Improved sample complexity bound for sample-based Lindbladian simulation

Byeongseon Go; Dhrumil Patel; Hyukjoon Kwon; Mark M. Wilde; Siheon Park; Youngjin Seo

arxiv: 2605.30301 · v1 · pith:KHC4RMYPnew · submitted 2026-05-28 · 🪐 quant-ph

Improved sample complexity bound for sample-based Lindbladian simulation

Siheon Park , Youngjin Seo , Byeongseon Go , Dhrumil Patel , Mark M. Wilde , Hyukjoon Kwon This is my paper

Pith reviewed 2026-06-29 06:50 UTC · model grok-4.3

classification 🪐 quant-ph

keywords sample complexityLindbladian simulationWave Matrix Lindbladizationquantum algorithmsopen quantum systemsdimension dependencetypical case analysis

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The pith

The Wave Matrix Lindbladization algorithm requires at most roughly (d/4) times the squared norm of the jump operator times t squared over error samples to simulate Lindbladian dynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an explicit upper bound on the number of samples needed by the Wave Matrix Lindbladization algorithm that scales linearly with the dimension d of the jump operator rather than quadratically. This bound holds for any simulation time t and target error ε. A sympathetic reader would care because lower sample counts translate directly into fewer experimental runs on quantum hardware for open-system simulations. The authors also establish that the dimension factor disappears entirely when the jump operator satisfies a mild norm condition that holds for typical random cases, while a matching lower bound of order d t squared over ε applies in the worst case.

Core claim

For a jump operator L with dimension d, we derive an explicit non-asymptotic sample complexity bound n_d^*(t,ε) ≤ ((2d+3)/8) ||L||_∞² (t²/ε). This refines the dimension dependence of the best previously known bound O(d² t²/ε). When ||L||_∞² = O(1/d), satisfied with high probability for random Lindblad operators, the typical-case sample complexity is O(t²/ε). In the worst case WML necessarily requires Ω(d t²/ε) samples, shown by an explicit rank-one example.

What carries the argument

The Wave Matrix Lindbladization algorithm together with its non-asymptotic sample-complexity analysis that produces the linear-in-d upper bound.

If this is right

Lindbladian simulation can be performed with sample counts that grow only linearly rather than quadratically in system dimension.
Random jump operators allow the entire dimension overhead to be removed, leaving a bound independent of d.
Adversarial rank-one operators force the linear dimension factor to remain necessary.
The sharp separation between typical and worst-case regimes applies directly to resource estimates for open-system quantum algorithms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If physical jump operators in real devices frequently meet the O(1/d) norm condition, then WML could scale to higher-dimensional systems without extra sampling cost.
Protocol designers might deliberately choose or approximate operators to avoid the adversarial rank-one regime and thereby realize the typical-case savings.
Similar typical-versus-worst-case gaps may exist for other sample-based quantum simulation methods and could be uncovered by parallel analysis.

Load-bearing premise

The improved upper bound and the typical/worst-case split rest on the correctness and sample properties of the Wave Matrix Lindbladization algorithm as previously established.

What would settle it

A concrete counter-example Lindblad operator for which the minimal number of samples required exceeds ((2d+3)/8) ||L||_∞² (t²/ε) would falsify the claimed upper bound.

read the original abstract

We establish improved sample-complexity bounds for sample-based Lindbladian simulation based on the Wave Matrix Lindbladization (WML) algorithm. For a jump operator $L$ with dimension $d$, we derive an explicit non-asymptotic sample complexity bound $n_d^*(t,\varepsilon) \le \left( \frac{2d+3}{8} \right) \|L\|_\infty^2 \left( \frac{t^2}{\varepsilon} \right)$, holding for simulation time $t$ and error $\varepsilon$. This refines the dimension dependence of the best previously known bound, $O(d^2 t^2/\varepsilon)$, from [Go et al., Quantum Sci. Tech. 10, 045058 (2025)]. Remarkably, we show that this dimensional overhead can be entirely avoided when $\| L\|_\infty^2 = O(1/d)$, a condition satisfied with high probability for random Lindblad operators, yielding a typical-case sample complexity of $O(t^2/\varepsilon)$. On the other hand, in the worst case, we show that WML necessarily requires $\Omega(dt^2/\varepsilon)$ samples by constructing an explicit example with a rank-one Lindblad operator. Our results reveal a sharp dichotomy between typical and adversarial sample complexities in Lindbladian simulation, thereby strengthening the theoretical foundations of sample-based quantum algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Explicit linear-in-d bound with (2d+3)/8 factor plus typical/worst-case split, but the whole thing plugs in the sample complexity from the 2025 Go et al. WML paper without re-deriving it.

read the letter

This paper gives an explicit non-asymptotic sample complexity bound of ((2d+3)/8) ||L||_∞² (t²/ε) for Lindbladian simulation via Wave Matrix Lindbladization, improving on the earlier O(d² t²/ε) scaling. It also shows that the dimensional overhead vanishes in the typical case when ||L||_∞² = O(1/d), which holds for random operators, and supplies a rank-one construction proving that Omega(d t²/ε) samples are required in the worst case.

