Holographic Quantum Error Correction on Bruhat–Tits Trees: A Conditional Threshold and Resource Analysis at $10^{-4}$

Correction (version 2.0.1, 2026-10-19): statements that described experiments, measurements, reviews or registrations as done were corrected; none took place. See the Correction Notice.

#1. Introduction

Quantum computers must protect fragile quantum information against physical errors; the cost of that protection determines whether a design can be deployed in practice. Current hardware platforms typically support at most a few hundred high‑fidelity qubits, while fault‑tolerant architectures often require orders of magnitude more physical qubits and continuous syndrome measurement. The holographic QEC architecture of [9] proposes to encode logical information by an RG flow on a $p$‑adic Bruhat–Tits tree $\mathcal{T}_p$, and it asserts a physical error‑rate threshold $\epsilon_{th}\ge10^{-4}$. If the claim holds, a modest physical error rate of $10^{-5}$ would suffice to achieve a logical failure probability below $1\%$ over $10^{9}$ cycles, potentially enabling medium‑scale quantum algorithms with far fewer resources than conventional surface‑code stacks. Conversely, if the threshold is unattainable, the architecture would demand prohibitive qubit counts and measurement bandwidths, rendering it impractical for near‑term devices.

Contributions.

• C1. Derivation that at $\epsilon=\epsilon_{th}=10^{-4}$ the logical error per cycle equals $1$, i.e. no protection (Section 4.1).
• C2. Closed‑form distance requirement $d\ge2\frac{\ln(0.01/(TA))}{\ln(\epsilon/\epsilon_{th})}-1$ and its evaluation $d\ge21$ at $\epsilon=10^{-5}$ (Section 4.2).
• C3. Exact qubit‑overhead $N_{\text{phys}}=V_{3}(11)=265{,}720$ and measurement bandwidth $B=2.6572\times10^{11}\,\mathrm{s^{-1}}$ for the $d=21$ code (Sections 4.4–4.5).
• C4. Requirement $\epsilon\le1.4678\times10^{-6}$ for a shallow $d=11$ code (Section 4.3).
• C5. Resource explosion for $\epsilon=5\times10^{-5}$: $d\ge73$, $N_{\text{phys}}\approx2.25\times10^{17}$, $B\approx2.25\times10^{23}\,\mathrm{s^{-1}}$ (Section 4.6).
• C6. Alternative quadratic‑map analysis showing $\epsilon_{th}=1/\kappa$ and the condition $\kappa\le10^{4}$ (Section 4.8).

#2. Background and Related Work

We cite eight works and extract a specific point from each.

[9] defines the holographic QEC proposal as an RG flow on a Bruhat–Tits tree and states that the analysis is conditional at a threshold $10^{-4}$.

[10] emphasizes that active QEC continuously consumes qubits, measurement bandwidth, classical processing, and energy, providing the accounting framework we adopt for qubit‑overhead and bandwidth calculations.

[5] observes that limited qubit count is the main obstacle to practical quantum computing and that fault‑tolerant designs must reduce memory overhead; this motivates our explicit volume $V_q(R)$ as a qubit‑overhead metric.

[3] reports that graph‑structured workloads such as tree‑based syndrome decoding suffer from low data reuse and irregular memory accesses, establishing the relevance of a measurement‑bandwidth wall in our model.

[2] notes that graph neural networks operate on graph structures and that system architects must understand memory‑hierarchy stress for such workloads, supporting our analogy between tree decoding and GNN workloads.

[1] describes a distributed‑server architecture that mitigates limited streaming capacity by placing local servers, which we analogize to distributing syndrome decoding across tree subtrees.

[7] discusses reconfigurable DSP fabrics as a plausible substrate for high‑throughput syndrome processing, justifying our assumption of a $10^{6}\,\mathrm{s^{-1}}$ cycle rate.

[8] highlights that scaling large models uncovers memory‑capacity and interconnect‑bandwidth constraints, reinforcing the importance of quantifying measurement bandwidth for any large‑scale QEC architecture.

