Quantum simulation algorithms based on quantum trajectories
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AbstractQuantum simulation has emerged as a key application of quantum computing, with significant progress made in algorithms for simulating both closed and open quantum systems. The simulation of open quantum systems, particularly those governed by the Lindblad master equation, has received attention recently with the current state-of-the-art algorithms having an input model query complexity of $O(T\mathrm{polylog}(T/\epsilon))$, where $T$ and $\epsilon$ are the desired time and precision of the simulation respectively. For the Hamiltonian simulation problem it has been show that the optimal Hamiltonian query complexity is $O(T + \log(1/\epsilon))$, which is additive in the two parameters, but for Lindbladian simulation this question remains open. In this work we show that the additive query complexity to a Lindbladian's jump operators is reachable for the simulation of a large class of Lindbladians by constructing a novel quantum algorithm based on quantum trajectories.Featured image: Trajectory CircuitPopular summarySimulating quantum mechanical systems has been a key target application for quantum computation ever since the introduction of quantum computation as a concept. This has been due to the difficulty in designing classical algorithms that can simulate quantum systems that are efficient and the belief that simulating quantum systems using a quantum mechanical system should be more "natural". In general quantum systems can be categorized into closed or open. Closed quantum mechanical systems are ones that are completely shielded from noise due to the environment, while open quantum systems are ones that are not shielded from such an environment. Quantum algorithms built to simulate closed quantum systems have been explored first, and it has been only recently that quantum algorithms built to simulate open quantum systems have been explored. When designing a quantum algorithm to simulate a quantum system one needs to give the quantum computer access to the data about the specific quantum mechanical system that one wants to simulate. Typically this data about the system to simulate is encoded in an oracle operation that the quantum algorithm can query. One can then measure how many times the algorithm needs to query this oracle operation as a type of resource cost. It has been shown that for various types of quantum simulation settings, if $T$ is the requested time of simulation, then in the worst-case an algorithm must query this oracle operation at least $T$ times. Such results have been called "no-fast-forwarding" theorems because they imply in the worst-case one cannot simulate a quantum mechanical system "faster" than nature can evolve it. In our work we focus on designing a quantum algorithm that can simulate open quantum mechanical systems that are modeled by the time-independent Lindblad master equation. More specifically, we design a quantum algorithm that can only simulate a restricted class of Lindblad master equations, but achieves a query complexity to the oracle operation encoding the Lindblad master equation that is $O(T)$. In-addition we also find that our algorithm saturates a corresponding "no-fast-forwarding" theorem for the restricted class of Lindblad master equations we considered.