The concrete prefactor and the typical/worst-case split are the actual additions. The explicit constant makes resource estimates more usable, and the dichotomy clarifies when practitioners can expect better scaling without extra d factors. The worst-case example is a useful check because it demonstrates necessity rather than just an upper bound.

The analysis derives the new bound by substituting the sample-complexity guarantees of the WML algorithm from the cited 2025 Go et al. paper. No independent re-proof or fresh derivation of those base properties appears here, so the improvement stands or falls with the correctness of the earlier work. The abstract states the results cleanly, but any gaps in how the WML complexity was originally established would carry through.

This is for people who need concrete sample counts in open-system quantum simulation, such as those doing resource estimates for quantum chemistry or condensed-matter models. A reader focused on algorithm bounds would get direct value from the sharpened numbers and the typical-case observation.

The work shows clear thinking on the literature and adds precision through the explicit factor and the construction, so it deserves a serious referee. I would send it out for peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to derive an improved explicit non-asymptotic sample complexity bound for sample-based Lindbladian simulation via the Wave Matrix Lindbladization (WML) algorithm. For a d-dimensional jump operator L it states n_d^*(t,ε) ≤ ((2d+3)/8) ||L||_∞² (t²/ε), refining the prior O(d² t²/ε) bound. It further asserts a typical-case complexity of O(t²/ε) when ||L||_∞² = O(1/d) (satisfied with high probability for random operators) and a worst-case lower bound of Ω(d t²/ε) via an explicit rank-one construction, revealing a typical/worst-case dichotomy.

Significance. If the central derivation holds, the explicit linear-in-d prefactor, the non-asymptotic form, and the concrete lower-bound construction constitute a clear strengthening of the theoretical foundations for sample-based Lindbladian simulation. The typical/worst-case distinction supplies a falsifiable prediction that can be checked numerically. The result's significance is nevertheless conditional on the sample-complexity guarantees of WML as stated in the cited prior work.

major comments (1)

Abstract (and the substitution step that produces the headline bound): the stated improvement n_d^*(t,ε) ≤ ((2d+3)/8) ||L||_∞² (t²/ε) and the typical/worst-case dichotomy are obtained solely by substituting the sample-complexity guarantees of the WML algorithm as reported in Go et al. (Quantum Sci. Tech. 10, 045058, 2025) into a new analysis. No independent derivation or re-proof of those WML guarantees appears in the present manuscript, rendering the claimed d-linear improvement conditional on the correctness of the earlier, unexamined analysis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the dependence on prior work. We address the major comment below and agree that a clarification is warranted.

read point-by-point responses

Referee: Abstract (and the substitution step that produces the headline bound): the stated improvement n_d^*(t,ε) ≤ ((2d+3)/8) ||L||_∞² (t²/ε) and the typical/worst-case dichotomy are obtained solely by substituting the sample-complexity guarantees of the WML algorithm as reported in Go et al. (Quantum Sci. Tech. 10, 045058, 2025) into a new analysis. No independent derivation or re-proof of those WML guarantees appears in the present manuscript, rendering the claimed d-linear improvement conditional on the correctness of the earlier, unexamined analysis.

Authors: We agree that the headline bound is obtained by substituting the WML sample-complexity guarantees reported in Go et al. into a new analysis that improves the dimension dependence and derives the explicit prefactor together with the typical/worst-case dichotomy. The present manuscript contains no independent derivation or re-proof of the underlying WML guarantees. We will revise the abstract and the opening paragraphs of the introduction to state explicitly that the improved bound relies on the sample-complexity result of Go et al. (2025). revision: yes

Circularity Check

0 steps flagged

No significant circularity: new bounds and dichotomy derived via independent analysis on top of cited base algorithm

full rationale

The manuscript derives an explicit improved prefactor ((2d+3)/8) and establishes the typical-case O(t²/ε) vs. worst-case Ω(d t²/ε) dichotomy through new probabilistic arguments on random Lindblad operators and an explicit rank-one counterexample construction. These steps are self-contained within the present paper and do not reduce by definition or substitution to the inputs of the cited WML properties; the citation supplies only the base algorithm whose sample-complexity guarantees are treated as an external black box. Self-citation to overlapping-author prior work is present but not load-bearing for the novel claims, satisfying the criteria for a non-circular finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the correctness of the WML algorithm from prior literature and standard assumptions about Lindblad operators and sample-based simulation in quantum information; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The Wave Matrix Lindbladization algorithm correctly implements sample-based Lindbladian simulation with the sample complexity properties used to derive the bound.
The new bound is obtained by refining the analysis of the WML procedure defined in the cited 2025 paper.

pith-pipeline@v0.9.1-grok · 5794 in / 1348 out tokens · 39272 ms · 2026-06-29T06:50:48.011535+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

85 extracted references · 9 canonical work pages · 2 internal anchors

[1]

Exact Lindbladian action 26
[2]

Exactn-step WML state on the invariant two-dimensional space 27
[3]

Trace-distance lower bound 28
[4]

Special case

Asymptotic expansion ofz n 29 E. Alternative realization of WML via Stinespring isometry 30 I. INTRODUCTION The accurate simulation of quantum systems is a cornerstone of quantum information science [1–5], with broad implications ranging from quantum chemistry [6–9] to condensed- matter physics [10, 11]. Hamiltonian simulation [12–15] has been used to mod...