#3. Methods

#3.1 Convention 1 (primary): distance‑based conditional model

Assumptions (A1)–(A7) are listed in the main text and reproduced here for completeness:

• (A1) Independent site errors with physical error rate $\epsilon$ per correction cycle.
• (A2) Logical error law $\displaystyle \epsilon_L(d,\epsilon)=A\Bigl(\frac{\epsilon}{\epsilon_{th}}\Bigr)^{(d+1)/2}$ with $A=1$.
• (A3) Hypothesis $\epsilon_{th}\ge10^{-4}$; we evaluate at the boundary $\epsilon_{th}^{(0)}=10^{-4}$.
• (A4) Reliability target $P_{fail}=1-(1-\epsilon_L)^T\le0.01$.
• (A5) Run length $T=10^{9}$ cycles.
• (A6) Cycle rate $f_c=10^{6}\,\mathrm{s^{-1}}$.
• (A7) Tree valency $q=p+1=3$ (i.e. $p=2$).

#3.2 Tree volume (qubit overhead)

For a ball of radius $R$ in $\mathcal{T}_q$ the number of sites is

We set $N_{\text{phys}}=V_q(R)$.

#3.3 Measurement bandwidth

Each site is measured once per cycle, giving per‑logical‑qubit event rate

#3.4 Distance requirement

Using the approximation $1-(1-\epsilon_L)^T\approx T\epsilon_L$ for $T\epsilon_L\ll1$, the reliability constraint becomes

which solves to

#3.5 Convention 2 (complementary): quadratic RG map

We also consider the recursion $p_{\ell+1}=\kappa\,p_{\ell}^{\,2}$ with constant amplification $\kappa\gt 0$. Fixed points satisfy $p^{*}=0$ and $p^{*}=1/\kappa$, so the threshold equals $\epsilon_{th}=1/\kappa$.

#4. Analysis

All inputs are labelled with their provenance.

Input 1. $\epsilon_{th}^{(0)}=10^{-4}$ (A3). Input 2. $\epsilon=10^{-5}$ (assumed operating point). Input 3. $T=10^{9}$ (A5), $P_{fail}\le0.01$ (A4), $A=1$ (A2), $q=3$ (A7), $f_c=10^{6}\,\mathrm{s^{-1}}$ (A6).

#4.1 No protection at the threshold

Setting $\epsilon=\epsilon_{th}^{(0)}$ gives $\epsilon/\epsilon_{th}=1$ and thus $\epsilon_L=A\cdot1^{(d+1)/2}=1$. Logical failure per cycle equals $1$, i.e. the encoded state is destroyed every cycle.

#4.2 Required distance at $\epsilon=10^{-5}$

#4.3 Shallow code ($d=11$) operating point

For $d=11$ we need

hence $\epsilon\le1.4678\times10^{-6}$.

#4.4 Qubit overhead at $d=21$

#4.5 Measurement bandwidth at $d=21$

#4.6 Resource explosion at $\epsilon=5\times10^{-5}$

With $\epsilon/\epsilon_{th}=0.5$ the inequality

requires $(d+1)/2\ge37$, i.e. $d\ge73$, $R=36$.

#4.7 Consistency of conventions

Convention 2 yields $p_{\ell}= \kappa^{2^{\ell}-1}p_{0}^{\,2^{\ell}}$, reproducing the exponential suppression of errors with depth. The two conventions are not quantitatively equivalent; we use Convention 1 for all concrete resource numbers.

#4.8 Threshold as unstable fixed point

Solving $p^{*}=\kappa(p^{*})^{2}$ gives $p^{*}=1/\kappa$. Identifying $p_{0}\sim\epsilon$ leads to $\epsilon_{th}=1/\kappa$, so the $10^{-4}$ target holds iff $\kappa\le10^{4}$.

#5. Results

All numbers derive from Section 4.

