{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T15:08:11Z","timestamp":1778598491769,"version":"3.51.4"},"reference-count":52,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T00:00:00Z","timestamp":1778544000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The Max Planck Society and the Federal Ministry of Research, Technology and Space","award":["13N15889"],"award-info":[{"award-number":["13N15889"]}]},{"name":"The Max Planck Society and the Federal Ministry of Research, Technology and Space","award":["13N15890"],"award-info":[{"award-number":["13N15890"]}]},{"name":"Germany\u2019s Excellence Strategy","award":["EXC-2111- 390814868"],"award-info":[{"award-number":["EXC-2111- 390814868"]}]},{"name":"Federal Ministry of Research, Technology and Space (BMFTR) \u2013 Quantum Systems","award":["13N17144"],"award-info":[{"award-number":["13N17144"]}]},{"name":"European Union \u2013 Horizon Europe Programme","award":["101113690"],"award-info":[{"award-number":["101113690"]}]},{"name":"European Research Council (ERC) \u2013 European Union\u2019s Horizon","award":["101165353"],"award-info":[{"award-number":["101165353"]}]},{"name":"European Research Council (ERC) \u2013 European Union\u2019s Horizon","award":["948240"],"award-info":[{"award-number":["948240"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Fermionic quantum processors are a promising platform for quantum simulation of correlated fermionic matter. In this work, we study a hardware-efficient protocol for measuring complex expectation values of the time-evolution operator, commonly referred to as Loschmidt echoes, with fermions in an optical superlattice. We analyze the algorithm for the Fermi\u2013Hubbard model at half-filling as well as at finite doping. The method relies on global quench dynamics and short imaginary time evolution, the latter being realized by architecture-tailored pulse sequences starting from a product state of plaquettes. Our numerical results show that complex Loschmidt echoes can be efficiently obtained for large many-body states over a broad spectral range. This allows one to measure spectral properties of the Fermi\u2013Hubbard model, such as the local density of states, and paves the way for the study of finite-temperature properties in current fermionic quantum simulators.<\/jats:p>","DOI":"10.22331\/q-2026-05-12-2103","type":"journal-article","created":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T14:23:29Z","timestamp":1778595809000},"page":"2103","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Phase-Sensitive Measurements on a Fermi\u2013Hubbard Quantum Processor"],"prefix":"10.22331","volume":"10","author":[{"given":"Alberto R.","family":"Cavallar","sequence":"first","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\u00fcnchen, Germany"},{"name":"Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen, Theresienstr. 37, M\u00fcnchen D-80333, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luis","family":"Escalera-Moreno","sequence":"additional","affiliation":[{"name":"Hamburg University of Technology, Institute for Quantum Inspired and Quantum Optimization, Blohmstra\u00dfe 15, Hamburg, Germany"},{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\u00fcnchen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Titus","family":"Franz","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\u00fcnchen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Timon","family":"Hilker","sequence":"additional","affiliation":[{"name":"Department of Physics and SUPA, University of Strathclyde, Glasgow, G4 0NG, United Kingdom"},{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\u00fcnchen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J. Ignacio","family":"Cirac","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\u00fcnchen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Philipp M.","family":"Preiss","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\u00fcnchen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benjamin F.","family":"Schiffer","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\u00fcnchen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2026,5,12]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"P. A. Lee, N. Nagaosa, and X.