A posteriori error estimates for the Lindblad master equation
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AbstractWe are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.► BibTeX data@article{Etienney2026posteriorierror, doi = {10.22331/q-2026-03-16-2031}, url = {https://doi.org/10.22331/q-2026-03-16-2031}, title = {A posteriori error estimates for the {L}indblad master equation}, author = {Etienney, Paul-Louis and Robin, R{\'{e}}mi and Rouchon, Pierre}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2031}, month = mar, year = {2026} }► References [1] Daniel Appelöand Yingda Cheng ``Kraus is king: High-order completely positive and trace preserving (CPTP) low rank method for the Lindblad master equation'' Journal of Computational Physics 534, 114036 (2025). https://doi.org/10.1016/j.jcp.2025.114036 https://www.sciencedirect.com/science/article/pii/S0021999125003195 [2] Sahel Ashhab, Felix Fischer, Davide Lonigro, Daniel Braak, and Daniel Burgarth, ``Finite-dimensional approximations of generalized squeezing'' Phys. Rev. A 113, 013703 (2026). https://doi.org/10.1103/9vwp-f35c [3] Rémi Azouit, Alain Sarlette, and Pierre Rouchon, ``Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping'' ESAIM: Control, Optimisation and Calculus of Variations 22, 1353–1369 (2016). https://doi.org/10.1051/cocv/2016050 [4] H.-P. Breuerand F. Petruccione ``The Theory of Open Quantum Systems'' Oxford University Press (2006). https://doi.org/10.1093/acprof:oso/9780199213900.001.0001 [5] Yu Caoand Jianfeng Lu ``Structure-Preserving Numerical Schemes for Lindblad Equations'' Journal of Scientific Computing 102, 27 (2024). https://doi.org/10.1007/s10915-024-02707-x [6] A. M Chebotarevand F Fagnola ``Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups'' Journal of Functional Analysis 153, 382–404 (1998). https://doi.org/10.1006/jfan.1997.3189 [7] Alexander Chebotarev, J. Garcia, and Roberto Quezada, ``On the Lindblad equation with unbounded time-dependent coefficients'' Mathematical Notes 61, 105–117 (1997). https://doi.org/10.1007/BF02355012 [8] Dariusz Chruscinskiand Saverio Pascazio ``A Brief History of the GKLS Equation'' Open Systems and Information Dynamics 24 (2017). https://doi.org/10.1142/S1230161217400017 [9] E. B. Davies ``Generators of dynamical semigroups'' Journal of Functional Analysis 34, 421–432 (1979). https://doi.org/10.1016/0022-1236(79)90085-5 [10] Felix Fischer, Daniel Burgarth, and Davide Lonigro, ``Quantum particle in the wrong box (or: the perils of finite-dimensional approximations)'' Quantum 10, 1985 (2026). https://doi.org/10.22331/q-2026-01-27-1985 [11] Paul Gondolf, Tim Möbus, and Cambyse Rouzé, ``Energy preserving evolutions over Bosonic systems'' Quantum 8, 1551 (2024). https://doi.org/10.22331/q-2024-12-04-1551 [12] Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan, ``Completely positive dynamical semigroups of N‐level systems'' Journal of Mathematical Physics 17, 821–825 (1976). https://doi.org/10.1063/1.522979 [13] Daniel Gottesman, Alexei Kitaev, and John Preskill, ``Encoding a qubit in an oscillator'' Phys. Rev. A 64, 012310 (2001). https://doi.org/10.1103/PhysRevA.64.012310 [14] Pierre Guilmin, Ronan Gautier, Adrien Bocquet, and Élie Genois, ``Dynamiqs: an open-source Python library for GPU-accelerated and differentiable simulation of quantum systems'' (2024). https://github.com/dynamiqs/dynamiqs [15] Timo Hillmannand Fernando Quijandría ``Quantum error correction with dissipatively stabilized squeezed-cat qubits'' Physical Review A 107 (2023). https://doi.org/10.1103/physreva.107.032423 [16] J.R. Johansson, P.D. Nation, and Franco Nori, ``QuTiP 2: A Python framework for the dynamics of open quantum