{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T19:38:37Z","timestamp":1769110717658,"version":"3.49.0"},"reference-count":92,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,20]],"date-time":"2023-02-20T00:00:00Z","timestamp":1676851200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["19-71-10091"],"award-info":[{"award-number":["19-71-10091"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["K1-2022-027"],"award-info":[{"award-number":["K1-2022-027"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National University of Science and Technology","award":["19-71-10091"],"award-info":[{"award-number":["19-71-10091"]}]},{"name":"National University of Science and Technology","award":["K1-2022-027"],"award-info":[{"award-number":["K1-2022-027"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Qubits, which are the quantum counterparts of classical bits, are used as basic information units for quantum information processing, whereas underlying physical information carriers, e.g., (artificial) atoms or ions, admit encoding of more complex multilevel states\u2014qudits. Recently, significant attention has been paid to the idea of using qudit encoding as a way for further scaling quantum processors. In this work, we present an efficient decomposition of the generalized Toffoli gate on five-level quantum systems\u2014so-called ququints\u2014that use ququints\u2019 space as the space of two qubits with a joint ancillary state. The basic two-qubit operation we use is a version of the controlled-phase gate. The proposed N-qubit Toffoli gate decomposition has O(N) asymptotic depth and does not use ancillary qubits. We then apply our results for Grover\u2019s algorithm, where we indicate on the sizable advantage of using the qudit-based approach with the proposed decomposition in comparison to the standard qubit case. We expect that our results are applicable for quantum processors based on various physical platforms, such as trapped ions, neutral atoms, protonic systems, superconducting circuits, and others.<\/jats:p>","DOI":"10.3390\/e25020387","type":"journal-article","created":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T02:07:44Z","timestamp":1676945264000},"page":"387","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover\u2019s Algorithm with Qudits"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8321-7103","authenticated-orcid":false,"given":"Anstasiia S.","family":"Nikolaeva","sequence":"first","affiliation":[{"name":"Russian Quantum Center, Skolkovo, Moscow 121205, Russia"},{"name":"National University of Science and Technology \u201cMISIS\u201d, Moscow 119049, Russia"}]},{"given":"Evgeniy O.","family":"Kiktenko","sequence":"additional","affiliation":[{"name":"Russian Quantum Center, Skolkovo, Moscow 121205, Russia"},{"name":"National University of Science and Technology \u201cMISIS\u201d, Moscow 119049, Russia"}]},{"given":"Aleksey K.","family":"Fedorov","sequence":"additional","affiliation":[{"name":"Russian Quantum Center, Skolkovo, Moscow 121205, Russia"},{"name":"National University of Science and Technology \u201cMISIS\u201d, Moscow 119049, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,20]]},"reference":[{"key":"ref_1","unstructured":"Manin, Y.I. (1980). The Computable and the Non-Computable. (Vychislimoe i Nevychislimoe), Sovetskoe Radio. (In Russian)."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1007\/BF02650179","article-title":"Simulating physics with computers","volume":"21","author":"Feynman","year":"1982","journal-title":"Int. J. Theor. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1007\/BF01886518","article-title":"Quantum mechanical computers","volume":"16","author":"Feynman","year":"1986","journal-title":"Found. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1098\/rspa.1985.0070","article-title":"Quantum theory, the Church-Turing principle and the universal quantum computer","volume":"400","author":"Deutsch","year":"1985","journal-title":"Proc. R. Soc. Lond. A. Math. Phys. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"11032","DOI":"10.1073\/pnas.95.19.11032","article-title":"Quantum computing","volume":"95","author":"Brassard","year":"1998","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","unstructured":"Preskill, J. (2012). Quantum computing and the entanglement frontier. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1126\/science.273.5278.1073","article-title":"Universal Quantum Simulators","volume":"273","author":"Lloyd","year":"1996","journal-title":"Science"},{"key":"ref_8","unstructured":"Shor, P. (1994, January 20\u201322). Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, USA."