{
  "id": "randomized-versus-quantum-query-complexity",
  "title": "Randomized versus quantum query complexity",
  "description": "How many input bits must an algorithm read to compute a Boolean function: deterministic, randomized and quantum query complexity. The quantum query model with a proof that measurements can be deferred; amplitude amplification; the adversary bound; Huang's theorem on the sensitivity graph; the inequality D(f) <= C Q(f)^4 for total functions, with complete proofs. Functions with quantum complexity about sqrt(m) and randomized complexity about m^(2-1/k), which show that the exponent 4 is optimal; the randomized lower bound uses one named hypothesis, the Forrelation lower bound of Bansal and Sinha. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "query-complexity-and-a-function-with-outward-pointers",
      "title": "Query complexity and a function with outward pointers",
      "reader": "query-complexity-and-a-function-with-outward-pointers.html",
      "source": "src/query-complexity-and-a-function-with-outward-pointers.md",
      "source_sha256": "3a3cd2eb0c000035e5ed65c2330ebdaa5cc031fb4a2c7d9d46d6ed2d4aa2620a",
      "status": "author_self_checked"
    },
    {
      "id": "the-quantum-query-model",
      "title": "The quantum query model",
      "reader": "the-quantum-query-model.html",
      "source": "src/the-quantum-query-model.md",
      "source_sha256": "ee59332fc6910fd383b5f03644dcfdeb05de598c80b7f646f2e56b765d6e4f95",
      "status": "author_self_checked"
    },
    {
      "id": "amplitude-amplification-and-the-forrelation-test",
      "title": "Amplitude amplification and the Forrelation test",
      "reader": "amplitude-amplification-and-the-forrelation-test.html",
      "source": "src/amplitude-amplification-and-the-forrelation-test.md",
      "source_sha256": "1a1d5f057d91d5e894b0c72ab6c7f102f228b890e20d80234d739b05c5a1be60",
      "status": "author_self_checked"
    },
    {
      "id": "a-tournament-with-probabilistic-comparisons",
      "title": "A tournament with probabilistic comparisons",
      "reader": "a-tournament-with-probabilistic-comparisons.html",
      "source": "src/a-tournament-with-probabilistic-comparisons.md",
      "source_sha256": "d57d99d6cbca1babaa542604e8ffb0fa45730641313561eabaa6e9afb0dbd8a5",
      "status": "author_self_checked"
    },
    {
      "id": "the-randomized-lower-bound",
      "title": "The randomized lower bound",
      "reader": "the-randomized-lower-bound.html",
      "source": "src/the-randomized-lower-bound.md",
      "source_sha256": "06c97a0b8137c58a94721ba0ef287fe0c402d87ab57413e5de0989ca0d00a31a",
      "status": "author_self_checked"
    },
    {
      "id": "the-quartic-upper-bound-and-the-optimal-exponent",
      "title": "The quartic upper bound and the optimal exponent",
      "reader": "the-quartic-upper-bound-and-the-optimal-exponent.html",
      "source": "src/the-quartic-upper-bound-and-the-optimal-exponent.md",
      "source_sha256": "438e0d8e5761841a28132866421921f0a3dc6badcabe4aab34ed88abfbc2f2fb",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Linear Algebra (B40): complex inner product spaces, unitary maps, eigenvalues of real symmetric matrices",
    "finite probability: conditional probabilities, means, the inequalities of Markov and Chebyshev",
    "the minimax theorem for finite games, proved in the course Large independent sets in three-colorable graphs (lesson Alignment of lists: patterns, minimax and Ramsey's theorem)",
    "the lessons prove every other fact they use, except the named hypothesis (BS) of the first lesson, which is used only for the randomized lower bound"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}