{
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  "slug": "measurement-and-feedforward-quantum-circuits-as-code-design-detectability-branch",
  "title": "Measurement-and-Feedforward Quantum Circuits as Code Design: Detectability, Branch Counting, and the Location of Nonstabilizerness",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Shallow quantum circuits augmented by mid-circuit measurements and classical feedforward can deterministically prepare long-range entangled states and implement global unitaries that no comparable-depth unitary circuit can reach. Recent work established a general correspondence between all such protocols and quantum error-correcting codes (QECCs): the unitary circuit preceding the measurements acts as an encoder, and unitary feedforward eliminates post-selection if and only if the measurement projectors are detectable errors on the codespace. We develop the quantitative consequences of this correspondence. We derive the branch-count identity N_br = 2^n-k for an [[n,k]] stabilizer code, the detectability fraction p_det = 1 - 2²-n-k (evaluated for the [[5,1,3]] and [[7,1,3]] codes as $0.9375$ and $0.984375$), and the magic accounting M₂(|T⟩) = log₂(4/3) ≈ 0.415037 bits per injected T state, with linear scaling M₂^(m) = m M₂(|T⟩). A worked post-selection comparison on the Steane code shows a depth-reduction factor of 80/17 ≈ 4.71 for an encoder of depth d=10. [...]",
  "abstract_markdown": "Shallow quantum circuits augmented by mid-circuit measurements and classical feedforward can deterministically prepare long-range entangled states and implement global unitaries that no comparable-depth unitary circuit can reach. Recent work established a general correspondence between all such protocols and quantum error-correcting codes (QECCs): the unitary circuit preceding the measurements acts as an encoder, and unitary feedforward eliminates post-selection if and only if the measurement projectors are detectable errors on the codespace. We develop the quantitative consequences of this correspondence. We derive the branch-count identity $N_{\\text{br}} = 2^{n-k}$ for an $[[n,k]]$ stabilizer code, the detectability fraction $p_{\\text{det}} = 1 - 2^{2-n-k}$ (evaluated for the $[[5,1,3]]$ and $[[7,1,3]]$ codes as $0.9375$ and $0.984375$), and the magic accounting $M_2(|T\\rangle) = \\log_2(4/3) \\approx 0.415037$ bits per injected $T$ state, with linear scaling $M_2^{(m)} = m\\,M_2(|T\\rangle)$. A worked post-selection comparison on the Steane code shows a depth-reduction factor of $80/17 \\approx 4.71$ for an encoder of depth $d=10$. [...]",
  "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/measurement-and-feedforward-quantum-circuits-as-code-design-detectability-branch",
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    "titles": [
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        "title": "Measurement-and-Feedforward Quantum Circuits as Code Design: Detectability, Branch Counting, and the Location of Nonstabilizerness"
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    "publicationYear": "2026",
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        "description": "Shallow quantum circuits augmented by mid-circuit measurements and classical feedforward can deterministically prepare long-range entangled states and implement global unitaries that no comparable-depth unitary circuit can reach. Recent work established a general correspondence between all such protocols and quantum error-correcting codes (QECCs): the unitary circuit preceding the measurements acts as an encoder, and unitary feedforward eliminates post-selection if and only if the measurement projectors are detectable errors on the codespace. We develop the quantitative consequences of this correspondence. We derive the branch-count identity N_br = 2^n-k for an [[n,k]] stabilizer code, the detectability fraction p_det = 1 - 2²-n-k (evaluated for the [[5,1,3]] and [[7,1,3]] codes as $0.9375$ and $0.984375$), and the magic accounting M₂(|T⟩) = log₂(4/3) ≈ 0.415037 bits per injected T state, with linear scaling M₂^(m) = m M₂(|T⟩). A worked post-selection comparison on the Steane code shows a depth-reduction factor of 80/17 ≈ 4.71 for an encoder of depth d=10. [...]",
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