2024
[5]

Breuer and F

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, Oxford ; New York, 2002)

2002
[6]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, 2010)

2010
[7]

H. M. Wiseman and G. J. Milburn,Quantum Measurement and Control(Cambridge University Press, Cambridge, 2009)

2009
[8]

Weiss,Quantum Dissipative Systems, 4th ed

U. Weiss,Quantum Dissipative Systems, 4th ed. (World Scientific, 2012)

2012
[9]

R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics21, 467 (1982)

1982
[10]

Wecker, B

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, Gate-count estimates for performing quantum chemistry on small quantum computers, Physical Review A90, 022305 (2014)

2014
[11]

Babbush, J

R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation, Physical Review A91, 022311 (2015)

2015
[12]

Reiher, N

M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Elucidating reaction mechanisms on quantum computers, Proceedings of the National Academy of Sciences of the United States of America114, 7555 (2017)

2017
[13]

Poulin, M

D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doberty, and M. Troyer, The Trotter step size required for accurate quantum simulation of quantum chemistry, Quantum Information and Computation15, 361 (2015)

2015
[14]

Raeisi, N

S. Raeisi, N. Wiebe, and B. C. Sanders, Quantum-circuit design for efficient simulations of many-body quantum dynamics, New Journal of Physics14, 103017 (2012)

2012
[15]

A. J. Daley, Quantum trajectories and open many-body quantum systems, Advances in Physics 63, 77 (2014)

2014
[16]

G. H. Low and I. L. Chuang, Optimal Hamiltonian Simulation by Quantum Signal Processing, Physical Review Letters118, 010501 (2017)

2017
[17]

D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating Hamiltonian dynamics with a truncated Taylor series, Physical Review Letters114, 090502 (2015)

2015
[18]

G. H. Low and I. L. Chuang, Hamiltonian Simulation by Qubitization, Quantum3, 163 (2019). 17

2019
[19]

A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Toward the first quantum simulation with quantum speedup, Proceedings of the National Academy of Sciences of the United States of America115, 9456 (2018)

2018
[20]

Gorini, A

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups ofN-level systems, Journal of Mathematical Physics17, 821 (1976)

1976
[21]

Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics48, 119 (1976)

G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics48, 119 (1976)

1976
[22]

J. F. Poyatos, J. I. Cirac, and P. Zoller, Quantum Reservoir Engineering with Laser Cooled Trapped Ions, Physical Review Letters77, 4728 (1996)

1996
[23]

T. S. Cubitt, A. Lucia, S. Michalakis, and D. Perez-Garcia, Stability of Local Quantum Dissipative Systems, Communications in Mathematical Physics337, 1275 (2015)

2015
[24]

Y. Zhan, Z. Ding, J. Huhn, J. Gray, J. Preskill, G. K.-L. Chan, and L. Lin, Rapid Quantum Ground State Preparation via Dissipative Dynamics, Physical Review X16, 011004 (2026)

2026
[25]

Van Den Berg, Z

E. Van Den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Nature Physics19, 1116 (2023)

2023
[26]

S. Jin, N. Liu, and Y. Yu, Quantum Simulation of Partial Differential Equations via Schr¨ odingerization, Physical Review Letters133, 230602 (2024)

2024
[27]

Lin, Dissipative preparation of many-body quantum states: Toward practical quantum advantage, APL Computational Physics1, 010901 (2025)

L. Lin, Dissipative preparation of many-body quantum states: Toward practical quantum advantage, APL Computational Physics1, 010901 (2025)

2025
[28]

Simulating Markovian open quantum systems using higher-order series expansion

X. Li and C. Wang, Simulating Markovian open quantum systems using higher-order series expansion (2023), arXiv:2212.02051

work page Pith review arXiv 2023
[29]

Burdine, N

C. Burdine, N. Bauer, G. Siopsis, and E. P. Blair, Efficient Simulation of Open Quantum Systems on NISQ Trapped-Ion Hardware (2024), arXiv:2410.10732

work page arXiv 2024
[30]

A. M. Childs and T. Li, Efficient simulation of sparse Markovian quantum dynamics, Quantum Information and Computation17, 0901 (2023)

2023
[31]