► BibTeX data@article{Borras2026quantumsimulation, doi = {10.22331/q-2026-04-13-2063}, url = {https://doi.org/10.22331/q-2026-04-13-2063}, title = {Quantum simulation algorithms based on quantum trajectories}, author = {Borras, Evan and Marvian, Milad}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2063}, month = apr, year = {2026} }► References [1] R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982). https://doi.org/10.1007/BF02650179 [2] D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, Communications in Mathematical Physics 270, 359 (2007). https://doi.org/10.1007/s00220-006-0150-x [3] D. W. Berry, R. Cleve, and S. Gharibian, Gate-efficient discrete simulations of continuous-time quantum query algorithms (2013), arXiv:1211.4637. https://doi.org/10.48550/arXiv.1211.4637 arXiv:1211.4637 [4] D. W. Berry, A. M. Childs, Childs, R. Cleve, R. Kothari, and R. D. Somma, in Proceedings of the forty-sixth annual ACM symposium on Theory of computing (ACM, 2014). https://doi.org/10.1145/2591796.2591854 [5] D. W. Berry, A. M. Childs, and R. Kothari, in 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (2015) pp. 792–809. https://doi.org/10.1109/FOCS.2015.54 [6] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Phys. Rev. Lett. 114, 090502 (2015b). https://doi.org/10.1103/PhysRevLett.114.090502 [7] E. Campbell, Phys. Rev. Lett. 123, 070503 (2019). https://doi.org/10.1103/PhysRevLett.123.070503 [8] A. M. Childs, A. Ostrander, and Y. Su, Quantum 3, 182 (2019). https://doi.org/10.22331/q-2019-09-02-182 [9] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Phys. Rev. X 11, 011020 (2021). https://doi.org/10.1103/PhysRevX.11.011020 [10] S. Lloyd, Science 273, 1073 (1996). https://doi.org/10.1126/science.273.5278.1073 [11] G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017). https://doi.org/10.1103/PhysRevLett.118.010501 [12] G. H. Low and I. L. Chuang, Quantum 3, 163 (2019). https://doi.org/10.22331/q-2019-07-12-163 [13] K. Nakaji, M. Bagherimehrab, and A. Aspuru-Guzik, PRX Quantum 5, 020330 (2024). https://doi.org/10.1103/PRXQuantum.5.020330 [14] D. Poulin, A. Qarry, R. Somma, and F. Verstraete, Phys. Rev. Lett. 106, 170501 (2011). https://doi.org/10.1103/PhysRevLett.106.170501 [15] G. Di Bartolomeo, M. Vischi, T. Feri, A. Bassi, and S. Donadi, Phys. Rev. Res. 6, 043321 (2024). https://doi.org/10.1103/PhysRevResearch.6.043321 [16] H. Chen, B. Li, J. Lu, and L. Ying, Quantum 9, 1917 (2025). https://doi.org/10.22331/q-2025-11-20-1917 [17] A. M. Childs and T. Li, Quantum Info. Comput. 17, 901–947 (2017). https://doi.org/10.26421/QIC17.11-12 [18] R. Cleve and C. Wang, Efficient quantum algorithms for simulating lindblad evolution (2019), arXiv:1612.09512. https://doi.org/10.48550/arXiv.1612.09512 arXiv:1612.09512 [19] I. J. David, I. Sinayskiy, and F. Petruccione, Faster quantum simulation of markovian open quantum systems via randomisation (2024), arXiv:2408.11683. https://doi.org/10.48550/arXiv.2408.11683 arXiv:2408.11683 [20] J. D. Guimarães, J. Lim, M. I. Vasilevskiy, S. F. Huelga, and M. B. Plenio, PRX Quantum 4, 040329 (2023). https://doi.org/10.1103/PRXQuantum.4.040329 [21] J. D. Guimarães, A. Ruiz-Molero, J. Lim, M. I. Vasilevskiy, S. F. Huelga, and M. B. Plenio, Phys. Rev. A 109, 052224 (2024). https://doi.org/10.1103/PhysRevA.109.052224 [22] Z. Hu, R. Xia, and S. Kais, Scientific Reports 10, 3301 (2020). https://doi.org/10.1038/s41598-020-60321-x [23] J. Joo and T. P. Spiller, New Journal of Physics 25, 083041 (2023). https://doi.org/10.1088/1367-2630/acf0e1 [24] M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert, Phys. Rev. Lett. 107, 120501 (2011). https://doi.org/10.1103/PhysRevLett.107.120501 [25] X. Li and C. Wang, in 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 261, edited by K. Etessami, U. Feige, and G. Puppis (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2023) pp. 87:1–87:20. https://doi.org/10.4230/LIPIcs.ICALP.2023.87 [26] X. Li and C. Wang, Communications in Mathematical Physics 401, 147–183 (2023b). https://doi.org/10.1007/s00220-023-04638-4 [27] H.