Quantum simulation algorithms based on quantum trajectories

Evan Borras; Milad Marvian

doi:10.22331/q-2026-04-13-2063

Quantum simulation algorithms based on quantum trajectories

Evan Borras^1,2 and Milad Marvian^1,2,3

¹Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA
²Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA
³Department of Electrical & Computer Engineering, University of New Mexico, Albuquerque, NM 87131, USA

Published:	2026-04-13, volume 10, page 2063
Editor:	Changpeng Shao
Eprint:	arXiv:2509.10425v2
Doi:	https://doi.org/10.22331/q-2026-04-13-2063
Citation:	Quantum 10, 2063 (2026).

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Abstract

Quantum 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 Circuit

Popular summary

Simulating 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}
}

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Cited by

[1] Shakib Daryanoosh, "Non-normality and dissipation in Markovian quantum dynamics: Implications for quantum simulation", Physical Review A 114 2, 022406 (2026).

[2] Zhong-Xia Shang, Dong An, and Changpeng Shao, "Exponential Lindbladian fast forwarding and exponential amplification of certain Gibbs state properties", Reports on Progress in Physics 89 5, 057602 (2026).

[3] Yu Cao, Mingfeng He, and Xiantao Li, "Dynamically optimal unraveling schemes for simulating Lindblad equations", Journal of Physics A Mathematical General 59 16, 165301 (2026).

[4] Anjali A. Agrawal, Evan Budd, Alexander F. Kemper, Vladimir V. Skokov, Andrey Tarasov, and Shaswat Tiwari, "JIMWLK on a quantum computer", Physical Review D 113 11, 114047 (2026).

[5] Shakib Daryanoosh, "Non-normality and dissipation in Markovian quantum dynamics: Implications for quantum simulation", arXiv:2604.16869, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 23:43:06) and SAO/NASA ADS (last updated successfully 2026-08-13 23:43:07). The list may be incomplete as not all publishers provide suitable and complete citation data.

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