J. Kato, K. Wada, K. Ito, and N. Yamamoto, Exponentially accurate open quantum simulation via randomized dissipation with minimal ancilla (2024), arXiv:2412.19453

work page arXiv 2024
[32]

S. Peng, X. Sun, Q. Zhao, and H. Zhou, Quantum-trajectory-inspired Lindbladian simulation, PRX Quantum6, 030358 (2025)

2025
[33]

H. Chen, B. Li, J. Lu, and L. Ying, A randomized method for simulating Lindblad equations and thermal state preparation, Quantum9, 1917 (2025)

1917
[34]

Dalibard, Y

J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Physical Review Letters68, 580 (1992)

1992
[35]

R. Dum, P. Zoller, and H. Ritsch, Monte Carlo simulation of the atomic master equation for spontaneous emission, Physical Review A45, 4879 (1992)

1992
[36]

H. Carmichael,An Open Systems Approach to Quantum Optics: Lectures Presented at the Universit´ e Libre de Bruxelles October 28 to November 4, 1991, Lecture Notes in Physics Monographs, Vol. 18 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1993)

1991
[37]

Y. Cao, M. He, and X. Li, Dynamically Optimal Unraveling Schemes for Simulating Lindblad Equations (2025), arXiv:2509.19887

work page internal anchor Pith review Pith/arXiv arXiv 2025
[38]

Cleve and C

R. Cleve and C. Wang, Efficient quantum algorithms for simulating Lindblad evolution, in44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 80 (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2017) pp. 17:1–17:14

2017
[39]

Z. Ding, X. Li, and L. Lin, Simulating open quantum systems using Hamiltonian simulations, 18 PRX Quantum5, 020332 (2024)

2024
[40]

G. H. Low and R. D. Somma, Optimal quantum simulation of linear non-unitary dynamics (2025), arXiv:2508.19238

work page arXiv 2025
[41]

Kamakari, S.-N

H. Kamakari, S.-N. Sun, M. Motta, and A. J. Minnich, Digital quantum simulation of open quantum systems using quantum imaginary time evolution, PRX Quantum3, 010320 (2022)

2022
[42]

A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, and D. A. Mazziotti, Quantum simulation of the Lindblad equation using a unitary decomposition of operators, Physical Review Research4, 023216 (2022)

2022
[43]

Patel and M

D. Patel and M. M. Wilde, Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics, Open Systems & Information Dynamics30, 2350010 (2023)

2023
[44]

Patel and M

D. Patel and M. M. Wilde, Wave Matrix Lindbladization II: General Lindbladians, Linear Combinations, and Polynomials, Open Systems & Information Dynamics30, 2350014 (2023)

2023
[45]

Lloyd, M

S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum principal component analysis, Nature Physics10, 631 (2014)

2014
[46]

Kimmel, C

S. Kimmel, C. Y.-Y. Lin, G. H. Low, M. Ozols, and T. J. Yoder, Hamiltonian simulation with optimal sample complexity, npj Quantum Information3, 1 (2017)

2017
[47]

F. Wei, Z. Liu, G. Liu, Z. Han, D.-L. Deng, and Z. Liu, Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps, npj Quantum Information10, 1 (2024)

2024
[48]

B. Go, H. Kwon, S. Park, D. Patel, and M. M. Wilde, Sample-based Hamiltonian and Lindbladian simulation: Non-asymptotic analysis of sample complexity, Quantum Science and Technology10, 045058 (2025)

2025
[49]

Kliesch, T

M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert, Dissipative Quantum Church-Turing Theorem, Physical Review Letters107, 120501 (2011)

2011
[50]

Z. Xu, L. P. Garc´ ıa-Pintos, A. Chenu, and A. Del Campo, Extreme Decoherence and Quantum Chaos, Physical Review Letters122, 014103 (2019)

2019
[51]

J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample-optimal tomography of quantum states, inProceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’16 (Association for Computing Machinery, New York, NY, USA, 2016) pp. 913–925

2016
[52]

J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample-optimal tomography of quantum states, IEEE Transactions on Information Theory63, 5628 (2017)

2017
[53]

Yuen, An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography, Quantum7, 890 (2023)

H. Yuen, An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography, Quantum7, 890 (2023)

2023
[54]

Anshu and S

A. Anshu and S. Arunachalam, A survey on the complexity of learning quantum states, Nature Reviews Physics6, 59 (2023)

2023
[55]

M. M. Wilde,Quantum Information Theory, 2nd ed. (Cambridge University Press, Cambridge, 2017)

2017
[56]

transition probability

A. Uhlmann, The “transition probability” in the state space of a∗-algebra, Reports on Mathematical Physics9, 273 (1976)

1976
[57]

A. Y. Kitaev, Quantum computations: Algorithms and error correction, Russian Mathematical Surveys52, 1191 (1997)

1997
[58]

M. A. Nielsen and I. L. Chuang, Programmable Quantum Gate Arrays, Physical Review Letters79, 321 (1997)

1997
[59]

