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  "slug": "error-confinement-theorem-for-ultrametric-decoders-and-its-implications-for-quan",
  "title": "Error Confinement Theorem for Ultrametric Decoders and Its Implications for Quantum Error Correction",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Fault‑tolerant quantum computing is impeded by the large qubit overhead required for active quantum error correction (QEC). Existing decoders either lack rigorous hierarchical guarantees or incur computational costs that scale with the full syndrome volume. We model a symmetric decoder with p≥ 3 internal states, i.i.d. depolarizing noise, and an ultrametric (tree) metric on the syndrome space. Under these assumptions we prove an Error Confinement Theorem: any error pattern that flips fewer than the ultrametric barrier B(d)=2^d leaves remains confined to a subtree of depth d, and the logical error probability is bounded by p^ B(d). Instantiating the bound with depth d=7 and physical error rate p=10^-3 yields a logical‑error bound of 10^-384. Based on the operation‑count figures reported in [9] for n=1023 qubits, the rank‑decomposition dynamic programming approach requires 1.07×10⁹ operations versus 4.35×10¹2 for tensor‑network contraction, implying a 4.06×10³‑fold speedup. These results indicate that ultrametric decoders can, in principle, suppress logical errors exponentially while dramatically reducing decoding complexity, thereby opening a pathway to lower QEC overheads if the ultrametric assumptions can be realized in hardware.",
  "abstract_markdown": "Fault‑tolerant quantum computing is impeded by the large qubit overhead required for active quantum error correction (QEC).  Existing decoders either lack rigorous hierarchical guarantees or incur computational costs that scale with the full syndrome volume.  We model a symmetric decoder with $p\\ge 3$ internal states, i.i.d. depolarizing noise, and an ultrametric (tree) metric on the syndrome space.  Under these assumptions we prove an **Error Confinement Theorem**: any error pattern that flips fewer than the ultrametric barrier $B(d)=2^{d}$ leaves remains confined to a subtree of depth $d$, and the logical error probability is bounded by $p^{\\,B(d)}$.  Instantiating the bound with depth $d=7$ and physical error rate $p=10^{-3}$ yields a logical‑error bound of $10^{-384}$.  Based on the operation‑count figures reported in [9] for $n=1023$ qubits, the rank‑decomposition dynamic programming approach requires $1.07\\times10^{9}$ operations versus $4.35\\times10^{12}$ for tensor‑network contraction, implying a $4.06\\times10^{3}$‑fold speedup.  These results indicate that ultrametric decoders can, in principle, suppress logical errors exponentially while dramatically reducing decoding complexity, thereby opening a pathway to lower QEC overheads if the ultrametric assumptions can be realized in hardware.",
  "date": "2026-10-21",
  "first_published": "2026-10-21",
  "last_revised": "2026-10-21T00:00:00Z",
  "modified": "2026-10-21T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/error-confinement-theorem-for-ultrametric-decoders-and-its-implications-for-quan",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Error Confinement Theorem for Ultrametric Decoders and Its Implications for Quantum Error Correction"
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        "description": "Fault‑tolerant quantum computing is impeded by the large qubit overhead required for active quantum error correction (QEC). Existing decoders either lack rigorous hierarchical guarantees or incur computational costs that scale with the full syndrome volume. We model a symmetric decoder with p≥ 3 internal states, i.i.d. depolarizing noise, and an ultrametric (tree) metric on the syndrome space. Under these assumptions we prove an Error Confinement Theorem: any error pattern that flips fewer than the ultrametric barrier B(d)=2^d leaves remains confined to a subtree of depth d, and the logical error probability is bounded by p^ B(d). Instantiating the bound with depth d=7 and physical error rate p=10^-3 yields a logical‑error bound of 10^-384. Based on the operation‑count figures reported in [9] for n=1023 qubits, the rank‑decomposition dynamic programming approach requires 1.07×10⁹ operations versus 4.35×10¹2 for tensor‑network contraction, implying a 4.06×10³‑fold speedup. These results indicate that ultrametric decoders can, in principle, suppress logical errors exponentially while dramatically reducing decoding complexity, thereby opening a pathway to lower QEC overheads if the ultrametric assumptions can be realized in hardware.",
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