{
  "annotation_id": "2eabc068-0cc8-400f-b0c5-9e35d70eab50",
  "date_created": "2026-03-02T18:02:45.242000Z",
  "date_modified": "2026-03-02T18:02:45.242000Z",
  "file_hash": "59614eb49a2772da6d07e81c639a873553ddfa95a37f41a7f41d516cbaca3851",
  "private": false,
  "record": {
    "abstract": "We define the model of quantum circuits with density matrices, where\nnon-unitary gates are allowed. Measurements in the middle of the computation,\nnoise and decoherence are implemented in a natural way in this model, which is\nshown to be equivalent in computational power to standard quantum circuits.\n  The main result in this paper is a solution for the subroutine problem: The\ngeneral function that a quantum circuit outputs is a probabilistic function,\nbut using pure state language, such a function can not be used as a black box\nin other computations. We give a natural definition of using general\nsubroutines, and analyze their computational power.\n  We suggest convenient metrics for quantum computing with mixed states. For\ndensity matrices we analyze the so called ``trace metric\u0027\u0027, and using this\nmetric, we define and discuss the ``diamond metric\u0027\u0027 on superoperators. These\nmetrics enable a formal discussion of errors in the computation.\n  Using a ``causality\u0027\u0027 lemma for density matrices, we also prove a simple\nlower bound for probabilistic functions.",
    "arxiv_id": "quant-ph/9806029",
    "authors": [
      "Dorit Aharonov",
      "Alexei Kitaev",
      "Noam Nisan"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Quantum Circuits with Mixed States",
    "url": "https://arxiv.org/abs/quant-ph/9806029"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "845e9c2a-790f-4e29-8c0b-489b10adf2b7",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}