{
  "id": "2205.06261",
  "version": "v1",
  "published": "2022-05-12T17:58:59.000Z",
  "updated": "2022-05-12T17:58:59.000Z",
  "title": "Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle",
  "authors": [
    "Zane M. Rossi",
    "Isaac L. Chuang"
  ],
  "comment": "20 pages + 4 figures + 10 page appendix",
  "categories": [
    "quant-ph"
  ],
  "abstract": "Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ans\\\"atze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, and say nothing about computing joint properties of two or more oracles; these can be far cheaper to compute given the ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. M-QSP is numerically stable and inherits the approximative efficiency of its single-variable analogue; however, its underlying theory is substantively more complex. M-QSP provides one bridge from quantum algorithms to a rich theory of algebraic geometry over multiple variables. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-12T17:58:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multivariable quantum signal processing",
      "two-headed oracle",
      "stable multivariable polynomial transformation",
      "multiple variables",
      "quantum algorithms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}