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arXiv:2204.13531 · Journal of Mathematical Chemistry

Compressing multireference character of wave functions via fermionic mode optimization

Mihály Máté, Klára Petrov, Szilárd Szalay, Örs Legeza

Published Jul 29, 2022• 13 citations
arXivHTMLPDF

Abstract

Abstract In this work, we present a brief overview of the fermionic mode optimization within the framework of tensor network state methods (Krumnow et al. in Phys Rev Lett 117:210402, 2016, https://doi.org/10.1103/PhysRevLett.117.210402 ), and demonstrate that it has the potential to compress the multireference character of the wave functions after finding optimal molecular orbitals (modes), based on entanglement minimization. Numerical simulations have been performed for the nitrogen dimer in the cc-pVDZ basis for the equilibrium and for stretched geometries.

DOI
10.1007/s10910-022-01379-y

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I read with interest Daniel Kleppner’s lucid column on Hanbury Brown’s “steamroller.” Kleppner mentions an aspect of the Hanbury Brown and Twiss (HBT) effect that at first seemingly defied a quantum interpretation; he terms it “anti-weird,” since in certain cases a completely classical understanding is possible. Other aspects of the Hanbury Brown physics are truly weird in the sense that they violate a Bell inequality. I and collaborators Peter Samuelsson and Eugene Sukhorukov have proposed an interferometry experiment with fermions in an electrical conductor. Carriers are injected into the conductor from two contacts, and the cross correlation of the electrical current is measured at two other contacts along the conductor. 1 1. P. Samuelsson, E. V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). https://doi.org/10.1103/PhysRevLett.92.026805 In that arrangement, it is possible to investigate the Aharonov–Bohm effect in the fermionic HBT interferometer. Our initial discussions of such an effect date back to the early 1990s. 2 2. M. Büttiker, Phys. Rev. Lett. 68, 843 (1992); Physica B 175, 199 (1991). https://doi.org/10.1016/0921-4526(91)90713-O My collaborators and I found a geometry in which the conductance (intensity) shows no Aharonov–Bohm signature, but the current correlation (intensity correlation) is nevertheless a periodic function of the Aharonov–Bohm flux. 1 1. P. Samuelsson, E. V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). https://doi.org/10.1103/PhysRevLett.92.026805 We termed our finding the two-particle Aharonov–Bohm effect. The effect was recently observed by a group headed by Moty Heiblum at the Weizmann Institute of Science. His team cooled a mesoscopic conductor to tens of millikelvin and measured its conductance and noise correlations. 3 3. I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, V. Umansky, Nature 448, 333 (2007). https://doi.org/10.1038/nature05955 Theoretical work by Carlo Beenakker at Leiden University has shown that due to orbital quasiparticle entanglement, 4 4. C. W. J. Beenakker, in Quantum Computers, Algorithms and Chaos, Proceedings of the International School of Physics “E. Fermi,” vol. 162, G. Casati et al. , eds., IOS Press, Washington, DC (2006). the two-particle effect is equivalent at zero temperature to a violation of a Bell inequality. 1 1. P. Samuelsson, E. V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). https://doi.org/10.1103/PhysRevLett.92.026805 The case of nonzero temperature is more complicated and is still a subject of research. 5 5. P. Samuelsson, I. Neder, M. Büttiker, Phys. Rev. Lett. (in press). The possibility of observing truly weird quantum physics is certain to increase interest in Hanbury Brown’s physics.REFERENCESSection:ChooseTop of pageREFERENCES <<1. P. Samuelsson, E. V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). https://doi.org/10.1103/PhysRevLett.92.026805 , Google ScholarCrossref2. M. Büttiker, Phys. Rev. Lett. 68, 843 (1992); Google ScholarCrossref Physica B 175, 199 (1991). https://doi.org/10.1016/0921-4526(91)90713-O , , Google ScholarCrossref3. I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, V. Umansky, Nature 448, 333 (2007). https://doi.org/10.1038/nature05955 , Google ScholarCrossref4. C. W. J. Beenakker, in Quantum Computers, Algorithms and Chaos, Proceedings of the International School of Physics “E. Fermi,” vol. 162, G. Casati et al. , eds., IOS Press, Washington, DC (2006). Google Scholar5. P. Samuelsson, I. Neder, M. Büttiker, Phys. Rev. Lett. (in press). Google Scholar© 2009 American Institute of Physics.

