{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T03:18:09Z","timestamp":1775186289907,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T00:00:00Z","timestamp":1772236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12475018"],"award-info":[{"award-number":["12475018"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["92365202"],"award-info":[{"award-number":["92365202"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12475011"],"award-info":[{"award-number":["12475011"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11921005"],"award-info":[{"award-number":["11921005"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"crossref","award":["2024YFA1409002"],"award-info":[{"award-number":["2024YFA1409002"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Shanghai Municipal Science and Technology Major Project","award":["2019SHZDZX01"],"award-info":[{"award-number":["2019SHZDZX01"]}]},{"name":"Innovation Program for Quantum Science and Technology","award":["2021ZD0302100"],"award-info":[{"award-number":["2021ZD0302100"]}]},{"name":"Shanghai Municipal Science and Technology Project","award":["25LZ2601100"],"award-info":[{"award-number":["25LZ2601100"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We analyze the power of the recently proposed Lorentz quantum computer (LQC), a theoretical model leveraging hyperbolic bits (hybits) governed by complex Lorentz transformations. We define the complexity class BLQP (bounded-error Lorentz quantum polynomial-time) and demonstrate its equivalence to the complexity class P\u266fP (the class of problems solvable by a deterministic polynomial-time Turing machine with access to a \u266fP oracle). LQC algorithms are shown to solve NP-hard problems, such as the maximum independent set (MIS), in polynomial time, thereby placing NP and co-NP within BLQP. Furthermore, we establish that LQC can efficiently simulate quantum computing with postselection (PostBQP), while the reverse is not possible, highlighting LQC\u2019s unique \u201csuper-postselection\u201d capability. By proving BLQP =P\u266fP, we situate the entire polynomial hierarchy (PH) within BLQP and reveal profound connections between computational complexity and physical frameworks like Lorentz quantum mechanics. These results underscore LQC\u2019s theoretical superiority over conventional quantum computing models and its potential to redefine boundaries in complexity theory.<\/jats:p>","DOI":"10.3390\/e28030266","type":"journal-article","created":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T14:06:56Z","timestamp":1772460416000},"page":"266","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Power of the Lorentz Quantum Computer"],"prefix":"10.3390","volume":"28","author":[{"given":"Qi","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Science, Liaoning Petrochemical University, Fushun 113001, China"},{"name":"Liaoning Provincial Key Laboratory of Novel Micro-Nano Functional Materials, Fushun 113001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9229-5894","authenticated-orcid":false,"given":"Biao","family":"Wu","sequence":"additional","affiliation":[{"name":"International Center for Quantum Materials, Peking University, Beijing 100871, China"},{"name":"Wilczek Quantum Center, Shanghai Institute for Advanced Studies, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Hefei National Laboratory, Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Arora, S., and Barak, B. (2009). Computational Complexity: A Modern Approach, Cambridge University Press.","DOI":"10.1017\/CBO9780511804090"},{"key":"ref_2","unstructured":"Papadimitriou, C.H. (1994). Computational Complexity, Addison-Wesley."},{"key":"ref_3","unstructured":"Knill, E. (2004). Fault-Tolerant Postselected Quantum Computation: Schemes. arXiv."},{"key":"ref_4","first-page":"3473","article-title":"Quantum computing, postselection, and probabilistic polynomial time","volume":"461","author":"Aaronson","year":"2005","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"040304","DOI":"10.1088\/1674-1056\/acad6a","article-title":"Lorentz quantum computer","volume":"32","author":"He","year":"2023","journal-title":"Chin. Phys. B"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1088\/1367-2630\/5\/1\/104","article-title":"Superfluidity of bose-einstein condensate in an optical lattice: Landau-zener tunnelling and dynamical instability","volume":"5","author":"Wu","year":"2003","journal-title":"New J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"013024","DOI":"10.1088\/1367-2630\/aa8496","article-title":"Lorentz quantum mechanics","volume":"20","author":"Zhang","year":"2018","journal-title":"New J. Phys."},{"key":"ref_8","unstructured":"Barch, B., and Lidar, D. (2025). Computational Complexity of Non-Hermitian Quantum Systems. arXiv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1098\/rspa.1942.0023","article-title":"The physical interpretation of quantum mechanics","volume":"180","author":"Dirac","year":"1942","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1103\/RevModPhys.15.175","article-title":"On Dirac\u2019s new method of field quantization","volume":"15","author":"Pauli","year":"1943","journal-title":"Rev. Mod. Phys."},{"key":"ref_11","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_12","unstructured":"Nielson, M.A., and Chuang, I.L. (2000). Quantum Computing and Quantum Information, Cambridge University Press."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bognar, J. (1974). Indefinite Inner Product Space, Springer.","DOI":"10.1007\/978-3-642-65567-8"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.ic.2017.06.001","article-title":"Exact algorithms for maximum independent sets","volume":"255","author":"Xiao","year":"2017","journal-title":"Inf. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"030304","DOI":"10.1088\/0256-307X\/38\/3\/030304","article-title":"Quantum algorithm for approximating maximum independent sets","volume":"38","author":"Yu","year":"2021","journal-title":"Chin. Phys. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"012318","DOI":"10.1103\/PhysRevA.101.012318","article-title":"Quantum independent-set problem and non-abelian adiabatic mixing","volume":"101","author":"Wu","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/0890-5401(91)90075-D","article-title":"On truth-table reducibility to sat","volume":"91","author":"Buss","year":"1991","journal-title":"Inf. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/0022-0000(89)90025-1","article-title":"The strong exponential hierarchy collapses","volume":"39","author":"Hemachandra","year":"1989","journal-title":"J. Comput. Syst. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1137\/0220053","article-title":"PP is as hard as the polynomial-time hierarchy","volume":"20","author":"Toda","year":"1991","journal-title":"SIAM J. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1137\/S0097539796300921","article-title":"Quantum complexity theory","volume":"26","author":"Bernstein","year":"1997","journal-title":"SIAM J. Comput."},{"key":"ref_21","unstructured":"(2024, April 02). Complexityzoo Homepage. Available online: http:\/\/www.complexityzoo.com."},{"key":"ref_22","unstructured":"Peres, A. (1995). Bell Inequalities with Postselection. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1073\/pnas.1522411112","article-title":"Quantum violation of the pigeonhole principle and the nature of quantum correlations","volume":"113","author":"Aharonov","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/266\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T05:19:41Z","timestamp":1772515181000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/266"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,28]]},"references-count":23,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["e28030266"],"URL":"https:\/\/doi.org\/10.3390\/e28030266","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,28]]}}}