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"030503","DOI":"10.1088\/2058-9565\/aab822","article-title":"Quantum optimization using variational algorithms on near-term quantum devices","volume":"3","author":"Moll","year":"2018","journal-title":"Quantum Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1038\/s41586-019-1666-5","article-title":"Quantum supremacy using a programmable superconducting processor","volume":"574","author":"Arute","year":"2019","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"180501","DOI":"10.1103\/PhysRevLett.127.180501","article-title":"Strong Quantum Computational Advantage Using a Superconducting Quantum Processor","volume":"127","author":"Wu","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1038\/s41586-021-04273-w","article-title":"Quantum logic with spin qubits crossing the surface code threshold","volume":"601","author":"Xue","year":"2022","journal-title":"Nature"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1038\/s41586-021-04292-7","article-title":"Precision tomography of a three-qubit donor quantum processor in silicon","volume":"601","author":"Madzik","year":"2022","journal-title":"Nature"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1038\/s41586-021-04182-y","article-title":"Fast universal quantum gate above the fault-tolerance threshold in silicon","volume":"601","author":"Noiri","year":"2022","journal-title":"Nature"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1460","DOI":"10.1126\/science.abe8770","article-title":"Quantum computational advantage using photons","volume":"370","author":"Zhong","year":"2020","journal-title":"Science"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1038\/s41586-022-04725-x","article-title":"Quantum computational advantage with a programmable photonic processor","volume":"606","author":"Madsen","year":"2022","journal-title":"Nature"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1038\/s41586-021-03582-4","article-title":"Quantum phases of matter on a 256-atom programmable quantum simulator","volume":"595","author":"Ebadi","year":"2021","journal-title":"Nature"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1038\/s41586-021-03585-1","article-title":"Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms","volume":"595","author":"Scholl","year":"2021","journal-title":"Nature"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"327","DOI":"10.22331\/q-2020-09-21-327","article-title":"Quantum computing with neutral atoms","volume":"4","author":"Henriet","year":"2020","journal-title":"Quantum"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1038\/s41586-022-04603-6","article-title":"Multi-qubit entanglement and algorithms on a neutral-atom quantum computer","volume":"604","author":"Graham","year":"2022","journal-title":"Nature"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1038\/nature24654","article-title":"Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator","volume":"551","author":"Zhang","year":"2017","journal-title":"Nature"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1038\/nphys2252","article-title":"Quantum simulations with trapped ions","volume":"8","author":"Blatt","year":"2012","journal-title":"Nat. Phys."},{"key":"ref_23","first-page":"031022","article-title":"Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator","volume":"8","author":"Hempel","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_24","unstructured":"Fedorov, A.K., Gisin, N., Beloussov, S.M., and Lvovsky, A.I. (2022). Quantum computing at the quantum advantage threshold: A down-to-business review. arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1126\/science.abo6587","article-title":"Quantum optimization of maximum independent set using Rydberg atom arrays","volume":"376","author":"Ebadi","year":"2022","journal-title":"Science"},{"key":"ref_26","first-page":"041041","article-title":"High-Coherence Fluxonium Qubit","volume":"9","author":"Nguyen","year":"2019","journal-title":"Phys. Rev. X"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"010502","DOI":"10.1103\/PhysRevLett.129.010502","article-title":"Fluxonium: An Alternative Qubit Platform for High-Fidelity Operations","volume":"129","author":"Bao","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1038\/s41534-022-00644-x","article-title":"High fidelity two-qubit gates on fluxoniums using a tunable coupler","volume":"8","author":"Moskalenko","year":"2022","journal-title":"Npj Quantum Inf."},{"key":"ref_29","unstructured":"Lloyd, S. (2018). Quantum approximate optimization is computationally universal. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1126\/science.261.5128.1569","article-title":"A Potentially Realizable Quantum Computer","volume":"261","author":"Lloyd","year":"1993","journal-title":"Science"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2403","DOI":"10.1103\/PhysRevA.57.2403","article-title":"Analog analogue of a digital quantum computation","volume":"57","author":"Farhi","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1134\/1.568130","article-title":"Multiqubit spin","volume":"70","author":"Ermakov","year":"1999","journal-title":"J. Exp. Theor. Phys. