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In this paper, we propose a method to efficiently verify such noisy intermediate-scale quantum computation. To this end, we first characterize small-scale quantum operations with respect to the diamond norm. Then by using these characterized quantum operations, we estimate the fidelity &#x27E8;<\/mml:mo>&#x03C8;<\/mml:mi>t<\/mml:mi><\/mml:msub>|<\/mml:mo><\/mml:mrow>&#x03C1;<\/mml:mi>&#x005E;<\/mml:mo><\/mml:mover><\/mml:mrow>o<\/mml:mi>u<\/mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msub>|<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>t<\/mml:mi><\/mml:msub>&#x27E9;<\/mml:mo><\/mml:math> between an actual n<\/mml:mi><\/mml:math>-qubit output state &#x03C1;<\/mml:mi>&#x005E;<\/mml:mo><\/mml:mover><\/mml:mrow>o<\/mml:mi>u<\/mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math> obtained from the noisy intermediate-scale quantum computation and the ideal output state (i.e., the target state) |<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>t<\/mml:mi><\/mml:msub>&#x27E9;<\/mml:mo><\/mml:math>. Although the direct fidelity estimation method requires O<\/mml:mi>(<\/mml:mo>2<\/mml:mn>n<\/mml:mi><\/mml:msup>)<\/mml:mo><\/mml:math> copies of &#x03C1;<\/mml:mi>&#x005E;<\/mml:mo><\/mml:mover><\/mml:mrow>o<\/mml:mi>u<\/mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math> on average, our method requires only O<\/mml:mi>(<\/mml:mo>D<\/mml:mi>3<\/mml:mn><\/mml:msup>2<\/mml:mn>12<\/mml:mn>D<\/mml:mi><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:math> copies even in the worst case, where D<\/mml:mi><\/mml:math> is the denseness of |<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>t<\/mml:mi><\/mml:msub>&#x27E9;<\/mml:mo><\/mml:math>. For logarithmic-depth quantum circuits on a sparse chip, D<\/mml:mi><\/mml:math> is at most O<\/mml:mi>(<\/mml:mo>log<\/mml:mi>&#x2061;<\/mml:mo>n<\/mml:mi><\/mml:mrow>)<\/mml:mo><\/mml:math>, and thus O<\/mml:mi>(<\/mml:mo>D<\/mml:mi>3<\/mml:mn><\/mml:msup>2<\/mml:mn>12<\/mml:mn>D<\/mml:mi><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:math> is a polynomial in n<\/mml:mi><\/mml:math>. By using the IBM Manila 5-qubit chip, we also perform a proof-of-principle experiment to observe the practical performance of our method.<\/jats:p>","DOI":"10.22331\/q-2022-07-07-758","type":"journal-article","created":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T12:26:52Z","timestamp":1657196812000},"page":"758","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["Divide-and-conquer verification method for noisy intermediate-scale quantum computation"],"prefix":"10.22331","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2428-7432","authenticated-orcid":false,"given":"Yuki","family":"Takeuchi","sequence":"first","affiliation":[{"name":"NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasuhiro","family":"Takahashi","sequence":"additional","affiliation":[{"name":"NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan"},{"name":"Faculty of Informatics, Gunma University, 4-2 Aramakimachi, Maebashi, Gunma 371-8510, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tomoyuki","family":"Morimae","sequence":"additional","affiliation":[{"name":"Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6041-1704","authenticated-orcid":false,"given":"Seiichiro","family":"Tani","sequence":"additional","affiliation":[{"name":"NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan"},{"name":"International Research Frontiers Initiative (IRFI), Tokyo Institute of Technology, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2022,7,7]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"J. 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