{
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  "slug": "the-scaling-paradox-of-surface-code-quantum-error-correction-exponential-suppres",
  "title": "The Scaling Paradox of Surface-Code Quantum Error Correction: Exponential Suppression, Logarithmic Overhead, and a Landauer Floor That Grows with Reliability",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Surface-code quantum error correction (QEC) is widely regarded as the leading route to fault-tolerant quantum computation, yet its resource story contains an under-examined tension we call the scaling paradox: the code suppresses logical errors exponentially in code distance d, so the physical-qubit overhead per logical qubit grows only as the square of a quantity that is itself logarithmic in the target reliability 1/p_L — while at the same time the thermodynamic cost of correction, priced by Landauer's principle, grows linearly in d and therefore also logarithmically in 1/p_L. Reliability is thus simultaneously cheap (polynomially, in qubits) and never free (the erasure floor per logical qubit per round, (n-k)/(k)k_B Tln 2, increases with every increment of d). We make this tension quantitative. Using the standard threshold ansatz p_L ≈ 0.1 (p/p_th)^(d+1)/2 with p_th = 10^-2, we show that reaching p_L = 10^-12 at physical error rate p = 10^-3 requires $d = 21$ and $n = 881$ physical qubits per logical qubit, whereas p = 10^-4 requires only $d = 10$ and $n = 199$ — a 4.44× overhead reduction. [...]",
  "abstract_markdown": "Surface-code quantum error correction (QEC) is widely regarded as the leading route to fault-tolerant quantum computation, yet its resource story contains an under-examined tension we call the scaling paradox: the code suppresses logical errors exponentially in code distance $d$, so the physical-qubit overhead per logical qubit grows only as the square of a quantity that is itself logarithmic in the target reliability $1/p_L$ — while at the same time the thermodynamic cost of correction, priced by Landauer's principle, grows linearly in $d$ and therefore also logarithmically in $1/p_L$. Reliability is thus simultaneously cheap (polynomially, in qubits) and never free (the erasure floor per logical qubit per round, $\\frac{n-k}{k}k_B T\\ln 2$, increases with every increment of $d$). We make this tension quantitative. Using the standard threshold ansatz $p_L \\approx 0.1\\,(p/p_{\\mathrm{th}})^{(d+1)/2}$ with $p_{\\mathrm{th}} = 10^{-2}$, we show that reaching $p_L = 10^{-12}$ at physical error rate $p = 10^{-3}$ requires $d = 21$ and $n = 881$ physical qubits per logical qubit, whereas $p = 10^{-4}$ requires only $d = 10$ and $n = 199$ — a $4.44\\times$ overhead reduction. [...]",
  "date": "2026-10-09",
  "first_published": "2026-10-09",
  "last_revised": "2026-10-09T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/the-scaling-paradox-of-surface-code-quantum-error-correction-exponential-suppres",
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    "titles": [
      {
        "title": "The Scaling Paradox of Surface-Code Quantum Error Correction: Exponential Suppression, Logarithmic Overhead, and a Landauer Floor That Grows with Reliability"
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    "publicationYear": "2026",
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        "description": "Surface-code quantum error correction (QEC) is widely regarded as the leading route to fault-tolerant quantum computation, yet its resource story contains an under-examined tension we call the scaling paradox: the code suppresses logical errors exponentially in code distance d, so the physical-qubit overhead per logical qubit grows only as the square of a quantity that is itself logarithmic in the target reliability 1/p_L — while at the same time the thermodynamic cost of correction, priced by Landauer's principle, grows linearly in d and therefore also logarithmically in 1/p_L. Reliability is thus simultaneously cheap (polynomially, in qubits) and never free (the erasure floor per logical qubit per round, (n-k)/(k)k_B Tln 2, increases with every increment of d). We make this tension quantitative. Using the standard threshold ansatz p_L ≈ 0.1 (p/p_th)^(d+1)/2 with p_th = 10^-2, we show that reaching p_L = 10^-12 at physical error rate p = 10^-3 requires $d = 21$ and $n = 881$ physical qubits per logical qubit, whereas p = 10^-4 requires only $d = 10$ and $n = 199$ — a 4.44× overhead reduction. [...]",
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