Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

Giovanni Cataldi1,2,3, Ashkan Abedi4, Giuseppe Magnifico1,2,3, Simone Notarnicola1,2,3, Nicola Dalla Pozza4, Vittorio Giovannetti5, and Simone Montangero1,2,3

1Dipartimento di Fisica e Astronomia ``G. Galilei'', Università di Padova, I-35131 Padova, Italy.
2Padua Quantum Technologies Research Center, Università degli Studi di Padova.
3Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova, I-35131 Padova, Italy.
4Scuola Normale Superiore, I-56127 Pisa, Italy.
5NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy.

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We present a novel mapping for studying 2D many-body quantum systems by solving an effective, one-dimensional long-range model in place of the original two-dimensional short-range one. In particular, we address the problem of choosing an efficient mapping from the 2D lattice to a 1D chain that optimally preserves the locality of interactions within the TN structure. By using Matrix Product States (MPS) and Tree Tensor Network (TTN) algorithms, we compute the ground state of the 2D quantum Ising model in transverse field with lattice size up to $64\times64$, comparing the results obtained from different mappings based on two space-filling curves, the snake curve and the Hilbert curve. We show that the locality-preserving properties of the Hilbert curve leads to a clear improvement of numerical precision, especially for large sizes, and turns out to provide the best performances for the simulation of 2D lattice systems via 1D TN structures.

Simulating quantum many-body systems is a very challenging problem since the memory required to exactly describe the state of the system grows exponentially with the number of degrees of freedom. Therefore, it is important to search for approximate and efficient representations of the quantum states of these systems. In the last decades, tensor networks have demonstrated remarkable capabilities to tackle this problem by efficiently retaining in memory only the relevant features of the state of the system.
In our work, we focus on two-dimensional systems where the search to improve tensor network representations is still ongoing. In particular, a standard strategy is to map the two-dimensional problem onto an effective one-dimensional one and then analyze it via established one-dimensional tensor network algorithms. Here, we show that the choice of the mapping drastically affects the numerical precision. In particular, we focus on two tensor network geometries and compare the results obtained with two possible mappings. We find that the mapping based on the Hilbert space-filling curve, thanks to its locality-preserving properties, guarantees a better precision with respect to the more standard mapping based on the snake curve.

► BibTeX data

► References

[1] Subir Sachdev. Quantum Phase Transitions. Cambridge University Press, 2 edition, 2011. https:/​/​doi.org/​10.1017/​CBO9780511973765.

[2] B. Andrei Bernevig and Taylor L. Hughes. Topological Insulators and Topological Superconductors. Princeton University Press, stu - student edition edition, 2013. ISBN 9780691151755. URL http:/​/​www.jstor.org/​stable/​j.ctt19cc2gc.

[3] Steven M. Girvin and Kun Yang. Modern Condensed Matter Physics. Cambridge University Press, 2019. https:/​/​doi.org/​10.1017/​9781316480649.

[4] T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers. Quantum hall physics: Hierarchies and conformal field theory techniques. Rev. Mod. Phys., 89: 025005, May 2017. https:/​/​doi.org/​10.1103/​RevModPhys.89.025005. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.89.025005.

[5] Benjamin Sacépé, Mikhail Feigel'man, and Teunis M. Klapwijk. Quantum breakdown of superconductivity in low-dimensional materials. Nature Physics, 16 (7): 734–746, July 2020. https:/​/​doi.org/​10.1038/​s41567-020-0905-x.

[6] Stephan Rachel. Interacting topological insulators: a review. Reports on Progress in Physics, 81 (11): 116501, oct 2018. 10.1088/​1361-6633/​aad6a6. URL https:/​/​doi.org/​10.1088/​1361-6633/​aad6a6.

[7] Lucile Savary and Leon Balents. Quantum spin liquids: a review. Reports on Progress in Physics, 80 (1): 016502, nov 2016. https:/​/​doi.org/​10.1088/​0034-4885/​80/​1/​016502. URL https:/​/​doi.org/​10.1088/​0034-4885/​80/​1/​016502.