systems'' Computer Physics Communications 184, 1234–1240 (2013). https://doi.org/10.1016/j.cpc.2012.11.019 https://www.sciencedirect.com/science/article/pii/S0010465512003955 [17] B.D. Josephson ``Possible new effects in superconductive tunnelling'' Physics Letters 1, 251–253 (1962). https://doi.org/10.1016/0031-9163(62)91369-0 https://www.sciencedirect.com/science/article/pii/0031916362913690 [18] A. Kossakowski ``On quantum statistical mechanics of non-Hamiltonian systems'' Reports on Mathematical Physics 3, 247–274 (1972). https://doi.org/10.1016/0034-4877(72)90010-9 https://www.sciencedirect.com/science/article/pii/0034487772900109 [19] Sebastian Krämer, David Plankensteiner, Laurin Ostermann, and Helmut Ritsch, ``QuantumOptics. jl: A Julia framework for simulating open quantum systems'' Computer Physics Communications 227, 109–116 (2018). https://doi.org/10.1016/j.cpc.2018.02.004 [20] G. Lindblad ``On the generators of quantum dynamical semigroups'' Communications in Mathematical Physics 48, 119–130 (1976). https://doi.org/10.1007/BF01608499 [21] Mazyar Mirrahimi, Zaki Leghtas, Victor V. Albert, Steven Touzard, Robert J. Schoelkopf, Liang Jiang, and Michel H. Devoret, ``Dynamically protected cat-qubits: a new paradigm for universal quantum computation'' New Journal of Physics 16, 045014 (2014). https://doi.org/10.1088/1367-2630/16/4/045014 [22] Bo Peng, Yuan Su, Daniel Claudino, Karol Kowalski, Guang Hao Low, and Martin Roetteler, ``Quantum simulation of boson-related Hamiltonians: techniques, effective Hamiltonian construction, and error analysis'' Quantum Science and Technology 10, 023002 (2025). https://doi.org/10.1088/2058-9565/adbf42 [23] Daniel Puzzuoli, Christopher J. Wood, Daniel J. Egger, Benjamin Rosand, and Kento Ueda, ``Qiskit Dynamics: A Python package for simulating the time dynamics of quantum systems'' Journal of Open Source Software 8, 5853 (2023). https://doi.org/10.21105/joss.05853 [24] Rémi Robinand Pierre Rouchon ``Convergence Analysis of Galerkin Approximations for the Lindblad Master Equation'' (2025). arXiv:2510.11416 [25] Rémi Robin, Pierre Rouchon, and Lev-Arcady Sellem, ``Convergence of Bipartite Open Quantum Systems Stabilized by Reservoir Engineering'' Annales Henri Poincaré (2024). https://doi.org/10.1007/s00023-024-01481-8 http://arxiv.org/abs/2311.10037 [26] Rémi Robin, Pierre Rouchon, and Lev-Arcady Sellem, ``Unconditionally stable time discretization of Lindblad master equations in infinite dimension using quantum channels'' (2025). arXiv:2503.01712 [27] L.-A. Sellem, A. Sarlette, Z. Leghtas, M. Mirrahimi, P. Rouchon, and P. Campagne-Ibarcq, ``Dissipative Protection of a GKP Qubit in a High-Impedance Superconducting Circuit Driven by a Microwave Frequency Comb'' Phys. Rev. X 15, 011011 (2025). https://doi.org/10.1103/PhysRevX.15.011011 [28] Yu Tong, Victor V. Albert, Jarrod R. McClean, John Preskill, and Yuan Su, ``Provably accurate simulation of gauge theories and bosonic systems'' Quantum 6, 816 (2022). https://doi.org/10.22331/q-2022-09-22-816 [29] Uri Vooland Michel Devoret ``Introduction to quantum electromagnetic circuits'' International Journal of Circuit Theory and Applications 45, 897–934 (2017). https://doi.org/10.1002/cta.2359 [30] M. P. Woods, M. Cramer, and M. B. Plenio, ``Simulating Bosonic Baths with Error Bars'' Phys. Rev. Lett. 115, 130401 (2015). https://doi.org/10.1103/PhysRevLett.115.130401Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-16 10:27:30: Could not fetch cited-by data for 10.22331/q-2026-03-16-2031 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-16 10:27:30: No response from ADS or unable to decode the received json data when getting the list of citing works.