-Y. Liu, X. Lin, Z.-Y. Chen, C. Xue, T.-P. Sun, Q.-S. Li, X.-N. Zhuang, Y.-J. Wang, Y.-C. Wu, M. Gong, and G.-P. Guo, Quantum 9, 1765 (2025). https://doi.org/10.22331/q-2025-06-05-1765 [28] S. Peng, X. Sun, Q. Zhao, and H. Zhou, PRX Quantum 6, 030358 (2025). https://doi.org/10.1103/ssrs-8x32 [29] A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, and D. A. Mazziotti, Phys. Rev. Lett. 127, 270503 (2021). https://doi.org/10.1103/PhysRevLett.127.270503 [30] A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, and D. A. Mazziotti, Phys. Rev. Res. 4, 023216 (2022a). https://doi.org/10.1103/PhysRevResearch.4.023216 [31] A. W. Schlimgen, K. Head-Marsden, L. M. Sager-Smith, P. Narang, and D. A. Mazziotti, Phys. Rev. A 106, 022414 (2022b). https://doi.org/10.1103/PhysRevA.106.022414 [32] N. Suri, J. Barreto, S. Hadfield, N. Wiebe, F. Wudarski, and J. Marshall, Quantum 7, 1002 (2023). https://doi.org/10.22331/q-2023-05-15-1002 [33] E. Borras and M. Marvian, Phys. Rev. Res. 7, 023076 (2025). https://doi.org/10.1103/PhysRevResearch.7.023076 [34] M. Pocrnic, D. Segal, and N. Wiebe, Journal of Physics A: Mathematical and Theoretical 58, 305302 (2025). https://doi.org/10.1088/1751-8121/adebc4 [35] F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Nature Physics 5, 633 (2009). https://doi.org/10.1038/nphys1342 [36] Z. Ding, M. Junge, P. Schleich, and P. Wu, Communications in Mathematical Physics 406, 60 (2025). https://doi.org/10.1007/s00220-025-05240-6 [37] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2007). https://doi.org/10.1093/acprof:oso/9780199213900.001.0001 [38] J. Watrous, Basic notions of quantum information, in The Theory of Quantum Information (Cambridge University Press, 2018) p. 58–123. https://doi.org/10.1017/9781316848142.003 [39] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010). https://doi.org/10.1017/CBO9780511976667 [40] C.-F. Chen, M. J. Kastoryano, F. G. S. L. Brandão, and A. Gilyén, Quantum thermal state preparation (2023a), arXiv:2303.18224. https://doi.org/10.48550/arXiv.2303.18224 arXiv:2303.18224 [41] C.-F. Chen, M. J. Kastoryano, and A. Gilyén, An efficient and exact noncommutative quantum gibbs sampler (2023b), arXiv:2311.09207. https://doi.org/10.48550/arXiv.2311.09207 arXiv:2311.09207 [42] O. Oreshkov, Continuous-time quantum error correction, in Quantum Error Correction, edited by D. A. Lidar and T. A. Brun (Cambridge University Press, 2013) p. 201–228. https://doi.org/10.1017/CBO9781139034807.010 [43] V. Tripathi, H. Chen, M. Khezri, K.-W. Yip, E. Levenson-Falk, and D. A. Lidar, Phys. Rev. Appl. 18, 024068 (2022). https://doi.org/10.1103/PhysRevApplied.18.024068 [44] A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC ’19 (ACM, 2019). https://doi.org/10.1145/3313276.3316366 [45] M. Mitzenmacher and E. Upfal, Balls, bins, and random graphs, in Probability and Computing: Randomized Algorithms and Probabilistic Analysis (Cambridge University Press, 2005) p. 90–125. https://doi.org/10.1017/CBO9780511813603.006 [46] M. Gao, Z. Ji, and C. Liu, Lévy-khintchine structure enables fast-forwardable lindbladian simulation (2026), arXiv:2511.10253. https://doi.org/10.48550/arXiv.2511.10253 arXiv:2511.10253 [47] F. vom Ende, Open Systems & Information Dynamics 30, 2350003 (2023). https://doi.org/10.1142/S1230161223500038 [48] Z.