[8] Antonio Acín, Immanuel Bloch, Harry Buhrman, Tommaso Calarco, Christopher Eichler, Jens Eisert, Daniel Esteve, Nicolas Gisin, Steffen J Glaser, Fedor Jelezko, Stefan Kuhr, Maciej Lewenstein, Max F Riedel, Piet O Schmidt, Rob Thew, Andreas Wallraff, Ian Walmsley, and Frank K Wilhelm. The quantum technologies roadmap: a european community view. New Journal of Physics, 20 (8): 080201, aug 2018. https:/​/​doi.org/​10.1088/​1367-2630/​aad1ea. URL https:/​/​doi.org/​10.1088/​1367-2630/​aad1ea.

[9] Yuri Alexeev, Dave Bacon, Kenneth R. Brown, Robert Calderbank, Lincoln D. Carr, Frederic T. Chong, Brian DeMarco, Dirk Englund, Edward Farhi, Bill Fefferman, Alexey V. Gorshkov, Andrew Houck, Jungsang Kim, Shelby Kimmel, Michael Lange, Seth Lloyd, Mikhail D. Lukin, Dmitri Maslov, Peter Maunz, Christopher Monroe, John Preskill, Martin Roetteler, Martin J. Savage, and Jeff Thompson. Quantum computer systems for scientific discovery. PRX Quantum, 2: 017001, Feb 2021. https:/​/​doi.org/​10.1103/​PRXQuantum.2.017001. URL https:/​/​link.aps.org/​doi/​10.1103/​PRXQuantum.2.017001.

[10] I. M. Georgescu, S. Ashhab, and Franco Nori. Quantum simulation. Rev. Mod. Phys., 86: 153–185, Mar 2014. https:/​/​doi.org/​10.1103/​RevModPhys.86.153. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.86.153.

[11] D.J. Scalapino. Numerical Methods for Lattice Quantum Many-Body Problems. Addison-Wesley Longman, Incorporated, 1998. ISBN 9780201156928. URL https:/​/​books.google.it/​books?id=yzUIPAAACAAJ.

[12] H. Fehske, R. Schneider, and A. Weiße. Computational Many-Particle Physics, volume 739. 2008. https:/​/​doi.org/​10.1007/​978-3-540-74686-7.

[13] H T Diep. Frustrated Spin Systems. World Scientific, 2nd edition, 2013. https:/​/​doi.org/​10.1142/​8676. URL https:/​/​www.worldscientific.com/​doi/​abs/​10.1142/​8676.

[14] J. P. F. LeBlanc, Andrey E. Antipov, Federico Becca, Ireneusz W. Bulik, Garnet Kin-Lic Chan, Chia-Min Chung, Youjin Deng, Michel Ferrero, Thomas M. Henderson, Carlos A. Jiménez-Hoyos, E. Kozik, Xuan-Wen Liu, Andrew J. Millis, N. V. Prokof'ev, Mingpu Qin, Gustavo E. Scuseria, Hao Shi, B. V. Svistunov, Luca F. Tocchio, I. S. Tupitsyn, Steven R. White, Shiwei Zhang, Bo-Xiao Zheng, Zhenyue Zhu, and Emanuel Gull. Solutions of the two-dimensional hubbard model: Benchmarks and results from a wide range of numerical algorithms. Phys. Rev. X, 5: 041041, Dec 2015. https:/​/​doi.org/​10.1103/​PhysRevX.5.041041. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.5.041041.

[15] Mingpu Qin, Chia-Min Chung, Hao Shi, Ettore Vitali, Claudius Hubig, Ulrich Schollwöck, Steven R. White, and Shiwei Zhang. Absence of superconductivity in the pure two-dimensional hubbard model. Phys. Rev. X, 10: 031016, Jul 2020. https:/​/​doi.org/​10.1103/​PhysRevX.10.031016. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.10.031016.

[16] Y. H. Matsuda, N. Abe, S. Takeyama, H. Kageyama, P. Corboz, A. Honecker, S. R. Manmana, G. R. Foltin, K. P. Schmidt, and F. Mila. Magnetization of ${\mathrm{srcu}}_{2}({\mathrm{bo}}_{3}{)}_{2}$ in ultrahigh magnetic fields up to 118 t. Phys. Rev. Lett., 111: 137204, Sep 2013. https:/​/​doi.org/​10.1103/​PhysRevLett.111.137204. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.111.137204.

