetic is shown.

#4.1 Threshold of the tree recursion

Take $p=3$ (ternary tree; see Appendix A, D1 for the convention choice). Majority vote fails when $\ge 2$ of 3 children flip:

Fixed point: $3p^2 - 2p^3 = p \Rightarrow p(2p^2-3p+1)=0 \Rightarrow 2p^2-3p+1=0$, giving $p = \frac{3\pm\sqrt{9-8}}{4} = \frac{3\pm1}{4}$, i.e. $p^\ast = 1/4$ (nontrivial, attracting for $0\lt p\lt 1/4$) and $p=1/2$ (unstable boundary). For a depolarizing channel with total rate $p_{\rm dep}$ split equally among $X,Y,Z$, the bit-flip-observable rate is $\frac{2}{3}p_{\rm dep}$ ($X$ and $Y$ both flip in the computational basis), so the depolarizing threshold is $\frac{2}{3}p_{\rm dep} \lt \frac14 \Rightarrow p_{\rm dep} \lt \frac{3}{8} = 3.75\times10^{-1}$.

Conclusion: the bare tree recursion has analytic threshold $3.75\times10^{-1}$ in total depolarizing rate. The claim that the full architecture achieves threshold $\ge 10^{-4}$ is therefore a conservative lower bound: the gap between $10^{-4}$ and $0.375$ must absorb measurement errors, leakage, correlated errors, and decoder imperfections.

#4.2 Cat-code inner layer: from hardware error to effective mode error

Inputs (sourced from [2]):

• Two-qubit gate fidelity 99.56% → $p_{2q} = 1 - 0.9956 = 4.4\times10^{-3}$.
• Single-qubit gate fidelity 99.90% → $p_{1q} = 1.0\times10^{-3}$.
• Readout fidelity 98.7% → $p_{\rm ro} = 1.3\times10^{-2}$.
• Cat mean photon number $\bar n = 6$ (design choice, labeled assumption).

Bit-flip suppression. A single photon loss maps even parity to odd parity; the bit-flip probability per loss event is the coherent-state overlap $p_{\rm bf} = e^{-2\bar n} = e^{-12}$. Computing: $e^{6} = 403.43$, so $e^{-6} = 2.4788\times10^{-3}$, and $e^{-12} = (2.4788\times10^{-3})^2 = 6.144\times10^{-6}$.

Effective per-mode error. Assume (labeled assumption) the dominant physical error channel occurs at rate $p_{2q} = 4.4\times10^{-3}$ per mode per cycle, and each occurrence produces a bit flip only with probability $p_{\rm bf}$. Then

(Arithmetic: $4.4\times6.144 = 27.03$; $10^{-3}\times10^{-6}=10^{-9}$; $27.03\times10^{-9} = 2.703\times10^{-8}$.)

Unsuppressed channel. Conservatively assign the full single-qubit error to dephasing-type errors the cat code does not suppress: $p_{\rm deph} = 1.0\times10^{-3}$ per mode per cycle. Phase flips are handled by the dual-basis repetition, which has the identical recursion and threshold $1/4$. Even charging the readout error as a per-cycle dephasing contribution, $p_{\rm ro} = 1.3\times10^{-2}$, the dual-basis tree margin is $0.25/0.013 = 19.2$. This is the tightest margin in the stack and the origin of our headline number.

Margin against the advertised threshold. The effective bit-flip rate $p_{\rm eff} = 2.703\times10^{-8}$ clears $10^{-4}$ by $10^{-4}/2.703\times10^{-8} = 3.70\times10^{3}$.

#4.3 Logical error under depth-2 concatenation (independence regime)

With $p_{\rm eff} = 2.703\times10^{-8}$ and $f(p) = 3p^2 - 2p^3$:

Level 1: $p_{\rm eff}^2 = 7.306\times10^{-16}$ (since $2.703^2 = 7.306$); $3p_{\rm eff}^2 = 2.192\times10^{-15}$; $2p_{\rm eff}^3 = 2\times1.975\times10^{-24} = 3.95\times10^{-24}$ (since $2.703^3 = 19.75$); so $f(p_{\rm eff}) \approx 2.192\times10^{-15}$.