Banchi, J

L. Banchi, J. Pereira, S. Lloyd, and S. Pirandola, Convex optimization of programmable quantum computers, npj Quantum Information6, 1 (2020)

2020
[60]

M. Jing, M. Guo, L. Zhu, H. Yao, and X. Wang, Programmable Open Quantum Systems 19 (2025), arXiv:2512.08279

work page arXiv 2025
[61]

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed State Entanglement and Quantum Error Correction, Physical Review A54, 3824 (1996)

1996
[62]

Bravyi and A

S. Bravyi and A. Kitaev, Universal Quantum Computation with ideal Clifford gates and noisy ancillas, Physical Review A71, 022316 (2005)

2005
[63]

X. Yuan, B. Regula, R. Takagi, and M. Gu, Virtual quantum resource distillation, Physical Review Letters132, 050203 (2024), 2303.00955

work page arXiv 2024
[64]

Aaronson, Quantum Copy-Protection and Quantum Money, in2009 24th Annual IEEE Conference on Computational Complexity(IEEE, Paris, France, 2009) pp

S. Aaronson, Quantum Copy-Protection and Quantum Money, in2009 24th Annual IEEE Conference on Computational Complexity(IEEE, Paris, France, 2009) pp. 229–242

2009
[65]

A. N. Sims, D. Patel, A. Philip, A. H. Rubin, R. Bandyopadhyay, M. Radulaski, and M. M. Wilde, Digital quantum simulations of the nonresonant open Tavis-Cummings model, Physical Review Research7, 043302 (2025)

2025
[66]

Rigol, V

M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452, 854 (2008)

2008
[67]

Bruzda, V

W. Bruzda, V. Cappellini, H.-J. Sommers, and K. ˙Zyczkowski, Random Quantum Operations, Physics Letters A373, 320 (2009)

2009
[68]

Denisov, T

S. Denisov, T. Laptyeva, W. Tarnowski, D. Chru´ sci´ nski, and K.˙Zyczkowski, Universal Spectra of Random Lindblad Operators, Physical Review Letters123, 140403 (2019)

2019
[69]

Kukulski, I

R. Kukulski, I. Nechita, L. Pawela, Z. Pucha la, and K. ˙Zyczkowski, Generating random quantum channels, Journal of Mathematical Physics62, 062201 (2021)

2021
[70]

L. S´ a, P. Ribeiro, and T. Prosen, Spectral and steady-state properties of random Liouvillians, Journal of Physics A: Mathematical and Theoretical53, 305303 (2020)

2020
[71]

Chru´ sci´ nski, S

D. Chru´ sci´ nski, S. Denisov, W. Tarnowski, and K.˙Zyczkowski, Spectral delineation of Markov Generators: Classical vs Quantum (2025), arXiv:2504.07903

work page arXiv 2025
[72]

Moske, P

L. Moske, P. Ribeiro, T. Prosen, S. Denysov, K. ˙Zyczkowski, and D. J. Luitz, Random matrix perspective on probabilistic error cancellation (2025), arXiv:2512.01957

work page internal anchor Pith review arXiv 2025
[73]

Nechita, Z

I. Nechita, Z. Pucha la, L. Pawela, and K. ˙Zyczkowski, Almost all quantum channels are equidistant, Journal of Mathematical Physics59, 052201 (2018)

2018
[74]

Watrous,The Theory of Quantum Information, 1st ed

J. Watrous,The Theory of Quantum Information, 1st ed. (Cambridge University Press, 2018)

2018
[75]

Zyczkowski and H.-J

K. Zyczkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, Journal of Physics A: Mathematical and General34, 7111 (2001)

2001
[76]

Vershynin,High-Dimensional Probability: An Introduction with Applications in Data Science, 1st ed

R. Vershynin,High-Dimensional Probability: An Introduction with Applications in Data Science, 1st ed. (Cambridge University Press, 2018)

2018
[77]

Laurent and P

B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection, Ann. Statist.28, 1302 (2000)

2000
[78]

NISO, Credit – contributor roles taxonomy,https://credit.niso.org/, Accessed 2026-05- 29

2026
[79]

Sweke, I

R. Sweke, I. Sinayskiy, D. Bernard, and F. Petruccione, Universal simulation of Markovian open quantum systems, Physical Review A91, 062308 (2015)

2015
[80]

Kossakowski matrix,

D. L. Hanson and F. T. Wright, A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables, The Annals of Mathematical Statistics42, 1079 (1971). 20 Appendix A: State to dynamics correspondence in sample-based Lindbladian simulation In this section, we explain why program state for dissipative Lindbladian dynamics needs to bed 2 dime...

1971

Showing first 80 references.