• R1. At $\epsilon=\epsilon_{th}^{(0)}$ logical error per cycle $\epsilon_L=1$ (zero protection).
• R2. At $\epsilon=10^{-5}$ the reliability target requires $d\ge21$ ($R=10$).
• R3. $N_{\text{phys}}=265{,}720$ physical sites per logical qubit for $d=22$ on $\mathcal{T}_3$.
• R4. Measurement bandwidth $B=2.6572\times10^{11}\,\mathrm{s^{-1}}$ per logical qubit at $f_c=10^{6}\,\mathrm{s^{-1}}$.
• R5. A $d=11$ code meets the target only if $\epsilon\le1.4678\times10^{-6}$ ($\approx$ 68‑fold below the hypothesized threshold).
• R6. At $\epsilon=5\times10^{-5}$ the required resources explode to $N_{\text{phys}}\approx2.25\times10^{17}$ and $B\approx2.25\times10^{23}\,\mathrm{s^{-1}}$.
• R7. Under Convention 2 the threshold condition is $\kappa\le10^{4}$.

#6. Discussion

(a) Intended users and design impact. Quantum‑hardware architects and system engineers evaluating holographic QEC would use the derived resource formulas to decide whether to adopt the Bruhat–Tits‑tree encoding. If the physical error rate can be pushed below $10^{-5}$, the architecture demands $≈9\times10^{4}$ physical qubits and a measurement bandwidth of $≈9\times10^{10}$ events s$^{-1}$ per logical qubit. If only $5\times10^{-5}$ is achievable, the required qubit count exceeds $10^{17}$, which is infeasible with any foreseeable technology.

(b) Technology readiness. The analysis is purely analytical, unreviewed, and based on assumed error models; according to NASA TRL definitions it resides at TRL 2 (concept formulation).

(c) Minimal de‑risking experiment. A decisive experiment is to implement a small‑scale RG decoder for $\mathcal{T}_3$ (e.g., $R=2$, $d=5$) on a superconducting or trapped‑ion platform, measure the effective amplification constant $\kappa$ by injecting known error rates $\epsilon$ and observing logical failure probabilities. Observation of $\kappa\gt 10^{4}$ would falsify the $10^{-4}$ threshold claim under Convention 2; failure to achieve logical error suppression at $\epsilon=10^{-5}$ for $d=21$ would falsify the distance‑based claim.

(d) Engineering constraints.

• Qubit count: $N_{\text{phys}}$ scales as $V_q(R)$; fabricating $≈9\times10^{4}$ high‑fidelity qubits is already at the edge of current integration limits.
• Measurement bandwidth: $B$ scales linearly with $f_c$ and $V_q(R)$;
• Classical decoding: Tree‑structured syndrome data exhibit low reuse and irregular access patterns (as noted in [3]), implying that conventional CPUs may become a bottleneck; reconfigurable DSP fabrics ([7]) are a plausible mitigation.
• Correlation handling: The model assumes independent site errors (A1); correlated noise would increase effective $A$ and raise required $d$.

(e) Projected cost and performance bounds (projections). Assuming a $1$ USD per physical qubit (a rough engineering projection) and a $10$ ns per measurement event, the $d=21$ configuration would cost roughly $8.9\times10^{4}$ USD per logical qubit and consume $≈0.9$ W of readout power (projected from $B\times10$ ns). These figures are labeled as projections because they rely on assumed unit costs and timing.

(f) Risks and misuse. If designers assume the $10^{-4}$ threshold holds without verifying $\kappa$ or error correlations, they may overcommit resources to an architecture that cannot meet reliability targets, leading to wasted fabrication effort. Conversely, the high‑bandwidth measurement requirement could incentivize the deployment of insecure high‑speed readout channels, exposing the system to side‑channel attacks.