-X. Shang, D. An, and C. Shao, Exponential lindbladian fast forwarding and exponential amplification of certain gibbs state properties (2025), arXiv:2509.09517. https://doi.org/10.48550/arXiv.2509.09517 arXiv:2509.09517Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-13 18:18:15: Could not fetch cited-by data for 10.22331/q-2026-04-13-2063 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-13 18:18:15: 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. AbstractQuantum simulation has emerged as a key application of quantum computing, with significant progress made in algorithms for simulating both closed and open quantum systems. The simulation of open quantum systems, particularly those governed by the Lindblad master equation, has received attention recently with the current state-of-the-art algorithms having an input model query complexity of $O(T\mathrm{polylog}(T/\epsilon))$, where $T$ and $\epsilon$ are the desired time and precision of the simulation respectively. For the Hamiltonian simulation problem it has been show that the optimal Hamiltonian query complexity is $O(T + \log(1/\epsilon))$, which is additive in the two parameters, but for Lindbladian simulation this question remains open. In this work we show that the additive query complexity to a Lindbladian's jump operators is reachable for the simulation of a large class of Lindbladians by constructing a novel quantum algorithm based on quantum trajectories.Featured image: Trajectory CircuitPopular summarySimulating quantum mechanical systems has been a key target application for quantum computation ever since the introduction of quantum computation as a concept. This has been due to the difficulty in designing classical algorithms that can simulate quantum systems that are efficient and the belief that simulating quantum systems using a quantum mechanical system should be more "natural". In general quantum systems can be categorized into closed or open. Closed quantum mechanical systems are ones that are completely shielded from noise due to the environment, while open quantum systems are ones that are not shielded from such an environment. Quantum algorithms built to simulate closed quantum systems have been explored first, and it has been only recently that quantum algorithms built to simulate open quantum systems have been explored. When designing a quantum algorithm to simulate a quantum system one needs to give the quantum computer access to the data about the specific quantum mechanical system that one wants to simulate. Typically this data about the system to simulate is encoded in an oracle operation that the quantum algorithm can query. One can then measure how many times the algorithm needs to query this oracle operation as a type of resource cost. It has been shown that for various types of quantum simulation settings, if $T$ is the requested time of simulation, then in the worst-case an algorithm must query this oracle operation at least $T$ times. Such results have been called "no-fast-forwarding" theorems because they imply in the worst-case one cannot simulate a quantum mechanical system "faster" than nature can evolve it. In our work we focus on designing a quantum algorithm that can simulate open quantum mechanical systems that are modeled by the time-independent Lindblad master equation. More specifically, we design a quantum algorithm that can only simulate a restricted class of Lindblad master equations, but achieves a query complexity to the oracle operation encoding the Lindblad master equation that is $O(T)$. In-addition we also find that our algorithm saturates a corresponding "no-fast-forwarding" theorem for the restricted class of Lindblad master equations we considered.