{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T16:50:09Z","timestamp":1769273409741,"version":"3.49.0"},"reference-count":29,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T00:00:00Z","timestamp":1645660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000015","name":"United States Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0019215"],"award-info":[{"award-number":["DE-SC0019215"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1955907"],"award-info":[{"award-number":["1955907"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We explore how to build quantum circuits that compute the lowest energy state corresponding to a given Hamiltonian within a symmetry subspace by explicitly encoding it into the circuit. We create an explicit unitary and a variationally trained unitary that maps any vector output by ansatz A(\u03b1\u2192) from a defined subspace to a vector in the symmetry space. The parameters are trained varitionally to minimize the energy, thus keeping the output within the labelled symmetry value. The method was tested for a spin XXZ Hamiltonian using rotation and reflection symmetry and H2 Hamiltonian within Sz=0 subspace using S2 symmetry. We have found the variationally trained unitary gives good results with very low depth circuits and can thus be used to prepare symmetry states within near term quantum computers.<\/jats:p>","DOI":"10.3390\/sym14030457","type":"journal-article","created":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T21:11:52Z","timestamp":1645737112000},"page":"457","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Variational Quantum Circuits to Prepare Low Energy Symmetry States"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4966-097X","authenticated-orcid":false,"given":"Raja","family":"Selvarajan","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manas","family":"Sajjan","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0574-5346","authenticated-orcid":false,"given":"Sabre","family":"Kais","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA"},{"name":"Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA"},{"name":"Purdue Quantum Science and Engineering Institute, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5162","DOI":"10.1103\/PhysRevLett.83.5162","article-title":"Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors","volume":"83","author":"Abrams","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","unstructured":"Nielsen, M., Chuang, I., and Chuang, I. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"79","DOI":"10.22331\/q-2018-08-06-79","article-title":"Quantum Computing in the NISQ era and beyond","volume":"2","author":"Preskill","year":"2018","journal-title":"Quantum"},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"20835","DOI":"10.1038\/s41598-021-00339-x","article-title":"Prime factorization using quantum variational imaginary time evolution","volume":"11","author":"Selvarajan","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_6","first-page":"031007","article-title":"Scalable Quantum Simulation of Molecular Energies","volume":"6","author":"Babbush","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1021\/acs.jctc.8b01004","article-title":"Generalized Unitary Coupled Cluster Wave functions for Quantum Computation","volume":"15","author":"Lee","year":"2018","journal-title":"J. Chem. Theory Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"495","DOI":"10.22331\/q-2021-07-05-495","article-title":"Nearly tight Trotterization of interacting electrons","volume":"5","author":"Su","year":"2021","journal-title":"Quantum"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Xia, R., and Kais, S. (2018). Quantum Machine Learning for Electronic Structure Calculations. Nat. Commun., 9.","DOI":"10.1038\/s41467-018-06598-z"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1038\/nature23879","article-title":"Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets","volume":"549","author":"Kandala","year":"2017","journal-title":"Nature"},{"key":"ref_11","unstructured":"Ryabinkin, I.G., Genin, S.N., and Izmaylov, A.F. (2021, December 22). Constrained Variational Quantum Eigensolver: Quantum Computer Search Engine in the Fock Space, Available online: http:\/\/xxx.lanl.gov\/abs\/1806.00461."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"022322","DOI":"10.1103\/PhysRevA.98.022322","article-title":"Quantum algorithms for electronic structure calculations: Particle\/hole Hamiltonian and optimized wavefunction expansions","volume":"98","author":"Barkoutsos","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Gard, B.T., Zhu, L., Barron, G.S., Mayhall, N.J., Economou, S.E., and Barnes, E. (2020). Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm. npj Quantum Inf., 6.","DOI":"10.1038\/s41534-019-0240-1"},{"key":"ref_14","unstructured":"Nam, Y., Chen, J.S., Pisenti, N.C., Wright, K., Delaney, C., Maslov, D., Brown, K.R., Allen, S., Amini, J.M., and Apisdorf, J. (2021, December 22). Ground-State Energy Estimation of the Water Molecule on a Trapped Ion Quantum Computer, Available online: http:\/\/xxx.lanl.gov\/abs\/1902.10171."},{"key":"ref_15","first-page":"011021","article-title":"Computation of Molecular Spectra on a Quantum Processor with an Error-Resilient Algorithm","volume":"8","author":"Colless","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_16","unstructured":"McCaskey, A.J., Parks, Z.P., Jakowski, J., Moore, S.V., Morris, T., Humble, T.S., and Pooser, R.C. (2021, December 22). Quantum Chemistry as a Benchmark for Near-Term Quantum Computers, Available online: http:\/\/xxx.lanl.gov\/abs\/1905.01534."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Shen, Y., Zhang, X., Zhang, S., Zhang, J.N., Yung, M.H., and Kim, K. (2017). Quantum implementation of the unitary coupled cluster for simulating molecular electronic structure. Phys. Rev. A, 95.","DOI":"10.1103\/PhysRevA.95.020501"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"013197","DOI":"10.1103\/PhysRevResearch.3.013197","article-title":"Penalty methods for a variational quantum eigensolver","volume":"3","author":"Kuroiwa","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Sajjan, M., Sureshbabu, S.H., and Kais, S. (2021). Quantum Machine-Learning for Eigenstate Filtration in Two-Dimensional Materials. J. Am. Chem. Soc., 143.","DOI":"10.1021\/jacs.1c06246"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., and Killoran, N. (2019). Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99.","DOI":"10.1103\/PhysRevA.99.032331"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Mitarai, K., Negoro, M., Kitagawa, M., and Fujii, K. (2018). Quantum circuit learning. Phys. Rev. A, 98.","DOI":"10.1103\/PhysRevA.98.032309"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"013006","DOI":"10.1103\/PhysRevResearch.1.013006","article-title":"Methodology for replacing indirect measurements with direct measurements","volume":"1","author":"Mitarai","year":"2019","journal-title":"Phys. Rev. Res."},{"key":"ref_23","unstructured":"Aleksandrowicz, G., Alexander, T., Barkoutsos, P., Bello, L., Ben-Haim, Y., Bucher, D., Cabrera-Hern\u00e1ndez, F.J., Carballo-Franquis, J., Chen, A., and Chen, C.-F. (2019). Qiskit: An Open-Source Framework for Quantum Computing. Zenodo."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1119\/1.4798343","article-title":"An introduction to the spectrum, symmetries, and dynamics of spin-1\/2 Heisenberg chains","volume":"81","author":"Joel","year":"2013","journal-title":"Am. J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Iten, R., Colbeck, R., Kukuljan, I., Home, J., and Christandl, M. (2016). Quantum circuits for isometries. Phys. Rev. A, 93.","DOI":"10.1103\/PhysRevA.93.032318"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zhang, F., Gomes, N., Berthusen, N.F., Orth, P.P., Wang, C.Z., Ho, K.M., and Yao, Y.X. (2021). Shallow-circuit variational quantum eigensolver based on symmetry-inspired Hilbert space partitioning for quantum chemical calculations. Phys. Rev. Res., 3.","DOI":"10.1103\/PhysRevResearch.3.013039"},{"key":"ref_27","unstructured":"Nakajima, Y., Kawano, Y., and Sekigawa, H. (2021, December 22). A New Algorithm for Producing Quantum Circuits Using KAK Decompositions, Available online: http:\/\/xxx.lanl.gov\/abs\/quant-ph\/0509196."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Larocca, M., Czarnik, P., Sharma, K., Muraleedharan, G., Coles, P.J., and Cerezo, M. (2021, December 22). Diagnosing Barren Plateaus with Tools from Quantum Optimal Control, Available online: http:\/\/xxx.lanl.gov\/abs\/2105.14377.","DOI":"10.22331\/q-2022-09-29-824"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1791","DOI":"10.1038\/s41467-021-21728-w","article-title":"Cost function dependent barren plateaus in shallow parametrized quantum circuits","volume":"12","author":"Cerezo","year":"2021","journal-title":"Nat. Commun."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/457\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:26:16Z","timestamp":1760135176000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/3\/457"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,24]]},"references-count":29,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["sym14030457"],"URL":"https:\/\/doi.org\/10.3390\/sym14030457","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,24]]}}}