[17] J. Stapmanns, P. Corboz, F. Mila, A. Honecker, B. Normand, and S. Wessel. Thermal critical points and quantum critical end point in the frustrated bilayer heisenberg antiferromagnet. Physical Review Letters, 121 (12), Sep 2018. ISSN 1079-7114. https:/​/​doi.org/​10.1103/​physrevlett.121.127201. URL http:/​/​dx.doi.org/​10.1103/​PhysRevLett.121.127201.

[18] P. A. Maksimov, Zhenyue Zhu, Steven R. White, and A. L. Chernyshev. Anisotropic-exchange magnets on a triangular lattice: Spin waves, accidental degeneracies, and dual spin liquids. Physical Review X, 9 (2), Apr 2019. ISSN 2160-3308. https:/​/​doi.org/​10.1103/​physrevx.9.021017. URL http:/​/​dx.doi.org/​10.1103/​PhysRevX.9.021017.

[19] Piotr Czarnik, Marek M. Rams, Philippe Corboz, and Jacek Dziarmaga. Tensor network study of the $m=\frac{1}{2}$ magnetization plateau in the shastry-sutherland model at finite temperature. Phys. Rev. B, 103: 075113, Feb 2021. https:/​/​doi.org/​10.1103/​PhysRevB.103.075113. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.103.075113.

[20] Alexander Wietek, Yuan-Yao He, Steven R. White, Antoine Georges, and E. Miles Stoudenmire. Stripes, antiferromagnetism, and the pseudogap in the doped hubbard model at finite temperature. Phys. Rev. X, 11: 031007, Jul 2021. https:/​/​doi.org/​10.1103/​PhysRevX.11.031007. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.11.031007.

[21] Pascal Scholl, Michael Schuler, Hannah J. Williams, Alexander A. Eberharter, Daniel Barredo, Kai-Niklas Schymik, Vincent Lienhard, Louis-Paul Henry, Thomas C. Lang, Thierry Lahaye, Andreas M. Läuchli, and Antoine Browaeys. Quantum simulation of 2d antiferromagnets with hundreds of rydberg atoms. Nature, 595 (7866): 233–238, Jul 2021. ISSN 1476-4687. https:/​/​doi.org/​10.1038/​s41586-021-03585-1.

[22] Sepehr Ebadi, Tout T. Wang, Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Dolev Bluvstein, Rhine Samajdar, Hannes Pichler, Wen Wei Ho, Soonwon Choi, Subir Sachdev, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature, 595 (7866): 227–232, Jul 2021. ISSN 1476-4687. https:/​/​doi.org/​10.1038/​s41586-021-03582-4.

[23] R. Blatt and C. F. Roos. Quantum simulations with trapped ions. Nat. Phys., 8 (4): 277–284, apr 2012. https:/​/​doi.org/​10.1038/​nphys2252.

[24] Loïc Henriet, Lucas Beguin, Adrien Signoles, Thierry Lahaye, Antoine Browaeys, Georges-Olivier Reymond, and Christophe Jurczak. Quantum computing with neutral atoms. Quantum, 4: 327, September 2020. ISSN 2521-327X. https:/​/​doi.org/​10.22331/​q-2020-09-21-327. URL https:/​/​doi.org/​10.22331/​q-2020-09-21-327.

[25] F. Verstraete, V. Murg, and J.I. Cirac. Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Advances in Physics, 57 (2): 143–224, 2008. https:/​/​doi.org/​10.1080/​14789940801912366. URL https:/​/​doi.org/​10.1080/​14789940801912366.

[26] Jutho Haegeman, Christian Lubich, Ivan Oseledets, Bart Vandereycken, and Frank Verstraete. Unifying time evolution and optimization with matrix product states. Phys. Rev. B, 94: 165116, Oct 2016. https:/​/​doi.org/​10.1103/​PhysRevB.94.165116. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.94.165116.