Level 2: $(2.192\times10^{-15})^2 = 4.805\times10^{-30}$; $3\times4.805\times10^{-30} = 1.44\times10^{-29}$; the cubic term $(2.192\times10^{-15})^3 = 1.053\times10^{-44}$ is negligible. Hence $f^{\circ 2}(p_{\rm eff}) = 1.4\times10^{-29}$.

Result (independence regime): $p_L = 1.4\times10^{-29}$ per logical qubit per cycle for a depth-2 ternary tree. We emphasize this is an upper-bound artifact of the independence assumption, reported to demonstrate the exponential suppression mechanism, not as a performance prediction. As a crude cross-check independent of the cat layer, the simple combinatorial bound "probability of $\ge2$ errors among $p=3$ physical inputs at rate $e$" is $1-[(1-e)^3 + 3e(1-e)^2]$; at $e = 4.4\times10^{-3}$ this gives $1 - (0.986862 + 0.013089) = 4.9\times10^{-5}$ — already below $10^{-4}$ even with no inner bosonic code, though with far less margin.

#4.4 Correlated-error floor

A correlated burst affecting all $p^L = 9$ boundary modes of one subtree simultaneously defeats the majority vote on that subtree. Assume (labeled assumption, deliberately pessimistic) $p_{\rm corr} = 10^{-3}$ per cycle per subtree, with the cat suppression still applying per photon-loss event within the burst. The floor is

(The factor 13 counts all subtrees of $\mathcal{T}_3^{(2)}$: vertices $= (3^3-1)/(3-1) = 13$.) Arithmetic: $6.144\times10^{-6}\times13 = 7.99\times10^{-5}$; $\times10^{-3} = 7.99\times10^{-8}$.

Result: $p_L^{\rm floor} \approx 8.0\times10^{-8}$ per cycle — a factor $10^{-4}/8.0\times10^{-8} = 1.25\times10^{3}$ below the advertised threshold. If instead the burst defeats the cat suppression entirely (worst case: direct bit flips, not photon losses), the floor becomes $p_{\rm corr}\times13 = 1.3\times10^{-2}$, which exceeds $10^{-4}$ — the genuine failure mode of the architecture. The honest statement is conditional: the $10^{-4}$ threshold holds iff correlated events are (a) rare, with the explicit condition derived from the threshold requirement $p_{\rm corr} \le 10^{-4}/13 = 7.7\times10^{-6}$ per subtree per cycle in the worst-case channel, or (b) themselves correctable by an outer layer.

#4.5 Resource comparison: photons per logical qubit

Holographic tree stack. Boundary modes: $p^L = 3^2 = 9$; photons per mode $\bar n = 6$; total $N_{\rm ph}^{\rm tree} = 9\times6 = 54$ photons per logical qubit.

Matched surface code. For a fair comparison, match a surface code without a bosonic inner layer operating at the raw two-qubit error $p_{2q} = 4.4\times10^{-3}$, using the standard circuit-level scaling $p_L \approx 0.1\,(p/p_{\rm th}^{\rm surf})^{(d+1)/2}$ with $p_{\rm th}^{\rm surf} = 10^{-2}$ (standard-literature assumption):

• $p/p_{\rm th}^{\rm surf} = 0.44$; require $0.1\times0.44^{(d+1)/2} \le 10^{-6} \Rightarrow 0.44^{(d+1)/2} \le 10^{-5}$.
• $\ln(0.44) = 3.7842 - 4.6052 = -0.8210$; $\ln(10^{-5}) = -11.5129$; $(d+1)/2 \ge 11.5129/0.8210 = 14.02 \Rightarrow d \ge 27.05$, so $d = 29$.
• Data qubits $d^2 = 841$; with syndrome ancillas $\sim 2d^2 = 1682$ physical qubits per logical qubit.

Against the tree stack's 54 photons, this is a reduction factor of $1682/54 \approx 31$ in physical carriers — consistent in order of magnitude with the 5–40× photon and ~100× mode advantages reported in [11] (which compares at $p_L = 10^{-6}$ with a $10^3$-qubit surface-code baseline, giving 25–200 photons per logical qubit, i.e. 2.5–20 modes at $\bar n = 10$; our $\bar n = 6$ design point sits inside that range).

#5. Results

All numbers below are computed in Section 4 or labeled projections.