► BibTeX data@article{Borras2026quantumsimulation, doi = {10.22331/q-2026-04-13-2063}, url = {https://doi.org/10.22331/q-2026-04-13-2063}, title = {Quantum simulation algorithms based on quantum trajectories}, author = {Borras, Evan and Marvian, Milad}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2063}, month = apr, year = {2026} }► References [1] R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982). https://doi.org/10.1007/BF02650179 [2] D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, Communications in Mathematical Physics 270, 359 (2007). https://doi.org/10.1007/s00220-006-0150-x [3] D. W. Berry, R. Cleve, and S. Gharibian, Gate-efficient discrete simulations of continuous-time quantum query algorithms (2013), arXiv:1211.4637. https://doi.org/10.48550/arXiv.1211.4637 arXiv:1211.4637 [4] D. W. Berry, A. M. Childs, Childs, R. Cleve, R. Kothari, and R. D. Somma, in Proceedings of the forty-sixth annual ACM symposium on Theory of computing (ACM, 2014). https://doi.org/10.1145/2591796.2591854 [5] D. W. Berry, A. M. Childs, and R. Kothari, in 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (2015) pp. 792–809. https://doi.org/10.1109/FOCS.2015.54 [6] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Phys. Rev. Lett. 114, 090502 (2015b). https://doi.org/10.1103/PhysRevLett.114.090502 [7] E. Campbell, Phys. Rev. Lett. 123, 070503 (2019). https://doi.org/10.1103/PhysRevLett.123.070503 [8] A. M. Childs, A. Ostrander, and Y. Su, Quantum 3, 182 (2019). https://doi.org/10.22331/q-2019-09-02-182 [9] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Phys. Rev. X 11, 011020 (2021). https://doi.org/10.1103/PhysRevX.11.011020 [10] S. Lloyd, Science 273, 1073 (1996). https://doi.org/10.1126/science.273.5278.1073 [11] G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017). https://doi.org/10.1103/PhysRevLett.118.010501 [12] G. H. Low and I. L. Chuang, Quantum 3, 163 (2019). https://doi.org/10.22331/q-2019-07-12-163 [13] K. Nakaji, M. Bagherimehrab, and A. Aspuru-Guzik, PRX Quantum 5, 020330 (2024). https://doi.org/10.1103/PRXQuantum.5.020330 [14] D. Poulin, A. Qarry, R. Somma, and F. Verstraete, Phys. Rev. Lett. 106, 170501 (2011). https://doi.org/10.1103/PhysRevLett.106.170501 [15] G. Di Bartolomeo, M. Vischi, T. Feri, A. Bassi, and S. Donadi, Phys. Rev. Res. 6, 043321 (2024). https://doi.org/10.1103/PhysRevResearch.6.043321 [16] H. Chen, B. Li, J. Lu, and L. Ying, Quantum 9, 1917 (2025). https://doi.org/10.22331/q-2025-11-20-1917 [17] A. M. Childs and T. Li, Quantum Info. Comput. 17, 901–947 (2017). https://doi.org/10.26421/QIC17.11-12 [18] R. Cleve and C. Wang, Efficient quantum algorithms for simulating lindblad evolution (2019), arXiv:1612.09512. https://doi.org/10.48550/arXiv.1612.09512 arXiv:1612.09512 [19] I. J. David, I. Sinayskiy, and F. Petruccione, Faster quantum simulation of markovian open quantum systems via randomisation (2024), arXiv:2408.11683. https://doi.org/10.48550/arXiv.2408.11683 arXiv:2408.11683 [20] J. D. Guimarães, J. Lim, M. I. Vasilevskiy, S. F. Huelga, and M. B. Plenio, PRX Quantum 4, 040329 (2023). https://doi.org/10.1103/PRXQuantum.4.040329 [21] J. D. Guimarães, A. Ruiz-Molero, J. Lim, M. I. Vasilevskiy, S. F. Huelga, and M. B. Plenio, Phys. Rev. A 109, 052224 (2024). https://doi.org/10.1103/PhysRevA.109.052224 [22] Z. Hu, R. Xia, and S. Kais, Scientific Reports 10, 3301 (2020). https://doi.org/10.1038/s41598-020-60321-x [23] J. Joo and T. P. Spiller, New Journal of Physics 25, 083041 (2023). https://doi.org/10.1088/1367-2630/acf0e1 [24] M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert, Phys. Rev. Lett. 107, 120501 (2011). https://doi.org/10.1103/PhysRevLett.107.120501 [25] X. Li and C. Wang, in 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 261, edited by K. Etessami, U. Feige, and G. Puppis (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2023) pp. 87:1–87:20. https://doi.org/10.4230/LIPIcs.ICALP.2023.87 [26] X. Li and C. Wang, Communications in Mathematical Physics 401, 147–183 (2023b). https://doi.org/10.1007/s00220-023-04638-4 [27] H.-Y. Liu, X. Lin, Z.-Y. Chen, C. Xue, T.-P. Sun, Q.-S. Li, X.