[27] Simone Montangero. Introduction to Tensor Network Methods. Springer International Publishing, 2018. ISBN 978-3-030-01408-7. https:/​/​doi.org/​10.1007/​978-3-030-01409-4.

[28] Román Orús. Tensor networks for complex quantum systems. Nature Reviews Physics, 1 (9): 538–550, August 2019. https:/​/​doi.org/​10.1038/​s42254-019-0086-7.

[29] J. Eisert, M. Cramer, and M. B. Plenio. Colloquium: Area laws for the entanglement entropy. Reviews of Modern Physics, 82 (1), Feb 2010. ISSN 1539-0756. https:/​/​doi.org/​10.1103/​revmodphys.82.277. URL http:/​/​dx.doi.org/​10.1103/​RevModPhys.82.277.

[30] M. Fannes, B. Nachtergaele, and R. F. Werner. Finitely correlated states on quantum spin chains. Communications in Mathematical Physics, 144 (3): 443–490, March 1992. https:/​/​doi.org/​10.1007/​BF02099178.

[31] A. Klumper, A. Schadschneider, and J. Zittartz. Equivalence and solution of anisotropic spin-1 models and generalized t-J fermion models in one dimension. Journal of Physics A Mathematical General, 24 (16): L955–L959, August 1991. https:/​/​doi.org/​10.1088/​0305-4470/​24/​16/​012.

[32] A. Klümper, A. Schadschneider, and J. Zittartz. Matrix product ground states for one-dimensional spin-1 quantum antiferromagnets. EPL (Europhysics Letters), 24: 293, November 1993. https:/​/​doi.org/​10.1209/​0295-5075/​24/​4/​010.

[33] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac. Criticality, the area law, and the computational power of projected entangled pair states. Phys. Rev. Lett., 96: 220601, Jun 2006. https:/​/​doi.org/​10.1103/​PhysRevLett.96.220601. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.96.220601.

[34] Frank Verstraete and J Ignacio Cirac. Renormalization algorithms for quantum-many body systems in two and higher dimensions. arXiv preprint cond-mat/​0407066, 2004. URL https:/​/​arxiv.org/​abs/​cond-mat/​0407066.

[35] Maurits S. J. Tepaske and David J. Luitz. Three-dimensional isometric tensor networks. Phys. Rev. Research, 3: 023236, Jun 2021. https:/​/​doi.org/​10.1103/​PhysRevResearch.3.023236. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.3.023236.

[36] Y.-Y. Shi, L.-M. Duan, and G. Vidal. Classical simulation of quantum many-body systems with a tree tensor network. Phys. Rev. A, 74: 022320, Aug 2006. https:/​/​doi.org/​10.1103/​PhysRevA.74.022320. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.74.022320.

[37] M. Gerster, P. Silvi, M. Rizzi, R. Fazio, T. Calarco, and S. Montangero. Unconstrained tree tensor network: An adaptive gauge picture for enhanced performance. Phys. Rev. B, 90: 125154, Sep 2014. https:/​/​doi.org/​10.1103/​PhysRevB.90.125154. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.90.125154.

[38] P. Silvi, V. Giovannetti, S. Montangero, M. Rizzi, J. I. Cirac, and R. Fazio. Homogeneous binary trees as ground states of quantum critical hamiltonians. Phys. Rev. A, 81: 062335, Jun 2010. https:/​/​doi.org/​10.1103/​PhysRevA.81.062335. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.81.062335.

[39] Pietro Silvi, Ferdinand Tschirsich, Matthias Gerster, Johannes Junemann, Daniel Jaschke, Matteo Rizzi, and Simone Montangero. The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems. SciPost Phys. Lect. Notes, page 8, 2019. https:/​/​doi.org/​10.21468/​SciPostPhysLectNotes.8. URL https:/​/​scipost.org/​10.21468/​SciPostPhysLectNotes.8.

[40] G. Vidal. Entanglement renormalization. Phys. Rev. Lett., 99: 220405, Nov 2007. https:/​/​doi.org/​10.1103/​PhysRevLett.99.220405. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.99.220405.