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1134\/1.568340","article-title":"Physical implementation of three-qubit gates on a separate quantum particle","volume":"71","author":"Ermakov","year":"2000","journal-title":"J. Exp. Theor. Phys. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"062322","DOI":"10.1103\/PhysRevA.66.062322","article-title":"Implementation schemes in NMR of quantum processors and the Deutsch-Jozsa algorithm by using virtual spin representation","volume":"66","author":"Kessel","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"052309","DOI":"10.1103\/PhysRevA.62.052309","article-title":"Multivalued logic gates for quantum computation","volume":"62","author":"Muthukrishnan","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"022317","DOI":"10.1103\/PhysRevA.66.022317","article-title":"Universal simulation of Hamiltonian dynamics for quantum systems with finite-dimensional state spaces","volume":"66","author":"Nielsen","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"042323","DOI":"10.1103\/PhysRevA.67.042323","article-title":"Entangling power and operator entanglement in qudit systems","volume":"67","author":"Wang","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"062313","DOI":"10.1103\/PhysRevA.67.062313","article-title":"Qutrit quantum computer with trapped ions","volume":"67","author":"Klimov","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"014303","DOI":"10.1103\/PhysRevA.67.014303","article-title":"Minimal measurements of the gate fidelity of a qudit map","volume":"67","author":"Bagan","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_40","first-page":"29","article-title":"Algebra of quantum computations with higher dimensional systems","volume":"Volume 5128","author":"Ozhigov","year":"2003","journal-title":"First International Symposium on Quantum Informatics"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1134\/S0021364008060143","article-title":"Implementation of the quantum order-finding algorithm on two qudits","volume":"87","author":"Zobov","year":"2008","journal-title":"JETP Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1134\/S0021364009050105","article-title":"Adder on ternary base elements for a quantum computer","volume":"89","author":"Zobov","year":"2009","journal-title":"JETP Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11128-022-03482-0","article-title":"Clustering by quantum annealing on the three-level quantum elements qutrits","volume":"21","author":"Zobov","year":"2022","journal-title":"Quantum Inf. Process."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"097901","DOI":"10.1103\/PhysRevLett.92.097901","article-title":"Maximizing the Hilbert Space for a Finite Number of Distinguishable Quantum States","volume":"92","author":"Greentree","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"032334","DOI":"10.1103\/PhysRevA.74.032334","article-title":"Parallelism for quantum computation with qudits","volume":"74","author":"Brennen","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"022313","DOI":"10.1103\/PhysRevA.75.022313","article-title":"Efficient Toffoli gates using qudits","volume":"75","author":"Ralph","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"060504","DOI":"10.1103\/PhysRevLett.100.060504","article-title":"Manipulating Biphotonic Qutrits","volume":"100","author":"Lanyon","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"012312","DOI":"10.1103\/PhysRevA.80.012312","article-title":"Generalized Toffoli gates using qudit catalysis","volume":"80","author":"Ionicioiu","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"062321","DOI":"10.1103\/PhysRevA.85.062321","article-title":"Time-efficient implementation of quantum search with qudits","volume":"85","author":"Ivanov","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2594","DOI":"10.1038\/srep02594","article-title":"Geometry of Quantum Computation with Qutrits","volume":"3","author":"Li","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"042312","DOI":"10.1103\/PhysRevA.91.042312","article-title":"Multilevel superconducting circuits as two-qubit systems: Operations, state preparation, and entropic inequalities","volume":"91","author":"Kiktenko","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.1016\/j.physleta.2015.03.023","article-title":"Single qudit realization of the Deutsch algorithm using superconducting many-level quantum circuits","volume":"379","author":"Kiktenko","year":"2015","journal-title":"Phys. Lett. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"062321","DOI":"10.1103\/PhysRevA.93.062321","article-title":"Generation of tree-type three-dimensional entangled states via adiabatic passage","volume":"93","author":"Song","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"012016","DOI":"10.1088\/1742-6596\/804\/1\/012016","article-title":"Determining quantum correlations in bipartite systems\u2014From qubit to qutrit and beyond","volume":"804","author":"Frydryszak","year":"2017","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"012306","DOI":"10.1103\/PhysRevA.96.012306","article-title":"Factoring with qutrits: Shor\u2019s algorithm on ternary and metaplectic quantum architectures","volume":"96","author":"Bocharov","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Gokhale, P., Baker, J.M., Duckering, C., Brown, N.C., Brown, K.R., and Chong, F.T. (2019, January 22\u201326). Asymptotic Improvements to Quantum Circuits via Qutrits. Proceedings of the 46th International Symposium on Computer Architecture (ISCA \u201919), Phoenix, AZ, USA.","DOI":"10.1145\/3307650.3322253"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"070505","DOI":"10.1103\/PhysRevLett.123.070505","article-title":"Quantum Teleportation in High Dimensions","volume":"123","author":"Luo","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"033128","DOI":"10.1103\/PhysRevResearch.2.033128","article-title":"Practical trapped-ion protocols for universal qudit-based quantum computing","volume":"2","author":"Low","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"10117","DOI":"10.1364\/OE.419568","article-title":"Dissipative preparation of qutrit entanglement via periodically modulated Rydberg double antiblockade","volume":"29","author":"Jin","year":"2021","journal-title":"Opt. Express"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1126\/science.1173440","article-title":"Emulation of a Quantum Spin with a Superconducting Phase Qudit","volume":"325","author":"Neeley","year":"2009","journal-title":"Science"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1038\/nphys1150","article-title":"Simplifying quantum logic using higher-dimensional Hilbert spaces","volume":"5","author":"Lanyon","year":"2009","journal-title":"Nat. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1038\/nature10713","article-title":"Implementation of a Toffoli gate with superconducting circuits","volume":"481","author":"Fedorov","year":"2012","journal-title":"Nature"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"022302","DOI":"10.1103\/PhysRevA.85.022302","article-title":"Control of inhomogeneous atomic ensembles of hyperfine qudits","volume":"85","author":"Mischuck","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"010501","DOI":"10.1103\/PhysRevLett.114.010501","article-title":"Coherence and Decay of Higher Energy Levels of a Superconducting Transmon Qubit","volume":"114","author":"Peterer","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"5617","DOI":"10.1038\/ncomms6617","article-title":"Hidden two-qubit dynamics of a four-level Josephson circuit","volume":"5","author":"Svetitsky","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"054523","DOI":"10.1103\/PhysRevB.91.054523","article-title":"Multiphoton dressing of an anharmonic superconducting many-level quantum circuit","volume":"91","author":"Cramer","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1038\/nature22986","article-title":"On-chip generation of high-dimensional entangled quantum states and their coherent control","volume":"546","author":"Kues","year":"2017","journal-title":"Nature"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"187702","DOI":"10.1103\/PhysRevLett.119.187702","article-title":"Operating Quantum States in Single Magnetic Molecules: Implementation of Grover\u2019s Quantum Algorithm","volume":"119","author":"Godfrin","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"013027","DOI":"10.1088\/1367-2630\/ab60f4","article-title":"Ultracold polar molecules as qudits","volume":"22","author":"Sawant","year":"2020","journal-title":"New J. Phys."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"032417","DOI":"10.1103\/PhysRevA.103.032417","article-title":"Quantum-Fourier-transform-based quantum arithmetic with qudits","volume":"103","author":"Pavlidis","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_71","unstructured":"Rambow, P., and Tian, M. (2021). Reduction of circuit depth by mapping qubit-based quantum gates to a qudit basis. arXiv."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1038\/s41467-022-28767-x","article-title":"A programmable qudit-based quantum processor","volume":"13","author":"Chi","year":"2022","journal-title":"Nat. Commun."