-N. Zhuang, Y.-J. Wang, Y.-C. Wu, M. Gong, and G.-P. Guo, Quantum 9, 1765 (2025). https://doi.org/10.22331/q-2025-06-05-1765 [28] S. Peng, X. Sun, Q. Zhao, and H. Zhou, PRX Quantum 6, 030358 (2025). https://doi.org/10.1103/ssrs-8x32 [29] A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, and D. A. Mazziotti, Phys. Rev. Lett. 127, 270503 (2021). https://doi.org/10.1103/PhysRevLett.127.270503 [30] A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, and D. A. Mazziotti, Phys. Rev. Res. 4, 023216 (2022a). https://doi.org/10.1103/PhysRevResearch.4.023216 [31] A. W. Schlimgen, K. Head-Marsden, L. M. Sager-Smith, P. Narang, and D. A. Mazziotti, Phys. Rev. A 106, 022414 (2022b). https://doi.org/10.1103/PhysRevA.106.022414 [32] N. Suri, J. Barreto, S. Hadfield, N. Wiebe, F. Wudarski, and J. Marshall, Quantum 7, 1002 (2023). https://doi.org/10.22331/q-2023-05-15-1002 [33] E. Borras and M. Marvian, Phys. Rev. Res. 7, 023076 (2025). https://doi.org/10.1103/PhysRevResearch.7.023076 [34] M. Pocrnic, D. Segal, and N. Wiebe, Journal of Physics A: Mathematical and Theoretical 58, 305302 (2025). https://doi.org/10.1088/1751-8121/adebc4 [35] F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Nature Physics 5, 633 (2009). https://doi.org/10.1038/nphys1342 [36] Z. Ding, M. Junge, P. Schleich, and P. Wu, Communications in Mathematical Physics 406, 60 (2025). https://doi.org/10.1007/s00220-025-05240-6 [37] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2007). https://doi.org/10.1093/acprof:oso/9780199213900.001.0001 [38] J. Watrous, Basic notions of quantum information, in The Theory of Quantum Information (Cambridge University Press, 2018) p. 58–123. https://doi.org/10.1017/9781316848142.003 [39] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010). https://doi.org/10.1017/CBO9780511976667 [40] C.-F. Chen, M. J. Kastoryano, F. G. S. L. Brandão, and A. Gilyén, Quantum thermal state preparation (2023a), arXiv:2303.18224. https://doi.org/10.48550/arXiv.2303.18224 arXiv:2303.18224 [41] C.-F. Chen, M. J. Kastoryano, and A. Gilyén, An efficient and exact noncommutative quantum gibbs sampler (2023b), arXiv:2311.09207. https://doi.org/10.48550/arXiv.2311.09207 arXiv:2311.09207 [42] O. Oreshkov, Continuous-time quantum error correction, in Quantum Error Correction, edited by D. A. Lidar and T. A. Brun (Cambridge University Press, 2013) p. 201–228. https://doi.org/10.1017/CBO9781139034807.010 [43] V. Tripathi, H. Chen, M. Khezri, K.-W. Yip, E. Levenson-Falk, and D. A. Lidar, Phys. Rev. Appl. 18, 024068 (2022). https://doi.org/10.1103/PhysRevApplied.18.024068 [44] A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC ’19 (ACM, 2019). https://doi.org/10.1145/3313276.3316366 [45] M. Mitzenmacher and E. Upfal, Balls, bins, and random graphs, in Probability and Computing: Randomized Algorithms and Probabilistic Analysis (Cambridge University Press, 2005) p. 90–125. https://doi.org/10.1017/CBO9780511813603.006 [46] M. Gao, Z. Ji, and C. Liu, Lévy-khintchine structure enables fast-forwardable lindbladian simulation (2026), arXiv:2511.10253. https://doi.org/10.48550/arXiv.2511.10253 arXiv:2511.10253 [47] F. vom Ende, Open Systems & Information Dynamics 30, 2350003 (2023). https://doi.org/10.1142/S1230161223500038 [48] Z.-X. Shang, D. An, and C. Shao, Exponential lindbladian fast forwarding and exponential amplification of certain gibbs state properties (2025), arXiv:2509.09517. https://doi.org/10.48550/arXiv.2509.09517 arXiv:2509.09517Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-13 18:18:15: Could not fetch cited-by data for 10.22331/q-2026-04-13-2063 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-13 18:18:15: 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.