[41] G. Evenbly and G. Vidal. Entanglement renormalization in two spatial dimensions. Phys. Rev. Lett., 102: 180406, May 2009. https:/​/​doi.org/​10.1103/​PhysRevLett.102.180406. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.102.180406.

[42] Ho N. Phien, Johann A. Bengua, Hoang D. Tuan, Philippe Corboz, and Román Orús. Infinite projected entangled pair states algorithm improved: Fast full update and gauge fixing. Phys. Rev. B, 92: 035142, Jul 2015. https:/​/​doi.org/​10.1103/​PhysRevB.92.035142. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.92.035142.

[43] Laurens Vanderstraeten, Jutho Haegeman, Philippe Corboz, and Frank Verstraete. Gradient methods for variational optimization of projected entangled-pair states. Phys. Rev. B, 94: 155123, Oct 2016. https:/​/​doi.org/​10.1103/​PhysRevB.94.155123. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.94.155123.

[44] M. T. Fishman, L. Vanderstraeten, V. Zauner-Stauber, J. Haegeman, and F. Verstraete. Faster methods for contracting infinite two-dimensional tensor networks. Phys. Rev. B, 98: 235148, Dec 2018. https:/​/​doi.org/​10.1103/​PhysRevB.98.235148. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.98.235148.

[45] Z. Y. Xie, H. J. Liao, R. Z. Huang, H. D. Xie, J. Chen, Z. Y. Liu, and T. Xiang. Optimized contraction scheme for tensor-network states. Phys. Rev. B, 96: 045128, Jul 2017. https:/​/​doi.org/​10.1103/​PhysRevB.96.045128. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.96.045128.

[46] Philippe Corboz, Piotr Czarnik, Geert Kapteijns, and Luca Tagliacozzo. Finite correlation length scaling with infinite projected entangled-pair states. Phys. Rev. X, 8: 031031, Jul 2018. https:/​/​doi.org/​10.1103/​PhysRevX.8.031031. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.8.031031.

[47] A. Kshetrimayum, M. Rizzi, J. Eisert, and R. Orús. Tensor network annealing algorithm for two-dimensional thermal states. Phys. Rev. Lett., 122: 070502, Feb 2019. https:/​/​doi.org/​10.1103/​PhysRevLett.122.070502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.122.070502.

[48] Timo Felser, Simone Notarnicola, and Simone Montangero. Efficient tensor network ansatz for high-dimensional quantum many-body problems. Phys. Rev. Lett., 126: 170603, Apr 2021. https:/​/​doi.org/​10.1103/​PhysRevLett.126.170603. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.126.170603.

[49] Giuseppe Magnifico, Timo Felser, Pietro Silvi, and Simone Montangero. Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks. Nature Communications, 12 (1): 3600, Jun 2021. ISSN 2041-1723. 10.1038/​s41467-021-23646-3. URL https:/​/​doi.org/​10.1038/​s41467-021-23646-3.

[50] Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69: 2863–2866, Nov 1992. https:/​/​doi.org/​10.1103/​PhysRevLett.69.2863. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.69.2863.

[51] U. Schollwöck. The density-matrix renormalization group. Rev. Mod. Phys., 77: 259–315, Apr 2005. https:/​/​doi.org/​10.1103/​RevModPhys.77.259. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.77.259.

[52] E.M. Stoudenmire and Steven R. White. Studying two-dimensional systems with the density matrix renormalization group. Annual Review of Condensed Matter Physics, 3 (1): 111–128, 2012. https:/​/​doi.org/​10.1146/​annurev-conmatphys-020911-125018. URL https:/​/​doi.org/​10.1146/​annurev-conmatphys-020911-125018.

[53] Ferdinand Tschirsich, Simone Montangero, and Marcello Dalmonte. Phase Diagram and Conformal String Excitations of Square Ice using Gauge Invariant Matrix Product States. SciPost Phys., 6: 28, 2019. https:/​/​doi.org/​10.21468/​SciPostPhys.6.3.028. URL https:/​/​scipost.org/​10.21468/​SciPostPhys.6.3.028.

[54] S. Yamada, T. Imamura, and M. Machida. Parallelization design on multi-core platforms in density matrix renormalization group toward 2-d quantum strongly-correlated systems. In SC '11: Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–10, 2011. https:/​/​doi.org/​10.1145/​2063384.2063467.

