{
  "id": "2411.12730",
  "version": "v1",
  "published": "2024-11-19T18:52:55.000Z",
  "updated": "2024-11-19T18:52:55.000Z",
  "title": "Testing classical properties from quantum data",
  "authors": [
    "Matthias C. Caro",
    "Preksha Naik",
    "Joseph Slote"
  ],
  "comment": "38 + 14 pages, 2 tables, 2 figures",
  "categories": [
    "quant-ph",
    "cs.CC",
    "cs.DS",
    "cs.LG"
  ],
  "abstract": "Many properties of Boolean functions can be tested far more efficiently than the function can be learned. However, this advantage often disappears when testers are limited to random samples--a natural setting for data science--rather than queries. In this work we investigate the quantum version of this scenario: quantum algorithms that test properties of a function $f$ solely from quantum data in the form of copies of the function state for $f$. For three well-established properties, we show that the speedup lost when restricting classical testers to samples can be recovered by testers that use quantum data. For monotonicity testing, we give a quantum algorithm that uses $\\tilde{\\mathcal{O}}(n^2)$ function state copies as compared to the $2^{\\Omega(\\sqrt{n})}$ samples required classically. We also present $\\mathcal{O}(1)$-copy testers for symmetry and triangle-freeness, comparing favorably to classical lower bounds of $\\Omega(n^{1/4})$ and $\\Omega(n)$ samples respectively. These algorithms are time-efficient and necessarily include techniques beyond the Fourier sampling approaches applied to earlier testing problems. These results make the case for a general study of the advantages afforded by quantum data for testing. We contribute to this project by complementing our upper bounds with a lower bound of $\\Omega(1/\\varepsilon)$ for monotonicity testing from quantum data in the proximity regime $\\varepsilon\\leq\\mathcal{O}(n^{-3/2})$. This implies a strict separation between testing monotonicity from quantum data and from quantum queries--where $\\tilde{\\mathcal{O}}(n)$ queries suffice when $\\varepsilon=\\Theta(n^{-3/2})$. We also exhibit a testing problem that can be solved from $\\mathcal{O}(1)$ classical queries but requires $\\Omega(2^{n/2})$ function state copies, complementing a separation of the same magnitude in the opposite direction derived from the Forrelation problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-19T18:52:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum data",
      "testing classical properties",
      "function state copies",
      "quantum algorithm",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}