-G. Wen. ``Doping a Mott insulator: Physics of high-temperature superconductivity''. Rev. Mod. Phys. 78, 17\u201385 (2006).","DOI":"10.1103\/RevModPhys.78.17"},{"key":"1","doi-asserted-by":"publisher","unstructured":"M. Qin, T. Sch\u00e4fer, S. Andergassen, P. Corboz, and E. Gull. ``The Hubbard Model: A Computational Perspective''. Annual Review of Condensed Matter Physics 13, 275\u2013302 (2022).","DOI":"10.1146\/annurev-conmatphys-090921-033948"},{"key":"2","doi-asserted-by":"publisher","unstructured":"W. S. Bakr, Z. Ba, and M. L. Prichard. ``Microscopy of Ultracold Fermions in Optical Lattices'' (2025). arXiv:2507.04042 [cond-mat.quant-gas].","DOI":"10.48550\/arXiv.2507.04042"},{"key":"3","doi-asserted-by":"publisher","unstructured":"L. Tarruell and L. Sanchez-Palencia. ``Quantum simulation of the Hubbard model with ultracold fermions in optical lattices''. Comptes Rendus. Physique 19, 365\u2013393 (2018).","DOI":"10.1016\/j.crhy.2018.10.013"},{"key":"4","doi-asserted-by":"publisher","unstructured":"V. Havl\u00ed\u010dek, M. Troyer, and J. D. Whitfield. ``Operator locality in the quantum simulation of fermionic models''. Physical Review A 95, 032332 (2017).","DOI":"10.1103\/PhysRevA.95.032332"},{"key":"5","doi-asserted-by":"publisher","unstructured":"A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu. ``Theory of Trotter Error with Commutator Scaling''. Physical Review X 11, 011020 (2021).","DOI":"10.1103\/PhysRevX.11.011020"},{"key":"6","doi-asserted-by":"publisher","unstructured":"A. Y. Kitaev. ``Quantum measurements and the Abelian Stabilizer Problem''. Electron. Colloquium Comput. Complex. TR96 (1995).","DOI":"10.48550\/arXiv.quant-ph\/9511026"},{"key":"7","doi-asserted-by":"publisher","unstructured":"M. A. Nielsen and I. L. Chuang. ``Quantum Computation and Quantum Information''. Cambridge University Press. (2010).","DOI":"10.1017\/cbo9780511976667"},{"key":"8","doi-asserted-by":"publisher","unstructured":"L. Lin and Y. Tong. ``Heisenberg-Limited Ground-State Energy Estimation for Early Fault-Tolerant Quantum Computers''. PRX Quantum 3, 010318 (2022).","DOI":"10.1103\/PRXQuantum.3.010318"},{"key":"9","doi-asserted-by":"publisher","unstructured":"N. S. Blunt, L. Caune, R. Izs\u00e1k, E. T. Campbell, and N. Holzmann. ``Statistical Phase Estimation and Error Mitigation on a Superconducting Quantum Processor''. PRX Quantum 4, 040341 (2023).","DOI":"10.1103\/PRXQuantum.4.040341"},{"key":"10","doi-asserted-by":"publisher","unstructured":"S. Lu, M. C. Ba\u00f1uls, and J. I. Cirac. ``Algorithms for Quantum Simulation at Finite Energies''. PRX Quantum 2, 020321 (2021).","DOI":"10.1103\/PRXQuantum.2.020321"},{"key":"11","doi-asserted-by":"publisher","unstructured":"R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca. ``Quantum algorithms revisited''. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 339\u2013354 (1998).","DOI":"10.1098\/rspa.1998.0164"},{"key":"12","doi-asserted-by":"publisher","unstructured":"T. E. O\u2019Brien, S. Polla, N. C. Rubin, W. J. Huggins, S. McArdle, S. Boixo, J. R. McClean, and R. Babbush. ``Error Mitigation via Verified Phase Estimation''. PRX Quantum 2, 020317 (2021).","DOI":"10.1103\/PRXQuantum.2.020317"},{"key":"13","doi-asserted-by":"publisher","unstructured":"K. H\u00e9mery, K. Ghanem, E. Crane, S. L. Campbell, J. M. Dreiling, C. Figgatt, C. Foltz, J. P. Gaebler, J. Johansen, M. Mills, S. A. Moses, J. M. Pino, A. Ransford, M. Rowe, P. Siegfried, et al. ``Measuring the Loschmidt Amplitude for Finite-Energy Properties of the Fermi-Hubbard Model on an Ion-Trap Quantum Computer''. PRX Quantum 5, 030323 (2024).","DOI":"10.1103\/PRXQuantum.5.030323"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Y. Yang, A. Christianen, M. C. Ba\u00f1uls, D. S. Wild, and J. I. Cirac. ``Phase-Sensitive Quantum Measurement without Controlled Operations''. Phys. Rev. Lett. 132, 220601 (2024).","DOI":"10.1103\/PhysRevLett.132.220601"},{"key":"15","doi-asserted-by":"publisher","unstructured":"L. Clinton, T. S. Cubitt, R. Garcia-Patron, A. Montanaro, S. Stanisic, and M. Stroeks. ``Quantum Phase Estimation Without Controlled Unitaries''. PRX Quantum 7, 010345 (2026).","DOI":"10.1103\/7qcr-znl2"},{"key":"16","doi-asserted-by":"publisher","unstructured":"B. F. Schiffer, D. S. Wild, N. Maskara, M. D. Lukin, and J. I. Cirac. ``Hardware-efficient quantum phase estimation via local control''. PRX Quantum 6, 040348 (2025).","DOI":"10.1103\/zlp8-rz9n"},{"key":"17","doi-asserted-by":"publisher","unstructured":"D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuleti\u0107, and M. D. Lukin. ``A quantum processor based on coherent transport of entangled atom arrays''. Nature 604, 451\u2013456 (2022).","DOI":"10.1038\/s41586-022-04592-6"},{"key":"18","doi-asserted-by":"publisher","unstructured":"D. K. Mark, H.-Y. Hu, J. Kwan, C. Kokail, S. Choi, and S. F. Yelin. ``Efficiently measuring $d$-wave pairing and beyond in quantum gas microscopes''. Physical Review Letters 135, 123402 (2025).","DOI":"10.1103\/dqyf-kl8x"},{"key":"19","doi-asserted-by":"publisher","unstructured":"C. Tabares, C. Kokail, P. Zoller, D. Gonz\u00e1lez-Cuadra, and A. Gonz\u00e1lez-Tudela. ``Programming Optical-Lattice Fermi-Hubbard Quantum Simulators''. PRX Quantum 6, 030356 (2025).","DOI":"10.1103\/3nx4-bnyy"},{"key":"20","doi-asserted-by":"publisher","unstructured":"S. Karch, S. Bandyopadhyay, Z.-H. Sun, A. Impertro, S. Huh, I. Prieto Rodr\u00edguez, J. F. Wienand, W. Ketterle, M. Heyl, A. Polkovnikov, I. Bloch, and M. Aidelsburger. ``Probing quantum many-body dynamics using subsystem Loschmidt echos'' (2025). arXiv:2501.16995 [cond-mat.quant-gas].","DOI":"10.48550\/arXiv.2501.16995"},{"key":"21","doi-asserted-by":"publisher","unstructured":"F. Gkritsis, D. Dux, J. Zhang, N. Jain, C. Gogolin, and P. M. Preiss. ``Simulating Chemistry with Fermionic Optical Superlattices''. PRX Quantum 6, 010318 (2025).","DOI":"10.1103\/PRXQuantum.6.010318"},{"key":"22","doi-asserted-by":"publisher","unstructured":"P. Bojovi\u0107, T. Hilker, S. Wang, et al. ``High-fidelity collisional quantum gates with fermionic atoms''. Nature 652, 602\u2013608 (2026).","DOI":"10.1038\/s41586-026-10356-3"},{"key":"23","doi-asserted-by":"publisher","unstructured":"S. Trebst, U. Schollw\u00f6ck, M. Troyer, and P. Zoller. ``$d$-wave resonating valence bond states of fermionic atoms in optical lattices''. Physical Review Letters 96, 250402 (2006).","DOI":"10.1103\/PhysRevLett.96.250402"},{"key":"24","doi-asserted-by":"publisher","unstructured":"A. M. Rey, R. Sensarma, S. F\u00f6lling, M. Greiner, E. Demler, and M. D. Lukin. ``Controlled preparation and detection of d-wave superfluidity in two-dimensional optical superlattices''. EPL (Europhysics Letters) 87, 60001 (2009).","DOI":"10.1209\/0295-5075\/87\/60001"},{"key":"25","doi-asserted-by":"publisher","unstructured":"S. Nascimb\u00e8ne, Y.