[55] Shoudan Liang and Hanbin Pang. Approximate diagonalization using the density matrix renormalization-group method: A two-dimensional-systems perspective. Phys. Rev. B, 49: 9214–9217, Apr 1994. https:/​/​doi.org/​10.1103/​PhysRevB.49.9214. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.49.9214.

[56] Tao Xiang, Jizhong Lou, and Zhaobin Su. Two-dimensional algorithm of the density-matrix renormalization group. Phys. Rev. B, 64: 104414, Aug 2001. https:/​/​doi.org/​10.1103/​PhysRevB.64.104414. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.64.104414.

[57] A. Kantian, M. Dolfi, M. Troyer, and T. Giamarchi. Understanding repulsively mediated superconductivity of correlated electrons via massively parallel density matrix renormalization group. Phys. Rev. B, 100: 075138, Aug 2019. https:/​/​doi.org/​10.1103/​PhysRevB.100.075138. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.100.075138.

[58] Greta Ghelli, Giuseppe Magnifico, Cristian Degli Esposti Boschi, and Elisa Ercolessi. Topological phases in two-legged heisenberg ladders with alternating interactions. Phys. Rev. B, 101: 085124, Feb 2020. https:/​/​doi.org/​10.1103/​PhysRevB.101.085124. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.101.085124.

[59] Mazen Ali. On the ordering of sites in the density matrix renormalization group using quantum mutual information. arXiv preprint arXiv:2103.01111, 2021. URL https:/​/​arxiv.org/​abs/​2103.01111.

[60] David Hilbert. Ueber die stetige abbildung einer linie auf ein flächenstück. Mathematische Annalen, 38: 459–460, 1891. URL http:/​/​www.digizeitschriften.de/​dms/​img/​?PPN=PPN235181684_0038&DMDID=dmdlog40.

[61] Sei Suzuki, Jun-ichi Inoue, and Bikas K. Chakrabarti. Quantum Ising Phases and Transitions in Transverse Ising Models, volume 859. 2013. https:/​/​doi.org/​10.1007/​978-3-642-33039-1.

[62] David J. Abel and David M. Mark. A comparative analysis of some two-dimensional orderings. International Journal of Geographical Information Systems, 4 (1): 21–31, 1990. https:/​/​doi.org/​10.1080/​02693799008941526. URL https:/​/​doi.org/​10.1080/​02693799008941526.

[63] H. V. Jagadish. Linear clustering of objects with multiple attributes. SIGMOD Rec., 19 (2): 332–342, May 1990. ISSN 0163-5808. https:/​/​doi.org/​10.1145/​93605.98742. URL https:/​/​doi.org/​10.1145/​93605.98742.

[64] B. Moon, H. V. Jagadish, C. Faloutsos, and J. H. Saltz. Analysis of the clustering properties of the hilbert space-filling curve. IEEE Transactions on Knowledge and Data Engineering, 13 (1): 124–141, 2001. https:/​/​doi.org/​10.1109/​69.908985.

[65] Todd Eavis and David Cueva. A hilbert space compression architecture for data warehouse environments. In Il Yeal Song, Johann Eder, and Tho Manh Nguyen, editors, Data Warehousing and Knowledge Discovery, pages 1–12, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. ISBN 978-3-540-74553-2. https:/​/​doi.org/​10.1007/​978-3-319-10160-6.

[66] Daniel Lemire and Owen Kaser. Reordering columns for smaller indexes. Information Sciences, 181 (12): 2550 – 2570, 2011. ISSN 0020-0255. https:/​/​doi.org/​10.1016/​j.ins.2011.02.002. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0020025511000788.

[67] Simon Anders. Visualization of genomic data with the Hilbert curve. Bioinformatics, 25 (10): 1231–1235, 03 2009. ISSN 1367-4803. https:/​/​doi.org/​10.1093/​bioinformatics/​btp152. URL https:/​/​doi.org/​10.1093/​bioinformatics/​btp152.