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"032621","DOI":"10.1103\/PhysRevA.105.032621","article-title":"Decomposing the generalized Toffoli gate with qutrits","volume":"105","author":"Nikolaeva","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"127902","DOI":"10.1103\/PhysRevLett.88.127902","article-title":"Security of Quantum Key Distribution Using d-Level Systems","volume":"88","author":"Cerf","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"033033","DOI":"10.1088\/1367-2630\/17\/3\/033033","article-title":"High-dimensional quantum cryptography with twisted light","volume":"17","author":"Mirhosseini","year":"2015","journal-title":"New J. Phys."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"010101","DOI":"10.1103\/PRXQuantum.3.010101","article-title":"Five Open Problems in Quantum Information Theory","volume":"3","author":"Horodecki","year":"2022","journal-title":"PRX Quantum"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"4714","DOI":"10.1103\/PhysRevLett.75.4714","article-title":"Demonstration of a Fundamental Quantum Logic Gate","volume":"75","author":"Monroe","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1038\/s41567-022-01658-0","article-title":"A universal qudit quantum processor with trapped ions","volume":"18","author":"Ringbauer","year":"2022","journal-title":"Nat. Phys."},{"key":"ref_79","unstructured":"Aksenov, M.A., Zalivako, I.V., Semerikov, I.A., Borisenko, A.S., Semenin, N.V., Sidorov, P.L., Fedorov, A.K., Khabarova, K.Y., and Kolachevsky, N.N. (2022). Realizing quantum gates with optically-addressable 171Yb+ ion qudits. arXiv."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"7481","DOI":"10.1038\/s41467-022-34851-z","article-title":"High-fidelity qutrit entangling gates for superconducting circuits","volume":"13","author":"Goss","year":"2022","journal-title":"Nat. Commun."},{"key":"ref_81","unstructured":"Hill, A.D., Hodson, M.J., Didier, N., and Reagor, M.J. (2021). Realization of arbitrary doubly-controlled quantum phase gates. arXiv."},{"key":"ref_82","unstructured":"Roy, T., Li, Z., Kapit, E., and Schuster, D.I. (2022). Realization of two-qutrit quantum algorithms on a programmable superconducting processor. arXiv."},{"key":"ref_83","unstructured":"Fang, C., Wang, Y., Sun, K., and Kim, J. (2023). Realization of Scalable Cirac-Zoller Multi-Qubit Gates. arXiv."},{"key":"ref_84","unstructured":"Nikolaeva, A.S., Kiktenko, E.O., and Fedorov, A.K. (2023). Compiling quantum circuits with qubits embedded in trapped-ion quidts. arXiv."},{"key":"ref_85","unstructured":"Nikolaeva, A.S., Kiktenko, E.O., and Fedorov, A.K. (2021). Efficient realization of quantum algorithms with qudits. arXiv."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"3457","DOI":"10.1103\/PhysRevA.52.3457","article-title":"Elementary gates for quantum computation","volume":"52","author":"Barenco","year":"1995","journal-title":"Phys. Rev. A"},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Grover, L.K. (1996, January 22\u201324). A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (STOC \u201996), Philadelphia, PA, USA.","DOI":"10.1145\/237814.237866"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1103\/PhysRevLett.79.325","article-title":"Quantum Mechanics Helps in Searching for a Needle in a Haystack","volume":"79","author":"Grover","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_89","doi-asserted-by":"crossref","unstructured":"Mato, K., Ringbauer, M., Hillmich, S., and Wille, R. (2022, January 18\u201323). Adaptive Compilation of Multi-Level Quantum Operations. Proceedings of the 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), Broomfield, CO, USA.","DOI":"10.1109\/QCE53715.2022.00070"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"160501","DOI":"10.1103\/PhysRevLett.129.160501","article-title":"Hardware Efficient Quantum Simulation of Non-Abelian Gauge Theories with Qudits on Rydberg Platforms","volume":"129","author":"Zache","year":"2022","journal-title":"Phys. Rev. Lett."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"4091","DOI":"10.1103\/PhysRevLett.74.4091","article-title":"Quantum computations with cold trapped ions","volume":"74","author":"Cirac","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"022304","DOI":"10.1103\/PhysRevA.101.022304","article-title":"Scalable quantum computing with qudits on a graph","volume":"101","author":"Kiktenko","year":"2020","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/2\/387\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:37:24Z","timestamp":1760121444000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/2\/387"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,20]]},"references-count":92,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["e25020387"],"URL":"https:\/\/doi.org\/10.3390\/e25020387","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,20]]}}}