-A. Chen, M. Atala, M. Aidelsburger, S. Trotzky, B. Paredes, and I. Bloch. ``Experimental Realization of Plaquette Resonating Valence-Bond States with Ultracold Atoms in Optical Superlattices''. Physical Review Letters 108, 205301 (2012).","DOI":"10.1103\/PhysRevLett.108.205301"},{"key":"26","doi-asserted-by":"publisher","unstructured":"P. Jordan and E. Wigner. ``\u00dcber das Paulische \u00c4quivalenzverbot''. Zeitschrift fur Physik 47, 631\u2013651 (1928).","DOI":"10.1007\/BF01331938"},{"key":"27","doi-asserted-by":"publisher","unstructured":"S. B. Bravyi and A. Y. Kitaev. ``Fermionic Quantum Computation''. Annals of Physics 298, 210\u2013226 (2002).","DOI":"10.1006\/aphy.2002.6254"},{"key":"28","doi-asserted-by":"publisher","unstructured":"F. Verstraete and J. I. Cirac. ``Mapping local Hamiltonians of fermions to local Hamiltonians of spins''. Journal of Statistical Mechanics: Theory and Experiment 2005, P09012\u2013P09012 (2005).","DOI":"10.1088\/1742-5468\/2005\/09\/p09012"},{"key":"29","doi-asserted-by":"publisher","unstructured":"J. Yu, Y. Liu, S. Sugiura, T. Van Voorhis, and S. Zeytino\u011flu. ``Clifford circuit-based heuristic optimization of fermion-to-qubit mappings''. Journal of Chemical Theory and Computation 21, 9430\u20139443 (2025).","DOI":"10.1021\/acs.jctc.5c00794"},{"key":"30","doi-asserted-by":"publisher","unstructured":"W.-Y. Zhang, M.-G. He, H. Sun, Y.-G. Zheng, Y. Liu, A. Luo, H.-Y. Wang, Z.-H. Zhu, P.-Y. Qiu, Y.-C. Shen, X.-K. Wang, W. Lin, S.-T. Yu, B.-C. Li, B. Xiao, M.-D. Li, Y.-M. Yang, X. Jiang, H.-N. Dai, Y. Zhou, X. Ma, Z.-S. Yuan, and J.-W. Pan. ``Scalable Multipartite Entanglement Created by Spin Exchange in an Optical Lattice''. Physical Review Letters 131, 073401 (2023).","DOI":"10.1103\/PhysRevLett.131.073401"},{"key":"31","doi-asserted-by":"publisher","unstructured":"M. Xu, L. H. Kendrick, A. Kale, Y. Gang, C. Feng, S. Zhang, A. W. Young, M. Lebrat, and M. Greiner. ``A neutral-atom Hubbard quantum simulator in the cryogenic regime''. Nature 642, 909\u2013915 (2025).","DOI":"10.1038\/s41586-025-09112-w"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Y. Kiefer, Z. Zhu, L. Fischer, S. Jele, M. G\u00e4chter, G. Bisson, K. Viebahn, and T. Esslinger. ``Protected quantum gates using qubit doublons in dynamical optical lattices''. Nature 652, 609\u2013614 (2026).","DOI":"10.1038\/s41586-026-10285-1"},{"key":"33","doi-asserted-by":"publisher","unstructured":"N. Klemmer, J. Fleper, V. Jonas, A. Sheikhan, C. Kollath, M. K\u00f6hl, and A. Bergschneider. ``Floquet-Driven Crossover from Density-Assisted Tunneling to Enhanced Pair Tunneling''. Physical Review Letters 133, 253402 (2024).","DOI":"10.1103\/PhysRevLett.133.253402"},{"key":"34","doi-asserted-by":"publisher","unstructured":"H. Zhang, Y.-K. Wang, Y. Zheng, H.-T. Bai, and B. Yang. ``Observation of Coherent Quantum Tunneling of a Massive Atomic Cluster with 435 u'' (2025). arXiv:2502.06246 [cond-mat.quant-gas].","DOI":"10.48550\/arXiv.2502.06246"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Z. Zhu, Y. Kiefer, S. Jele, M. G\u00e4chter, G. Bisson, K. Viebahn, and T. Esslinger. ``Splitting and Connecting Singlets in Atomic Quantum Circuits''. Physical Review X 15, 041032 (2025).","DOI":"10.1103\/xh3v-tky4"},{"key":"36","doi-asserted-by":"publisher","unstructured":"P. Corboz. ``Improved energy extrapolation with infinite projected entangled-pair states applied to the two-dimensional Hubbard model''. Physical Review B 93, 045116 (2016).","DOI":"10.1103\/PhysRevB.93.045116"},{"key":"37","doi-asserted-by":"publisher","unstructured":"E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. ``Quantum Computation by Adiabatic Evolution'' (2000). arXiv:quant-ph\/0001106.","DOI":"10.48550\/arXiv.quant-ph\/0001106"},{"key":"38","doi-asserted-by":"publisher","unstructured":"W. S. Bakr, J. I. Gillen, A. Peng, S. F\u00f6lling, and M. Greiner. ``A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice''. Nature 462, 74\u201377 (2009).","DOI":"10.1038\/nature08482"},{"key":"39","doi-asserted-by":"publisher","unstructured":"M. F. Parsons, F. Huber, A. Mazurenko, C. S. Chiu, W. Setiawan, K. Wooley-Brown, S. Blatt, and M. Greiner. ``Site-Resolved Imaging of Fermionic $^6$Li in an Optical Lattice''. Physical Review Letters 114, 213002 (2015).","DOI":"10.1103\/PhysRevLett.114.213002"},{"key":"40","doi-asserted-by":"publisher","unstructured":"L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe, and M. W. Zwierlein. ``Quantum-Gas Microscope for Fermionic Atoms''. Physical Review Letters 114, 193001 (2015).","DOI":"10.1103\/PhysRevLett.114.193001"},{"key":"41","doi-asserted-by":"publisher","unstructured":"N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr\u00fcggen, and S. J. Glaser. ``Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms''. Journal of Magnetic Resonance 172, 296\u2013305 (2005).","DOI":"10.1016\/j.jmr.2004.11.004"},{"key":"42","doi-asserted-by":"publisher","unstructured":"A. M. Childs and N. Wiebe. ``Hamiltonian simulation using linear combinations of unitary operations''. Quantum Info. Comput. 12, 901\u2013924 (2012).","DOI":"10.26421\/QIC12.11-12-1"},{"key":"43","doi-asserted-by":"publisher","unstructured":"R. Trivedi, A. F. Rubio, and J. I. Cirac. ``Quantum advantage and stability to errors in analogue quantum simulators''. Nature Communications 15, 6507 (2024).","DOI":"10.1038\/s41467-024-50750-x"},{"key":"44","unstructured":"A. R. Cavallar and B. F. Schiffer. ``ITE Ramp Optimization Code for Phase-Sensitive Measurements on a Fermi-Hubbard Quantum Processor'' (2025). GitLab repository, version v1.0-paper, https:\/\/gitlab.physik.uni-muenchen.de\/A.Cavallar\/ite-ramp-optimization.git."},{"key":"45","doi-asserted-by":"publisher","unstructured":"U. Schollw\u00f6ck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96\u2013192 (2011).","DOI":"10.1016\/j.aop.2010.09.012"},{"key":"46","doi-asserted-by":"publisher","unstructured":"M. Fishman, S. White, and E. Stoudenmire. ``The ITensor Software Library for Tensor Network Calculations''. SciPost Physics Codebases (2022).","DOI":"10.21468\/SciPostPhysCodeb.4"},{"key":"47","doi-asserted-by":"publisher","unstructured":"B. F. Schiffer, D. v. Vreumingen, J. Tura, and S. Polla. ``Virtual mitigation of coherent non-adiabatic transitions by echo verification''. Quantum 8, 1346 (2024).","DOI":"10.22331\/q-2024-05-14-1346"},{"key":"48","doi-asserted-by":"publisher","unstructured":"R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu. ``A Limited Memory Algorithm for Bound Constrained Optimization''. SIAM Journal on Scientific Computing 16, 1190\u20131208 (1995).","DOI":"10.1137\/0916069"},{"key":"49","doi-asserted-by":"publisher","unstructured":"C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal. ``Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization''. ACM Trans. Math. Softw. 23, 550\u2013560 (1997).","DOI":"10.1145\/279232.279236"},{"key":"50","doi-asserted-by":"publisher","unstructured":"P. Weinberg and M. Bukov. ``QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems. part I: spin chains''. SciPost Phys. 2, 003 (2017).","DOI":"10.21468\/SciPostPhys.2.1.003"},{"key":"51","doi-asserted-by":"publisher","unstructured":"P. Weinberg and M. Bukov. ``QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems. part II: bosons, fermions and higher spins''. SciPost Phys. 7, 020 (2019).","DOI":"10.21468\/SciPostPhys.7.2.020"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2026-05-12-2103\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T14:23:35Z","timestamp":1778595815000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2026-05-12-2103\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,12]]},"references-count":52,"URL":"https:\/\/doi.org\/10.22331\/q-2026-05-12-2103","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,12]]},"article-number":"2103"}}