1. Raw physical errors (from [2], §4.2): single-qubit $1.0\times10^{-3}$; two-qubit $4.4\times10^{-3}$; readout $1.3\times10^{-2}$.
2. Analytic tree threshold (§4.1): bit-flip $p^\ast = 1/4$; depolarizing $3/8 = 3.75\times10^{-1}$.
3. Cat-code suppression (§4.2): $p_{\rm bf} = e^{-12} = 6.14\times10^{-6}$ per loss event at $\bar n = 6$.
4. Effective bit-flip rate (§4.2): $p_{\rm eff} = 2.703\times10^{-8}$ per mode per cycle; margin over $10^{-4}$: $3.70\times10^{3}$.
5. Tightest channel margin (§4.2): readout-charged dephasing $1.3\times10^{-2}$ vs dual-basis threshold $0.25$: margin $19.2$.
6. Depth-2 logical error, independence regime (§4.3): $1.4\times10^{-29}$ (mechanism demonstration, not a prediction).
7. Conservative no-inner-code cross-check (§4.3): $\ge2$-of-3 error bound $4.9\times10^{-5}$ at $e = 4.4\times10^{-3}$ — below $10^{-4}$ with thin margin.
8. Correlated-error floor (§4.4): $8.0\times10^{-8}$ per cycle at $p_{\rm corr} = 10^{-3}$ with cat suppression intact; survivability condition $p_{\rm corr} \le 7.7\times10^{-6}$ per subtree per cycle in the worst-case channel.
9. Resource budget (§4.5): 54 photons per logical qubit (tree stack) vs $\sim$1682 physical qubits (matched $d=29$ surface code), reduction $\approx31\times$; consistent with [11]'s 5–40× / ~100× figures.

Summary of the headline claim: the threshold of at least $10^{-4}$ is arithmetically consistent with the central-case hardware budget, with the binding practical constraint the readout margin of $19.2\times$; it is conditional on correlated events satisfying $p_{\rm corr} \le 7.7\times10^{-6}$ per subtree per cycle (or being correctable by an outer layer).

#6. Discussion

#Limitations

The analysis is a budget calculation, not a simulation or experiment. Three assumptions carry the result. First, the cat-code operating point $\bar n = 6$ and the mapping "one $p_{2q}$-rate event per mode per cycle" are design choices, not measurements; the effective error scales linearly in both. Second, the decorrelation assumption is critical: superconducting platforms exhibit correlated leakage and crosstalk, and a burst that defeats cat suppression entirely produces a floor of $1.3\times10^{-2}$, far above threshold. Third, the independence-regime figure $1.4\times10^{-29}$ is an artifact demonstrating exponential suppression; real prefactors grow with depth and erode such margins.

#Failure modes

The model fails if: (a) correlated bursts defeat the cat suppression at rates above $p_{\rm corr} \approx 7.7\times10^{-6}$ per subtree per cycle; (b) the RG fixed-point subspace on $\mathcal{T}_p$ is not rigid — local perturbations fail to map to correctable syndromes, which would show up as failure of the CV-flow conditions [8] on the truncated tree graph; (c) bosonic suppression at $\bar n = 6$ is unattainable in superconducting cavities; (d) the $p$-adic structure is not actually realized in the hardware dynamics — i.e., the tree graph is implemented only as a classical decoding layer while the physical noise acts on an ordinary Euclidean lattice, in which case the ultrametric protection claimed in [9] is absent and the analysis reduces to ordinary concatenated repetition coding; or (e) the CV-flow conditions of [8] fail on the truncated tree, making the majority-vote decoder non-deterministic at some depth.

#Outlook and falsifiable tests

The architecture stands or falls on two measurable propositions. First, the survivability condition $p_{\rm corr} \le 7.7\times10^{-6}$ per subtree per cycle in the worst (readout-dominated) channel can be tested directly by crosstalk tomography on the Tianyan-287 platform [2]: if measured correlated-event rates exceed this bound, the $10^{-4}$ threshold claim is falsified unless an outer correctable layer is added. Second, the ultrametric signature of the encoding — the hierarchical clustering of logical syndromes predicted by the $p$-adic geometry — is accept-or-reject testable on trapped-ion simulators along the lines of the program organized in [10]. A negative result on either front confines the proposal to the class of conventional concatenated codes with a geometric interpretation, rather than a genuinely holographic one.

#References

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