{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T03:04:23Z","timestamp":1774667063474,"version":"3.50.1"},"reference-count":42,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2020,6,4]],"date-time":"2020-06-04T00:00:00Z","timestamp":1591228800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on ann<\/mml:mi><\/mml:math>-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on\u2308<\/mml:mo>log<\/mml:mi>3<\/mml:mn><\/mml:msub>\u2061<\/mml:mo>(<\/mml:mo>2<\/mml:mn>n<\/mml:mi>+<\/mml:mo>1<\/mml:mn>)<\/mml:mo>\u2309<\/mml:mo><\/mml:math>qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less thanlog<\/mml:mi>3<\/mml:mn><\/mml:msub>\u2061<\/mml:mo>(<\/mml:mo>2<\/mml:mn>n<\/mml:mi>)<\/mml:mo><\/mml:math>qubits on average. We apply it to the problem of learningk<\/mml:mi><\/mml:math>-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of allk<\/mml:mi><\/mml:math>-fermion RDMs in parallel, to precision\u03f5<\/mml:mi><\/mml:math>, by repeating a single quantum circuit for\u2272<\/mml:mo>(<\/mml:mo>2<\/mml:mn>n<\/mml:mi>+<\/mml:mo>1<\/mml:mn>)<\/mml:mo>k<\/mml:mi><\/mml:msup>\u03f5<\/mml:mi>\u2212<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>times. This result is based on a method we develop here that allows one to determine individual elements of allk<\/mml:mi><\/mml:math>-qubit RDMs in parallel, to precision\u03f5<\/mml:mi><\/mml:math>, by repeating a single quantum circuit for\u2272<\/mml:mo>3<\/mml:mn>k<\/mml:mi><\/mml:msup>\u03f5<\/mml:mi>\u2212<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>times, independent of the system size. This improves over existing schemes for determining qubit RDMs.<\/jats:p>","DOI":"10.22331\/q-2020-06-04-276","type":"journal-article","created":{"date-parts":[[2020,6,4]],"date-time":"2020-06-04T11:48:29Z","timestamp":1591271309000},"page":"276","source":"Crossref","is-referenced-by-count":65,"title":["Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning"],"prefix":"10.22331","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0435-655X","authenticated-orcid":false,"given":"Zhang","family":"Jiang","sequence":"first","affiliation":[{"name":"Google Research, Venice, CA 90291"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6618-9622","authenticated-orcid":false,"given":"Amir","family":"Kalev","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742-2420, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8497-6363","authenticated-orcid":false,"given":"Wojciech","family":"Mruczkiewicz","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9681-6746","authenticated-orcid":false,"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2020,6,4]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"R. 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