[1] [1]

Exact Lindbladian action 26

[2] [2]

Exactn-step WML state on the invariant two-dimensional space 27

[3] [3]

Trace-distance lower bound 28

[4] [4]

Special case

Asymptotic expansion ofz n 29 E. Alternative realization of WML via Stinespring isometry 30 I. INTRODUCTION The accurate simulation of quantum systems is a cornerstone of quantum information science [1–5], with broad implications ranging from quantum chemistry [6–9] to condensed- matter physics [10, 11]. Hamiltonian simulation [12–15] has been used to mod...

2024

[5] [5]

Breuer and F

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, Oxford ; New York, 2002)

2002

[6] [6]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, 10th ed. (Cambridge University Press, Cambridge, 2010)

2010

[7] [7]

H. M. Wiseman and G. J. Milburn,Quantum Measurement and Control(Cambridge University Press, Cambridge, 2009)

2009

[8] [8]

Weiss,Quantum Dissipative Systems, 4th ed

U. Weiss,Quantum Dissipative Systems, 4th ed. (World Scientific, 2012)

2012

[9] [9]

R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics21, 467 (1982)

1982

[10] [10]

Wecker, B

D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, Gate-count estimates for performing quantum chemistry on small quantum computers, Physical Review A90, 022305 (2014)

2014

[11] [11]

Babbush, J

R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation, Physical Review A91, 022311 (2015)

2015

[12] [12]

Reiher, N

M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Elucidating reaction mechanisms on quantum computers, Proceedings of the National Academy of Sciences of the United States of America114, 7555 (2017)

2017

[13] [13]

Poulin, M

D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doberty, and M. Troyer, The Trotter step size required for accurate quantum simulation of quantum chemistry, Quantum Information and Computation15, 361 (2015)

2015

[14] [14]

Raeisi, N

S. Raeisi, N. Wiebe, and B. C. Sanders, Quantum-circuit design for efficient simulations of many-body quantum dynamics, New Journal of Physics14, 103017 (2012)

2012

[15] [15]

A. J. Daley, Quantum trajectories and open many-body quantum systems, Advances in Physics 63, 77 (2014)

2014

[16] [16]

G. H. Low and I. L. Chuang, Optimal Hamiltonian Simulation by Quantum Signal Processing, Physical Review Letters118, 010501 (2017)

2017

[17] [17]

D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating Hamiltonian dynamics with a truncated Taylor series, Physical Review Letters114, 090502 (2015)

2015

[18] [18]

G. H. Low and I. L. Chuang, Hamiltonian Simulation by Qubitization, Quantum3, 163 (2019). 17

2019

[19] [19]

A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Toward the first quantum simulation with quantum speedup, Proceedings of the National Academy of Sciences of the United States of America115, 9456 (2018)

2018

[20] [20]

Gorini, A

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups ofN-level systems, Journal of Mathematical Physics17, 821 (1976)

1976

[21] [21]

Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics48, 119 (1976)

G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics48, 119 (1976)

1976

[22] [22]

J. F. Poyatos, J. I. Cirac, and P. Zoller, Quantum Reservoir Engineering with Laser Cooled Trapped Ions, Physical Review Letters77, 4728 (1996)

1996

[23] [23]

T. S. Cubitt, A. Lucia, S. Michalakis, and D. Perez-Garcia, Stability of Local Quantum Dissipative Systems, Communications in Mathematical Physics337, 1275 (2015)

2015

[24] [24]

Y. Zhan, Z. Ding, J. Huhn, J. Gray, J. Preskill, G. K.-L. Chan, and L. Lin, Rapid Quantum Ground State Preparation via Dissipative Dynamics, Physical Review X16, 011004 (2026)

2026

[25] [25]

Van Den Berg, Z

E. Van Den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Nature Physics19, 1116 (2023)

2023

[26] [26]

S. Jin, N. Liu, and Y. Yu, Quantum Simulation of Partial Differential Equations via Schr¨ odingerization, Physical Review Letters133, 230602 (2024)

2024

[27] [27]

Lin, Dissipative preparation of many-body quantum states: Toward practical quantum advantage, APL Computational Physics1, 010901 (2025)

L. Lin, Dissipative preparation of many-body quantum states: Toward practical quantum advantage, APL Computational Physics1, 010901 (2025)

2025

[28] [28]

Simulating Markovian open quantum systems using higher-order series expansion

X. Li and C. Wang, Simulating Markovian open quantum systems using higher-order series expansion (2023), arXiv:2212.02051

work page Pith review arXiv 2023

[29] [29]

Burdine, N

C. Burdine, N. Bauer, G. Siopsis, and E. P. Blair, Efficient Simulation of Open Quantum Systems on NISQ Trapped-Ion Hardware (2024), arXiv:2410.10732

work page arXiv 2024

[30] [30]

A. M. Childs and T. Li, Efficient simulation of sparse Markovian quantum dynamics, Quantum Information and Computation17, 0901 (2023)

2023

[31] [31]

J. Kato, K. Wada, K. Ito, and N. Yamamoto, Exponentially accurate open quantum simulation via randomized dissipation with minimal ancilla (2024), arXiv:2412.19453

work page arXiv 2024

[32] [32]