[68] Zvi Friedman. Ising model with a transverse field in two dimensions: Phase diagram and critical properties from a real-space renormalization group. Phys. Rev. B, 17: 1429–1432, Feb 1978. https:/​/​doi.org/​10.1103/​PhysRevB.17.1429. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.17.1429.

[69] Sheng-Hao Li and Guo-Ping Lei. Quantum phase transition in a two-dimensional quantum ising model: Tensor network states and ground-state fidelity. Journal of Physics: Conference Series, 1087: 052011, sep 2018. https:/​/​doi.org/​10.1088/​1742-6596/​1087/​5/​052011. URL https:/​/​doi.org/​10.1088/​1742-6596/​1087/​5/​052011.

[70] J A Mydosh. Spin glasses: redux: an updated experimental/​materials survey. Reports on Progress in Physics, 78 (5): 052501, apr 2015. https:/​/​doi.org/​10.1088/​0034-4885/​78/​5/​052501. URL https:/​/​doi.org/​10.1088/​0034-4885/​78/​5/​052501.

[71] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik. Quantum chemistry in the age of quantum computing. Chemical Reviews, 119 (19): 10856–10915, Oct 2019. ISSN 0009-2665. https:/​/​doi.org/​10.1021/​acs.chemrev.8b00803. URL https:/​/​doi.org/​10.1021/​acs.chemrev.8b00803.

[72] Johannes Hauschild and Frank Pollmann. Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy). SciPost Phys. Lect. Notes, page 5, 2018. https:/​/​doi.org/​10.21468/​SciPostPhysLectNotes.5. URL https:/​/​scipost.org/​10.21468/​SciPostPhysLectNotes.5. Code available from https:/​/​github.com/​tenpy/​tenpy.

[73] Ulrich Schollwöck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326 (1): 96–192, Jan 2011. ISSN 0003-4916. https:/​/​doi.org/​10.1016/​j.aop.2010.09.012. URL http:/​/​dx.doi.org/​10.1016/​j.aop.2010.09.012.

[74] R. B. Lehoucq, D. C. Sorensen, and C. Yang. Arpack users guide: Solution of large scale eigenvalue problems by implicitly restarted arnoldi methods., 1997.

Cited by

[1] Guglielmo Lami, Giuseppe Carleo, and Mario Collura, "Matrix product states with backflow correlations", Physical Review B 106 8, L081111 (2022).

[2] Poetri Sonya Tarabunga, Emanuele Tirrito, Titas Chanda, and Marcello Dalmonte, "Many-Body Magic Via Pauli-Markov Chains—From Criticality to Gauge Theories", PRX Quantum 4 4, 040317 (2023).

[3] Andreas Haller, Solofo Groenendijk, Alireza Habibi, Andreas Michels, and Thomas L. Schmidt, "Quantum skyrmion lattices in Heisenberg ferromagnets", Physical Review Research 4 4, 043113 (2022).

[4] Giovanni Ferrari, Giuseppe Magnifico, and Simone Montangero, "Adaptive-weighted tree tensor networks for disordered quantum many-body systems", Physical Review B 105 21, 214201 (2022).

[5] Xiangjian Qian and Mingpu Qin, "Augmenting Density Matrix Renormalization Group with Disentanglers", Chinese Physics Letters 40 5, 057102 (2023).

[6] Mari Carmen Bañuls, "Tensor Network Algorithms: A Route Map", Annual Review of Condensed Matter Physics 14 1, 173 (2023).

[7] Massimo Bortone, Yannic Rath, and George H. Booth, "Impact of conditional modelling for a universal autoregressive quantum state", Quantum 8, 1245 (2024).

[8] Mezache Zinelabiddine, Abdullah Alzahrani, Abdullah Alwabli, Amar Jaffar, Enas Ali, and Mohammed S. Alzaidi, "Miniaturization and Fabrication of a Novel Cross-Fractal Biosensor and Sensor for Characterizing 3D Printing Electromagnetic Properties in Polylactic Acid", IEEE Access 12, 33045 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 06:10:11) and SAO/NASA ADS (last updated successfully 2024-06-22 06:10:12). The list may be incomplete as not all publishers provide suitable and complete citation data.