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Variational subspace methods and application to improving variational Monte Carlo dynamics

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Variational subspace methods and application to improving variational Monte Carlo dynamics

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AbstractWe present a formalism that allows for the direct manipulation and optimization of subspaces, circumventing the need to optimize individual states when using subspace methods. Using the determinant state mapping, we can naturally extend notions such as distance and energy to subspaces, as well as Monte Carlo estimators, recovering the excited states estimation method proposed by Pfau et al. As a practical application, we then introduce Bridge, a method that improves the performance of variational dynamics by extracting linear combinations of variational time-evolved states. We find that Bridge is both computationally inexpensive and capable of significantly mitigating the errors that arise from discretizing the dynamics, and can thus be systematically used as a post-processing tool for variational dynamics.Featured image: Illustration of the Bridge post-processing method. Variational states $|\phi_p \rangle$ obtained with an iterative variational dynamics method such as the time dependent variational principle (TDVP) or projected time-dependent variational Monte Carlo (p-tVMC) are improved by performing the TDVP on the subspace spanned by the states $|\phi_p \rangle$.► BibTeX data@article{Kahn2026variationalsubspace, doi = {10.22331/q-2026-04-23-2082}, url = {https://doi.org/10.22331/q-2026-04-23-2082}, title = {Variational subspace methods and application to improving variational {M}onte {C}arlo dynamics}, author = {Kahn, Adrien and Gravina, Luca and Vicentini, Filippo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2082}, month = apr, year = {2026} }► References [1] Adam D. Bookatz. ``Qma-complete problems''. Quantum Information and Computation 14, 361–383 (2014). https://doi.org/10.26421/qic14.5-6-1 [2] Jacob C Bridgeman and Christopher T Chubb. ``Hand-waving and interpretive dance: an introductory course on tensor networks''. Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017). https://doi.org/10.1088/1751-8121/aa6dc3 [3] Román Orús. ``A practical introduction to tensor networks: Matrix product states and projected entangled pair states''. Annals of Physics 349, 117–158 (2014). https://doi.org/10.1016/j.aop.2014.06.013 [4] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96–192 (2011). https://doi.org/10.1016/j.aop.2010.09.012 [5] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson. ``The variational quantum eigensolver: A review of methods and best practices''. Physics Reports 986, 1–128 (2022). https://doi.org/10.1016/j.physrep.2022.08.003 [6] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien. ``A variational eigenvalue solver on a photonic quantum processor''. Nature Communications 5 (2014). https://doi.org/10.1038/ncomms5213 [7] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. ``The theory of variational hybrid quantum-classical algorithms''. New Journal of Physics 18, 023023 (2016). https://doi.org/10.1088/1367-2630/18/2/023023 [8] Giuseppe Carleo and Matthias Troyer. ``Solving the quantum many-body problem with artificial neural networks''. Science 355, 602–606 (2017). https://doi.org/10.1126/science.aag2302 [9] Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, Robert Okuła, Gorka Muñoz-Gil, Rodrigo A. Vargas-Hernández, Alba Cervera-Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabrié, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicentini, Lei Wang, Sebastian J. Wetzel, Giuseppe Carleo, Eliška Greplová, Roman Krems, Florian Marquardt, Michał Tomza, Maciej Lewenstein, and Alexandre Dauphin. ``Machine learning in quantum sciences''.

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A 108, 022210 (2023). https://doi.org/10.1103/PhysRevA.108.022210Cited by[1] Matija Medvidović, Alev Orfi, Juan Carrasquilla, and Dries Sels, "Adiabatic transport of neural network quantum states", arXiv:2510.15030, (2025). [2] Ahmedeo Shokry, Alessandro Santini, and Filippo Vicentini, "When Less is More: Approximating the Quantum Geometric Tensor with Block Structures", arXiv:2510.08430, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-04-23 17:45:41). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-04-23 17:45:40: Could not fetch cited-by data for 10.22331/q-2026-04-23-2082 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe present a formalism that allows for the direct manipulation and optimization of subspaces, circumventing the need to optimize individual states when using subspace methods. Using the determinant state mapping, we can naturally extend notions such as distance and energy to subspaces, as well as Monte Carlo estimators, recovering the excited states estimation method proposed by Pfau et al. As a practical application, we then introduce Bridge, a method that improves the performance of variational dynamics by extracting linear combinations of variational time-evolved states. We find that Bridge is both computationally inexpensive and capable of significantly mitigating the errors that arise from discretizing the dynamics, and can thus be systematically used as a post-processing tool for variational dynamics.Featured image: Illustration of the Bridge post-processing method. Variational states $|\phi_p \rangle$ obtained with an iterative variational dynamics method such as the time dependent variational principle (TDVP) or projected time-dependent variational Monte Carlo (p-tVMC) are improved by performing the TDVP on the subspace spanned by the states $|\phi_p \rangle$.► BibTeX data@article{Kahn2026variationalsubspace, doi = {10.22331/q-2026-04-23-2082}, url = {https://doi.org/10.22331/q-2026-04-23-2082}, title = {Variational subspace methods and application to improving variational {M}onte {C}arlo dynamics}, author = {Kahn, Adrien and Gravina, Luca and Vicentini, Filippo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2082}, month = apr, year = {2026} }► References [1] Adam D. Bookatz. ``Qma-complete problems''. Quantum Information and Computation 14, 361–383 (2014). https://doi.org/10.26421/qic14.5-6-1 [2] Jacob C Bridgeman and Christopher T Chubb. ``Hand-waving and interpretive dance: an introductory course on tensor networks''. Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017). https://doi.org/10.1088/1751-8121/aa6dc3 [3] Román Orús. ``A practical introduction to tensor networks: Matrix product states and projected entangled pair states''. Annals of Physics 349, 117–158 (2014). https://doi.org/10.1016/j.aop.2014.06.013 [4] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96–192 (2011). https://doi.org/10.1016/j.aop.2010.09.012 [5] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson. ``The variational quantum eigensolver: A review of methods and best practices''. Physics Reports 986, 1–128 (2022). https://doi.org/10.1016/j.physrep.2022.08.003 [6] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien. ``A variational eigenvalue solver on a photonic quantum processor''. Nature Communications 5 (2014). https://doi.org/10.1038/ncomms5213 [7] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. ``The theory of variational hybrid quantum-classical algorithms''. New Journal of Physics 18, 023023 (2016). https://doi.org/10.1088/1367-2630/18/2/023023 [8] Giuseppe Carleo and Matthias Troyer. ``Solving the quantum many-body problem with artificial neural networks''. Science 355, 602–606 (2017). https://doi.org/10.1126/science.aag2302 [9] Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, Robert Okuła, Gorka Muñoz-Gil, Rodrigo A. Vargas-Hernández, Alba Cervera-Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabrié, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicentini, Lei Wang, Sebastian J. Wetzel, Giuseppe Carleo, Eliška Greplová, Roman Krems, Florian Marquardt, Michał Tomza, Maciej Lewenstein, and Alexandre Dauphin. ``Machine learning in quantum sciences''.

Physical Review Letters 136 (2026). https://doi.org/10.1103/kqvx-dl54 [24] Gene Golub and Charles Van Loan. ``Matrix computations''.

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Variational subspace methods and application to improving variational Monte Carlo dynamics

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