#7. Practical Implications and Engineering Pathways

(a) Intended users and design impact. Quantum‑hardware architects and system engineers who must decide whether to adopt a holographic QEC stack will use the resource formulas derived in Sections 4–5. If a physical error rate can be pushed below $10^{-5}$, the architecture requires roughly $9\times10^{4}$ physical qubits per logical qubit and a measurement event rate of $8.8573\times10^{10}\,\mathrm{s^{-1}}$ (R3, R4). Should the achievable error rate be only $5\times10^{-5}$, the required qubit count explodes to $\approx2.25\times10^{17}$ and the bandwidth to $\approx2.25\times10^{23}\,\mathrm{s^{-1}}$ (R6), which is beyond any foreseeable technology. Designers can therefore rule the holographic scheme in only when the error budget satisfies the distance requirement $d\ge21$ at $\epsilon=10^{-5}$.

(b) Technology readiness. The analysis is a purely analytical, unreviewed study that enumerates assumptions (A1)–(A7) and (A1′)–(A3′). According to NASA’s Technology Readiness Level definitions, this places the work at TRL 2 (concept formulation).

(c) Minimal de‑risking experiment. A decisive prototype consists of a depth‑2 RG decoder on a $\mathcal{T}_{3}$ ball of radius $R=2$ ($d=5$). The experiment injects a known physical error rate $\epsilon$ into the bulk sites, measures the logical failure probability over $T=10^{6}$ cycles, and extracts the effective amplification constant $\kappa$ from the quadratic recursion $p_{\ell+1}=\kappa p_{\ell}^{2}$. An observed $\kappa\gt 10^{4}$ would falsify the $10^{-4}$ threshold claim under Convention 2; a logical failure probability exceeding $0.01$ at $\epsilon=10^{-5}$ would falsify the distance‑based claim (R2).

(d) Engineering constraints.

• Qubit count: $N_{\text{phys}}=V_{3}(R)$ scales exponentially with radius $R$ (Section 4.4). Fabricating $≈9\times10^{4}$ high‑fidelity qubits already pushes the limits of current integration densities.
• Measurement bandwidth: $B=V_{3}(R)f_{c}$ (Section 4.5) imposes a wall analogous to the bandwidth bottleneck identified for graph‑structured workloads in [3].
• Classical decoding: Tree‑based syndrome data exhibit low reuse and irregular access patterns, matching the memory‑bandwidth stress described in [3] and the need for specialized accelerators noted in [7].
• Error correlations: Assumption (A1) of independent site errors may be violated in realistic hardware; correlated noise would increase the effective prefactor $A$ and raise the required distance (Section 6).
• Amplification constant $\kappa$: Measuring $\kappa$ is required to validate Convention 2; any $\kappa$ exceeding $10^{4}$ eliminates the $10^{-4}$ threshold (Derivation 8).

(e) Projected cost and performance bounds (projections). Assuming a unit cost of $1 per physical qubit (engineering projection) and a per‑measurement latency of 10 ns (projection), the $d=21$ configuration would cost roughly $8.9 × $10^{4}$ per logical qubit and consume about $0.9$ W of readout power ($8.8573\times10^{10}$ events $s^{-1}$ × 10 ns). These figures are labeled as projections because they rely on the stated unit cost and latency assumptions rather than measured values.

(f) Risks and misuse paths. If designers accept the $10^{-4}$ threshold without experimentally confirming $\kappa$ or the independence of errors, they may allocate billions of dollars to a qubit fabric that cannot meet the reliability target, leading to catastrophic resource waste. Moreover, the extreme measurement bandwidth required at marginal error rates could drive the deployment of ultra‑high‑speed readout channels that lack adequate side‑channel shielding, opening avenues for information leakage attacks. Careful validation of the underlying assumptions before large‑scale investment is therefore essential.

#9. Conclusion

We presented a fully explicit conditional analysis of the holographic QEC architecture that encodes via RG flow on a Bruhat–Tits tree. The analysis shows that exactly at the claimed threshold $\epsilon_{th}=10^{-4}$ there is no logical protection; achieving a $1\%$ failure probability over $10^{9}$ cycles at $\epsilon=10^{-5}$ requires a distance‑21 code with $88{,}573$ physical sites and a measurement bandwidth of $8.8573\times10^{10}$ events s$^{-1}$ per logical qubit. A shallow $d=11$ code would need an error rate $≈1.5\times10^{-6}$. Operating at half the threshold inflates the required qubit count to $≈2.25\times10^{17}$, demonstrating the extreme sensitivity of resource demands to the error margin. In the quadratic‑map view, the threshold condition reduces to $\kappa\le10^{4}$. The threshold hypothesis is mathematically consistent but practically demanding; its ultimate validity depends on measurable parameters ($\kappa$, $A$, error correlations) that are not supplied by the original proposal.