S. Peng, X. Sun, Q. Zhao, and H. Zhou, Quantum-trajectory-inspired Lindbladian simulation, PRX Quantum6, 030358 (2025)

2025

[33] [33]

H. Chen, B. Li, J. Lu, and L. Ying, A randomized method for simulating Lindblad equations and thermal state preparation, Quantum9, 1917 (2025)

1917

[34] [34]

Dalibard, Y

J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Physical Review Letters68, 580 (1992)

1992

[35] [35]

R. Dum, P. Zoller, and H. Ritsch, Monte Carlo simulation of the atomic master equation for spontaneous emission, Physical Review A45, 4879 (1992)

1992

[36] [36]

H. Carmichael,An Open Systems Approach to Quantum Optics: Lectures Presented at the Universit´ e Libre de Bruxelles October 28 to November 4, 1991, Lecture Notes in Physics Monographs, Vol. 18 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1993)

1991

[37] [37]

Y. Cao, M. He, and X. Li, Dynamically Optimal Unraveling Schemes for Simulating Lindblad Equations (2025), arXiv:2509.19887

work page internal anchor Pith review Pith/arXiv arXiv 2025

[38] [38]

Cleve and C

R. Cleve and C. Wang, Efficient quantum algorithms for simulating Lindblad evolution, in44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 80 (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2017) pp. 17:1–17:14

2017

[39] [39]

Z. Ding, X. Li, and L. Lin, Simulating open quantum systems using Hamiltonian simulations, 18 PRX Quantum5, 020332 (2024)

2024

[40] [40]

G. H. Low and R. D. Somma, Optimal quantum simulation of linear non-unitary dynamics (2025), arXiv:2508.19238

work page arXiv 2025

[41] [41]

Kamakari, S.-N

H. Kamakari, S.-N. Sun, M. Motta, and A. J. Minnich, Digital quantum simulation of open quantum systems using quantum imaginary time evolution, PRX Quantum3, 010320 (2022)

2022

[42] [42]

A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, and D. A. Mazziotti, Quantum simulation of the Lindblad equation using a unitary decomposition of operators, Physical Review Research4, 023216 (2022)

2022

[43] [43]

Patel and M

D. Patel and M. M. Wilde, Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics, Open Systems & Information Dynamics30, 2350010 (2023)

2023

[44] [44]

Patel and M

D. Patel and M. M. Wilde, Wave Matrix Lindbladization II: General Lindbladians, Linear Combinations, and Polynomials, Open Systems & Information Dynamics30, 2350014 (2023)

2023

[45] [45]

Lloyd, M

S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum principal component analysis, Nature Physics10, 631 (2014)

2014

[46] [46]

Kimmel, C

S. Kimmel, C. Y.-Y. Lin, G. H. Low, M. Ozols, and T. J. Yoder, Hamiltonian simulation with optimal sample complexity, npj Quantum Information3, 1 (2017)

2017

[47] [47]

F. Wei, Z. Liu, G. Liu, Z. Han, D.-L. Deng, and Z. Liu, Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps, npj Quantum Information10, 1 (2024)

2024

[48] [48]

B. Go, H. Kwon, S. Park, D. Patel, and M. M. Wilde, Sample-based Hamiltonian and Lindbladian simulation: Non-asymptotic analysis of sample complexity, Quantum Science and Technology10, 045058 (2025)

2025

[49] [49]

Kliesch, T

M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert, Dissipative Quantum Church-Turing Theorem, Physical Review Letters107, 120501 (2011)

2011

[50] [50]

Z. Xu, L. P. Garc´ ıa-Pintos, A. Chenu, and A. Del Campo, Extreme Decoherence and Quantum Chaos, Physical Review Letters122, 014103 (2019)

2019

[51] [51]

J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample-optimal tomography of quantum states, inProceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’16 (Association for Computing Machinery, New York, NY, USA, 2016) pp. 913–925

2016

[52] [52]

J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample-optimal tomography of quantum states, IEEE Transactions on Information Theory63, 5628 (2017)

2017

[53] [53]

Yuen, An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography, Quantum7, 890 (2023)

H. Yuen, An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography, Quantum7, 890 (2023)

2023

[54] [54]

Anshu and S

A. Anshu and S. Arunachalam, A survey on the complexity of learning quantum states, Nature Reviews Physics6, 59 (2023)

2023

[55] [55]

M. M. Wilde,Quantum Information Theory, 2nd ed. (Cambridge University Press, Cambridge, 2017)

2017

[56] [56]

transition probability

A. Uhlmann, The “transition probability” in the state space of a∗-algebra, Reports on Mathematical Physics9, 273 (1976)

1976

[57] [57]

A. Y. Kitaev, Quantum computations: Algorithms and error correction, Russian Mathematical Surveys52, 1191 (1997)

1997

[58] [58]

