{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T22:49:59Z","timestamp":1775688599244,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,28]],"date-time":"2022-02-28T00:00:00Z","timestamp":1646006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CHE-1836497"],"award-info":[{"award-number":["CHE-1836497"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["PHY-1748958"],"award-info":[{"award-number":["PHY-1748958"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"McDevitt bequest at Georgetown University","award":["not applicable"],"award-info":[{"award-number":["not applicable"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The chemistry community has long sought the exact relationship between the conventional and the unitary coupled cluster ansatz for a single-reference system, especially given the interest in performing quantum chemistry on quantum computers. In this work, we show how one can use the operator manipulations given by the exponential disentangling identity and the Hadamard lemma to relate the factorized form of the unitary coupled-cluster approximation to a factorized form of the conventional coupled cluster approximation (the factorized form is required, because some amplitudes are operator-valued and do not commute with other terms). By employing the Trotter product formula, one can then relate the factorized form to the standard form of the unitary coupled cluster ansatz. The operator dependence of the factorized form of the coupled cluster approximation can also be removed at the expense of requiring even more higher-rank operators, finally yielding the conventional coupled cluster. The algebraic manipulations of this approach are daunting to carry out by hand, but can be automated on a computer for small enough systems.<\/jats:p>","DOI":"10.3390\/sym14030494","type":"journal-article","created":{"date-parts":[[2022,2,28]],"date-time":"2022-02-28T20:11:57Z","timestamp":1646079117000},"page":"494","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Operator Relationship between Conventional Coupled Cluster and Unitary Coupled Cluster"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6232-9165","authenticated-orcid":false,"given":"James K.","family":"Freericks","sequence":"first","affiliation":[{"name":"Department of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1103\/RevModPhys.79.291","article-title":"Coupled-cluster theory in quantum chemistry","volume":"79","author":"Bartlett","year":"2007","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kutzelnigg, W. (1977). Pair Correlation Theories. Methods of Electronic Structure Theory, Springer. Chapter 5.","DOI":"10.1007\/978-1-4757-0887-5_5"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/S0009-2614(89)87372-5","article-title":"Alternative coupled-cluster ansatze II. The unitary coupled-cluster method","volume":"155","author":"Bartlett","year":"1989","journal-title":"Chem. Phys. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1142\/9789812819529_0006","article-title":"Unitary Group Based Coupled Cluster Methods and Calculation of Molecular Properties","volume":"Volume 3","author":"Bartlett","year":"1997","journal-title":"Recent Advances in Coupled-Cluster Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"044107","DOI":"10.1063\/1.5011033","article-title":"On the difference between variational and unitary coupled cluster theories","volume":"148","author":"Harsha","year":"2018","journal-title":"J. Chem. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"143002","DOI":"10.1103\/PhysRevLett.97.143002","article-title":"Anti-Hermitian Contracted Schr\u00f6dinger Equation: Direct Determination of the Two-Electron Reduced Density Matrices of Many-Electron Molecules","volume":"97","author":"Mazziotti","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4213","DOI":"10.1038\/ncomms5213","article-title":"A variational eigenvalue solver on a photonic quantum processor","volume":"5","author":"Peruzzo","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3007","DOI":"10.1038\/s41467-019-10988-2","article-title":"An adaptive variational algorithm for exact molecular simulations on a quantum computer","volume":"10","author":"Grimsley","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"244112","DOI":"10.1063\/1.5133059","article-title":"Exact parameterization of fermionic wave functions via unitary coupled cluster theory","volume":"151","author":"Evangelista","year":"2019","journal-title":"J. Chem. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2040049","DOI":"10.1142\/S0217984920400497","article-title":"Test of the unitary coupled-cluster variational quantum eigensolver for a simple strongly correlated condensed-matter system","volume":"34","author":"Xu","year":"2020","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1021\/acs.jctc.0c01052","article-title":"Quantum-Inspired Algorithm for the Factorized Form of Unitary Coupled Cluster Theory","volume":"17","author":"Chen","year":"2021","journal-title":"J. Chem. Theor. Comp."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2211","DOI":"10.1103\/PhysRevA.6.2211","article-title":"Atomic Coherent States in Quantum Optics","volume":"6","author":"Arecchi","year":"1972","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"234102","DOI":"10.1063\/1.3520564","article-title":"Benchmark studies of variational, unitary and extended coupled cluster methods","volume":"133","author":"Cooper","year":"2010","journal-title":"J. Chem. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"044106","DOI":"10.1063\/5.0074311","article-title":"Flexibility of the factorized form of the unitary coupled cluster ansatz","volume":"156","author":"Chen","year":"2022","journal-title":"J. Chem. Phys."},{"key":"ref_15","unstructured":"Jacobson, N. (1962). Lie Algebras, Interscience Publishers."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Galindo, A., and Pascual, P. (1990). Quantum Mechanics I, Springer.","DOI":"10.1007\/978-3-642-83854-5"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"12980","DOI":"10.1039\/D0CP01707H","article-title":"On the order problem in construction of unitary operators for the variational quantum eigensolver","volume":"22","author":"Izmaylov","year":"2020","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1119\/1.17149","article-title":"Finite-dimensional matrix representations as calculational tools in quantum optics","volume":"61","author":"Mufti","year":"1991","journal-title":"Am. J. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"204108","DOI":"10.1063\/1.4718704","article-title":"A sequential transformation approach to the internally contracted multireference coupled cluster method","volume":"126","author":"Evangelista","year":"2012","journal-title":"J. Chem. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/494\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:29:19Z","timestamp":1760135359000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/494"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,28]]},"references-count":19,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["sym14030494"],"URL":"https:\/\/doi.org\/10.3390\/sym14030494","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,28]]}}}