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 are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.► BibTeX data@article{Etienney2026posteriorierror, doi = {10.22331/q-2026-03-16-2031}, url = {https://doi.org/10.22331/q-2026-03-16-2031}, title = {A posteriori error estimates for the {L}indblad master equation}, author = {Etienney, Paul-Louis and Robin, R{\'{e}}mi and Rouchon, Pierre}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2031}, month = mar, year = {2026} }► References [1] Daniel Appelöand Yingda Cheng ``Kraus is king: High-order completely positive and trace preserving (CPTP) low rank method for the Lindblad master equation'' Journal of Computational Physics 534, 114036 (2025). https://doi.org/10.1016/j.jcp.2025.114036 https://www.sciencedirect.com/science/article/pii/S0021999125003195 [2] Sahel Ashhab, Felix Fischer, Davide Lonigro, Daniel Braak, and Daniel Burgarth, ``Finite-dimensional approximations of generalized squeezing'' Phys. Rev. A 113, 013703 (2026). https://doi.org/10.1103/9vwp-f35c [3] Rémi Azouit, Alain Sarlette, and Pierre Rouchon, ``Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping'' ESAIM: Control, Optimisation and Calculus of Variations 22, 1353–1369 (2016). https://doi.org/10.1051/cocv/2016050 [4] H.-P. Breuerand F. Petruccione ``The Theory of Open Quantum Systems'' Oxford University Press (2006). https://doi.org/10.1093/acprof:oso/9780199213900.001.0001 [5] Yu Caoand Jianfeng Lu ``Structure-Preserving Numerical Schemes for Lindblad Equations'' Journal of Scientific Computing 102, 27 (2024). https://doi.org/10.1007/s10915-024-02707-x [6] A. M Chebotarevand F Fagnola ``Sufficient Conditions for Conservativity of Minimal Quantum Dynamical Semigroups'' Journal of Functional Analysis 153, 382–404 (1998). https://doi.org/10.1006/jfan.1997.3189 [7] Alexander Chebotarev, J. Garcia, and Roberto Quezada, ``On the Lindblad equation with unbounded time-dependent coefficients'' Mathematical Notes 61, 105–117 (1997). https://doi.org/10.1007/BF02355012 [8] Dariusz Chruscinskiand Saverio Pascazio ``A Brief History of the GKLS Equation'' Open Systems and Information Dynamics 24 (2017). https://doi.org/10.1142/S1230161217400017 [9] E. B. Davies ``Generators of dynamical semigroups'' Journal of Functional Analysis 34, 421–432 (1979). https://doi.org/10.1016/0022-1236(79)90085-5 [10] Felix Fischer, Daniel Burgarth, and Davide Lonigro, ``Quantum particle in the wrong box (or: the perils of finite-dimensional approximations)'' Quantum 10, 1985 (2026). https://doi.org/10.22331/q-2026-01-27-1985 [11] Paul Gondolf, Tim Möbus, and Cambyse Rouzé, ``Energy preserving evolutions over Bosonic systems'' Quantum 8, 1551 (2024). https://doi.org/10.22331/q-2024-12-04-1551 [12] Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan, ``Completely positive dynamical semigroups of N‐level systems'' Journal of Mathematical Physics 17, 821–825 (1976). https://doi.org/10.1063/1.522979 [13] Daniel Gottesman, Alexei Kitaev, and John Preskill, ``Encoding a qubit in an oscillator'' Phys. Rev. A 64, 012310 (2001). https://doi.org/10.1103/PhysRevA.64.012310 [14] Pierre Guilmin, Ronan Gautier, Adrien Bocquet, and Élie Genois, ``Dynamiqs: an open-source Python library for GPU-accelerated and differentiable simulation of quantum systems'' (2024). https://github.com/dynamiqs/dynamiqs [15] Timo Hillmannand Fernando Quijandría ``Quantum error correction with dissipatively stabilized squeezed-cat qubits'' Physical Review A 107 (2023). https://doi.org/10.1103/physreva.107.032423 [16] J.R. Johansson, P.D. Nation, and Franco Nori, ``QuTiP 2: A Python framework for the dynamics of open quantum systems'' Computer Physics Communications 184, 1234–1240 (2013). https://doi.org/10.1016/j.cpc.2012.11.019 https://www.sciencedirect.com/science/article/pii/S0010465512003955 [17] B.D. Josephson ``Possible new effects in superconductive tunnelling'' Physics Letters 1, 251–253 (1962). https://doi.org/10.1016/0031-9163(62)91369-0 https://www.sciencedirect.com/science/article/pii/0031916362913690 [18] A. Kossakowski ``On quantum statistical mechanics of non-Hamiltonian systems'' Reports on Mathematical Physics 3, 247–274 (1972). https://doi.org/10.1016/0034-4877(72)90010-9 https://www.sciencedirect.com/science/article/pii/0034487772900109 [19] Sebastian Krämer, David Plankensteiner, Laurin Ostermann, and Helmut Ritsch, ``QuantumOptics. jl: A Julia framework for simulating open quantum systems'' Computer Physics Communications 227, 109–116 (2018). https://doi.org/10.1016/j.cpc.2018.02.004 [20] G. Lindblad ``On the generators of quantum dynamical semigroups'' Communications in Mathematical Physics 48, 119–130 (1976). https://doi.org/10.1007/BF01608499 [21] Mazyar Mirrahimi, Zaki Leghtas, Victor V. Albert, Steven Touzard, Robert J. Schoelkopf, Liang Jiang, and Michel H. Devoret, ``Dynamically protected cat-qubits: a new paradigm for universal quantum computation'' New Journal of Physics 16, 045014 (2014). https://doi.org/10.1088/1367-2630/16/4/045014 [22] Bo Peng, Yuan Su, Daniel Claudino, Karol Kowalski, Guang Hao Low, and Martin Roetteler, ``Quantum simulation of boson-related Hamiltonians: techniques, effective Hamiltonian construction, and error analysis'' Quantum Science and Technology 10, 023002 (2025). https://doi.org/10.1088/2058-9565/adbf42 [23] Daniel Puzzuoli, Christopher J. Wood, Daniel J. Egger, Benjamin Rosand, and Kento Ueda, ``Qiskit Dynamics: A Python package for simulating the time dynamics of quantum systems'' Journal of Open Source Software 8, 5853 (2023). https://doi.org/10.21105/joss.05853 [24] Rémi Robinand Pierre Rouchon ``Convergence Analysis of Galerkin Approximations for the Lindblad Master Equation'' (2025). arXiv:2510.11416 [25] Rémi Robin, Pierre Rouchon, and Lev-Arcady Sellem, ``Convergence of Bipartite Open Quantum Systems Stabilized by Reservoir Engineering'' Annales Henri Poincaré (2024). https://doi.org/10.1007/s00023-024-01481-8 http://arxiv.org/abs/2311.10037 [26] Rémi Robin, Pierre Rouchon, and Lev-Arcady Sellem, ``Unconditionally stable time discretization of Lindblad master equations in infinite dimension using quantum channels'' (2025). arXiv:2503.01712 [27] L.-A. Sellem, A. Sarlette, Z. Leghtas, M. Mirrahimi, P. Rouchon, and P. Campagne-Ibarcq, ``Dissipative Protection of a GKP Qubit in a High-Impedance Superconducting Circuit Driven by a Microwave Frequency Comb'' Phys. Rev. X 15, 011011 (2025). https://doi.org/10.1103/PhysRevX.15.011011 [28] Yu Tong, Victor V. Albert, Jarrod R. McClean, John Preskill, and Yuan Su, ``Provably accurate simulation of gauge theories and bosonic systems'' Quantum 6, 816 (2022). https://doi.org/10.22331/q-2022-09-22-816 [29] Uri Vooland Michel Devoret ``Introduction to quantum electromagnetic circuits'' International Journal of Circuit Theory and Applications 45, 897–934 (2017). https://doi.org/10.1002/cta.2359 [30] M. P. Woods, M. Cramer, and M. B. Plenio, ``Simulating Bosonic Baths with Error Bars'' Phys. Rev. Lett. 115, 130401 (2015). https://doi.org/10.1103/PhysRevLett.115.130401Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-16 10:27:30: Could not fetch cited-by data for 10.22331/q-2026-03-16-2031 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-16 10:27:30: No response from ADS or unable to decode the received json data when getting the list of citing works.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.