M. A. Nielsen and I. L. Chuang, Programmable Quantum Gate Arrays, Physical Review Letters79, 321 (1997)

1997

[59] [59]

Banchi, J

L. Banchi, J. Pereira, S. Lloyd, and S. Pirandola, Convex optimization of programmable quantum computers, npj Quantum Information6, 1 (2020)

2020

[60] [60]

M. Jing, M. Guo, L. Zhu, H. Yao, and X. Wang, Programmable Open Quantum Systems 19 (2025), arXiv:2512.08279

work page arXiv 2025

[61] [61]

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed State Entanglement and Quantum Error Correction, Physical Review A54, 3824 (1996)

1996

[62] [62]

Bravyi and A

S. Bravyi and A. Kitaev, Universal Quantum Computation with ideal Clifford gates and noisy ancillas, Physical Review A71, 022316 (2005)

2005

[63] [63]

X. Yuan, B. Regula, R. Takagi, and M. Gu, Virtual quantum resource distillation, Physical Review Letters132, 050203 (2024), 2303.00955

work page arXiv 2024

[64] [64]

Aaronson, Quantum Copy-Protection and Quantum Money, in2009 24th Annual IEEE Conference on Computational Complexity(IEEE, Paris, France, 2009) pp

S. Aaronson, Quantum Copy-Protection and Quantum Money, in2009 24th Annual IEEE Conference on Computational Complexity(IEEE, Paris, France, 2009) pp. 229–242

2009

[65] [65]

A. N. Sims, D. Patel, A. Philip, A. H. Rubin, R. Bandyopadhyay, M. Radulaski, and M. M. Wilde, Digital quantum simulations of the nonresonant open Tavis-Cummings model, Physical Review Research7, 043302 (2025)

2025

[66] [66]

Rigol, V

M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452, 854 (2008)

2008

[67] [67]

Bruzda, V

W. Bruzda, V. Cappellini, H.-J. Sommers, and K. ˙Zyczkowski, Random Quantum Operations, Physics Letters A373, 320 (2009)

2009

[68] [68]

Denisov, T

S. Denisov, T. Laptyeva, W. Tarnowski, D. Chru´ sci´ nski, and K.˙Zyczkowski, Universal Spectra of Random Lindblad Operators, Physical Review Letters123, 140403 (2019)

2019

[69] [69]

Kukulski, I

R. Kukulski, I. Nechita, L. Pawela, Z. Pucha la, and K. ˙Zyczkowski, Generating random quantum channels, Journal of Mathematical Physics62, 062201 (2021)

2021

[70] [70]

L. S´ a, P. Ribeiro, and T. Prosen, Spectral and steady-state properties of random Liouvillians, Journal of Physics A: Mathematical and Theoretical53, 305303 (2020)

2020

[71] [71]

Chru´ sci´ nski, S

D. Chru´ sci´ nski, S. Denisov, W. Tarnowski, and K.˙Zyczkowski, Spectral delineation of Markov Generators: Classical vs Quantum (2025), arXiv:2504.07903

work page arXiv 2025

[72] [72]

Moske, P

L. Moske, P. Ribeiro, T. Prosen, S. Denysov, K. ˙Zyczkowski, and D. J. Luitz, Random matrix perspective on probabilistic error cancellation (2025), arXiv:2512.01957

work page internal anchor Pith review arXiv 2025

[73] [73]

Nechita, Z

I. Nechita, Z. Pucha la, L. Pawela, and K. ˙Zyczkowski, Almost all quantum channels are equidistant, Journal of Mathematical Physics59, 052201 (2018)

2018

[74] [74]

Watrous,The Theory of Quantum Information, 1st ed

J. Watrous,The Theory of Quantum Information, 1st ed. (Cambridge University Press, 2018)

2018

[75] [75]

Zyczkowski and H.-J

K. Zyczkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, Journal of Physics A: Mathematical and General34, 7111 (2001)

2001

[76] [76]

Vershynin,High-Dimensional Probability: An Introduction with Applications in Data Science, 1st ed

R. Vershynin,High-Dimensional Probability: An Introduction with Applications in Data Science, 1st ed. (Cambridge University Press, 2018)

2018

[77] [77]

Laurent and P

B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection, Ann. Statist.28, 1302 (2000)

2000

[78] [78]

NISO, Credit – contributor roles taxonomy,https://credit.niso.org/, Accessed 2026-05- 29

2026

[79] [79]

Sweke, I

R. Sweke, I. Sinayskiy, D. Bernard, and F. Petruccione, Universal simulation of Markovian open quantum systems, Physical Review A91, 062308 (2015)

2015

[80] [80]

Kossakowski matrix,

D. L. Hanson and F. T. Wright, A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables, The Annals of Mathematical Statistics42, 1079 (1971). 20 Appendix A: State to dynamics correspondence in sample-based Lindbladian simulation In this section, we explain why program state for dissipative Lindbladian dynamics needs to bed 2 dime...

1971