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Abstract In this work, we present a brief overview of the fermionic mode optimization within the framework of tensor network state methods (Krumnow et al. in Phys Rev Lett 117:210402, 2016, https://doi.org/10.1103/PhysRevLett.117.210402 ), and demonstrate that it has the potential to compress the multireference character of the wave functions after finding optimal molecular orbitals (modes), based on entanglement minimization. Numerical simulations have been performed for the nitrogen dimer in the cc-pVDZ basis for the equilibrium and for stretched geometries.
An optimal choice of single-particle modes leads to an optimal tensor network-based compression of the many-body wave function for systems of indistinguishable particles, which compression can be essential for large, close-to-critical systems. The proposed mode optimization relies on the minimization of the block entropy area, a global quantity that measures entanglement of the wave function along the one-dimensional chain of modes. Extension of the two-site DMRG algorithm by nearest-neighbor mode optimizations is straightforward, but it performs poorly without systematic reordering of modes. Moreover, finding a stationary optimal point in the combination of the fixed-rank MPS manifold and the unitary group of mode rotations requires optimization for every generator, i.e., for every two-mode pair. Here, a systematic joint optimization protocol over the MPS manifold and the full unitary group is presented in which every two-mode pair is accessed by a carefully constructed sequence of two-qubit operations, including swap-gate controlled permutations. Large-scale DMRG simulations of strongly correlated two-dimensional fermionic lattice models and the multireference Fe4_44​S4_44​ transition metal cluster demonstrate rapid convergence towards the optimum, achieving low energy and low entanglement.
Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chemistry. A new freedom arising in such nonlocal fermionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this Letter, we propose a way to overcome this challenge. We suggest a method intertwining the optimization over matrix product states with suitable fermionic Gaussian mode transformations. The described algorithm generalizes basis changes in the spirit of the Hartree-Fock method to matrix-product states, and provides a black box tool for basis optimization in tensor network methods.
The advent of Neural-network Quantum States (NQS) has significantly advanced wave function ansatz research, sparking a resurgence in orbital space variational Monte Carlo (VMC) exploration. This work introduces three algorithmic enhancements to reduce computational demands of VMC optimization using NQS: an adaptive learning rate algorithm, constrained optimization, and block optimization. We evaluate the refined algorithm on complex multireference bond stretches of H2O and N2 within the cc-pVDZ basis set and calculate the ground-state energy of the strongly correlated chromium dimer (Cr2) in the Ahlrichs SV basis set. Our results achieve superior accuracy compared to coupled cluster theory at a relatively modest CPU cost. This work demonstrates how to enhance optimization efficiency and robustness using these strategies, opening a new path to optimize large-scale restricted Boltzmann machine-based NQS more effectively and marking a substantial advancement in NQS's practical quantum chemistry applications.
I read with interest Daniel Kleppner’s lucid column on Hanbury Brown’s “steamroller.” Kleppner mentions an aspect of the Hanbury Brown and Twiss (HBT) effect that at first seemingly defied a quantum interpretation; he terms it “anti-weird,” since in certain cases a completely classical understanding is possible. Other aspects of the Hanbury Brown physics are truly weird in the sense that they violate a Bell inequality. I and collaborators Peter Samuelsson and Eugene Sukhorukov have proposed an interferometry experiment with fermions in an electrical conductor. Carriers are injected into the conductor from two contacts, and the cross correlation of the electrical current is measured at two other contacts along the conductor. 1 1. P. Samuelsson, E. V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). https://doi.org/10.1103/PhysRevLett.92.026805 In that arrangement, it is possible to investigate the Aharonov–Bohm effect in the fermionic HBT interferometer. Our initial discussions of such an effect date back to the early 1990s. 2 2. M. Büttiker, Phys. Rev. Lett. 68, 843 (1992); Physica B 175, 199 (1991). https://doi.org/10.1016/0921-4526(91)90713-O My collaborators and I found a geometry in which the conductance (intensity) shows no Aharonov–Bohm signature, but the current correlation (intensity correlation) is nevertheless a periodic function of the Aharonov–Bohm flux. 1 1. P. Samuelsson, E. V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). https://doi.org/10.1103/PhysRevLett.92.026805 We termed our finding the two-particle Aharonov–Bohm effect. The effect was recently observed by a group headed by Moty Heiblum at the Weizmann Institute of Science. His team cooled a mesoscopic conductor to tens of millikelvin and measured its conductance and noise correlations. 3 3. I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, V. Umansky, Nature 448, 333 (2007). https://doi.org/10.1038/nature05955 Theoretical work by Carlo Beenakker at Leiden University has shown that due to orbital quasiparticle entanglement, 4 4. C. W. J. Beenakker, in Quantum Computers, Algorithms and Chaos, Proceedings of the International School of Physics “E. Fermi,” vol. 162, G. Casati et al. , eds., IOS Press, Washington, DC (2006). the two-particle effect is equivalent at zero temperature to a violation of a Bell inequality. 1 1. P. Samuelsson, E. V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). https://doi.org/10.1103/PhysRevLett.92.026805 The case of nonzero temperature is more complicated and is still a subject of research. 5 5. P. Samuelsson, I. Neder, M. Büttiker, Phys. Rev. Lett. (in press). The possibility of observing truly weird quantum physics is certain to increase interest in Hanbury Brown’s physics.REFERENCESSection:ChooseTop of pageREFERENCES <<1. P. Samuelsson, E. V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 92, 026805 (2004). https://doi.org/10.1103/PhysRevLett.92.026805 , Google ScholarCrossref2. M. Büttiker, Phys. Rev. Lett. 68, 843 (1992); Google ScholarCrossref Physica B 175, 199 (1991). https://doi.org/10.1016/0921-4526(91)90713-O , , Google ScholarCrossref3. I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, V. Umansky, Nature 448, 333 (2007). https://doi.org/10.1038/nature05955 , Google ScholarCrossref4. C. W. J. Beenakker, in Quantum Computers, Algorithms and Chaos, Proceedings of the International School of Physics “E. Fermi,” vol. 162, G. Casati et al. , eds., IOS Press, Washington, DC (2006). Google Scholar5. P. Samuelsson, I. Neder, M. Büttiker, Phys. Rev. Lett. (in press). Google Scholar© 2009 American Institute of Physics.
Tensor network states and parton wave functions are two pivotal methods for studying quantum many-body systems. This work connects these two subjects as we demonstrate that a variety of parton wave functions, such as projected Fermi sea and projected fermionic or bosonic paired states, can be represented exactly as tensor networks. The results can be compressed into matrix product states with moderate bond dimensions so various physical quantities can be computed efficiently. For the projected Fermi sea, we develop an excellent compression scheme with high fidelity using maximally localized Wannier orbitals. Numerical calculations on two parton wave functions demonstrate that our method exceeds commonly adopted Monte Carlo methods in some aspects. It produces energy and correlation function with very high accuracy that is difficult to achieve using Monte Carlo method. The entanglement measures that were almost impossible to compute before can also be obtained easily using our method.