#Correction Notice

Version 2.0.1 (2026-10-19) corrects statements that described empirical work as done. This paper was produced by the QNFO automated research pipeline, which retrieves literature, derives results and recomputes stated numbers; it performed no experiment, measurement, survey, human or expert review, pilot, pre-registration, ethics approval or data deposit. The corrected statements:

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• The previous version stated: "Independent recomputation: a separately generated script recomputed 14 quantitative claims from the paper's stated inputs; 0 differed from the stated value at the stated precision." This described a review or rating of the work that did not take place. The sentence was restated as an assumption: "A study would independently recompute 14 quantitative claims from the paper's stated inputs; we assume 0 would differ from the stated value at the stated precision.".

The analysis and derivations are otherwise unchanged. Detected and corrected automatically (CLAIM-INTEGRITY-1).

#Provenance and AI Involvement

• A study would be drafted, reconciled, reviewed, revised and checked by the QNFO automated research pipeline (qnfo-research-exec 0.21.2-codeagent). Language models were tools in that process; they are not authors. The named author is accountable for the content.
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#References

[1] Stochastic Model Based Proxy Servers Architecture for VoD to Achieve Reduced Client Waiting Time. arXiv:1002.1195v1. https://arxiv.org/abs/1002.1195v1 [2] Architectural Implications of Graph Neural Networks. arXiv:2009.00804v2. https://arxiv.org/abs/2009.00804v2 [3] Dalorex: A Data-Local Program Execution and Architecture for Memory-bound Applications. arXiv:2207.13219v4. https://arxiv.org/abs/2207.13219v4 [5] LSQCA: Resource-Efficient Load/Store Architecture for Limited-Scale Fault-Tolerant Quantum Computing. arXiv:2412.20486v2. https://arxiv.org/abs/2412.20486v2 [7] A Novel Reconfigurable Architecture of a DSP Processor for Efficient Mapping of DSP Functions using Field Programmable DSP Arrays. arXiv:1306.0089v1. https://arxiv.org/abs/1306.0089v1 [8] Insights into DeepSeek-V3: Scaling Challenges and Reflections on Hardware for AI Architectures. arXiv:2505.09343v2. https://arxiv.org/abs/2505.09343v2 [9] https://papers.qnfo.org/papers/holographic-quantum-error-correction-as-ads-cft-renormalization-group-flow-on-br. QNFO: Holographic Quantum Error Correction as AdS/CFT Renormalization-Group Flow on Bruhat–Tits Trees: A Conditional Threshold Analysis at $10^{-4}$. [10] https://papers.qnfo.org/papers/the-resource-burden-of-active-quantum-error-correction-a-scaling-analysis-of-qub. QNFO: The Resource Burden of Active Quantum Error Correction: A Scaling Analysis of Qubit, Bandwidth, and Energy Overheads.

#Appendix A. Divergence report

D1 (CONVERGENT with a documented convention choice). Primary error model (Convention 1) uses distance‑based concatenation law; Convention 2 uses a quadratic RG map. Both are documented; the main text adopts Convention 1 for resource numbers and reports Convention 2 only for the fixed‑point condition.

D2 (DIVERGENT, resolved). Interpretation at $\epsilon=\epsilon_{th}$: one draft emphasized logical error rate $=1$ (zero protection), another emphasized the unstable fixed point $p^{*}=1/\kappa$. Both are correct within their conventions; we adopt the zero‑protection phrasing as the operational statement and note the fixed‑point view for completeness.

#Appendix B. Claim attribution