{
  "id": "unitary-synthesis-from-a-boolean-oracle",
  "title": "Unitary synthesis from a Boolean oracle",
  "description": "One quantum oracle circuit, generated in polynomial time from n alone, with polynomially many qubits, gates H, T, T*, CNOT and oracle calls, approximates every n-qubit unitary within diamond-norm distance 1/2 once a suitable Boolean oracle is chosen. The proof: programmable slots; a diagonal block of small norm from random signs and Walsh-Hadamard transforms; the transfer function of the block matrix and two nearly isometric field maps with few nonzero Fourier coefficients per row; amplitude amplification and a recursion that calls the smaller circuit once per level; compilation of every slot into words in H, T, T* read from one oracle, with the phases kept. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "oracle-circuits-and-the-synthesis-theorem",
      "title": "Oracle circuits and the synthesis theorem",
      "reader": "oracle-circuits-and-the-synthesis-theorem.html",
      "source": "src/oracle-circuits-and-the-synthesis-theorem.md",
      "source_sha256": "9540d3153e4b0caea4fdd5885f773844b1f13cd20d969f6578c713ee5b6efc94",
      "status": "author_self_checked"
    },
    {
      "id": "programmable-slots",
      "title": "Programmable slots",
      "reader": "programmable-slots.html",
      "source": "src/programmable-slots.md",
      "source_sha256": "acfcfbb4a94ee83f60f9336c2f12058a865bb9c6e80b794ae7789704520a7fc4",
      "status": "author_self_checked"
    },
    {
      "id": "a-contracting-block",
      "title": "A contracting block",
      "reader": "a-contracting-block.html",
      "source": "src/a-contracting-block.md",
      "source_sha256": "2af5f48ebf508e23b01b31d7e1ce4e70a171a55f6a4c8d274754a21e6c557669",
      "status": "author_self_checked"
    },
    {
      "id": "fourier-fields-for-a-smaller-unitary",
      "title": "Fourier fields for a smaller unitary",
      "reader": "fourier-fields-for-a-smaller-unitary.html",
      "source": "src/fourier-fields-for-a-smaller-unitary.md",
      "source_sha256": "d1e79d30085964edf90411cd38f39bd6352358b2e842a927efeb896f486059b2",
      "status": "author_self_checked"
    },
    {
      "id": "encoders-and-the-recursion",
      "title": "Encoders and the recursion",
      "reader": "encoders-and-the-recursion.html",
      "source": "src/encoders-and-the-recursion.md",
      "source_sha256": "94b11971f09eaf97ba8d180cea29065983f7b0da91fcdc7021188f349ca9ee50",
      "status": "author_self_checked"
    },
    {
      "id": "compilation-into-one-boolean-oracle",
      "title": "Compilation into one Boolean oracle",
      "reader": "compilation-into-one-boolean-oracle.html",
      "source": "src/compilation-into-one-boolean-oracle.md",
      "source_sha256": "e0c1698ed4b4edbe6793bf8573c2f06ab7d9331c2c735f689aaac135d4bbe9e7",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Real Analysis II (Lebl, Basic Analysis II): the inverse function theorem (Theorem 8.5.1); compactness and continuity in finite dimensions",
    "linear algebra over the complex numbers: inner products, operator norms, unitary and Hermitian matrices, tensor products",
    "probability on finite sets and Gaussian integrals: independence, expectation, Markov's and Jensen's inequalities, Fubini's theorem, integration by parts"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}