pith. sign in

arxiv: 2605.16195 · v1 · pith:7JAT3MB5new · submitted 2026-05-15 · 🪐 quant-ph

Efficient quantum algorithm for linear matrix differential equations and applications to open quantum systems

Sophia Simon , Dominic W. Berry , Rolando D. Somma This is my paper

Pith reviewed 2026-05-20 17:49 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum algorithmmatrix differential equationsopen quantum systemsdissipative dynamicsquantum simulationquery complexitylinear algebra
5
0 comments X

The pith

A quantum algorithm computes entries of solutions to linear matrix differential equations with query complexity scaling as O(ν L t / ε) for unitary or dissipative dynamics.

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

The paper develops a quantum algorithm for solving linear matrix differential equations that computes a single entry of the solution matrix rather than encoding the full solution as a quantum state. Prior approaches suffered from exponentially small amplitudes that forced exponential runtime, but this method uses query access to evolution operators and integrates their norm bounds to reach a query complexity of roughly O(ν L t / ε). For unitary dynamics the factor ν L grows linearly with t, while for dissipative dynamics it can remain constant. The result includes a matching lower bound proving near-optimality up to logs and is illustrated on dissipative dynamics of non-interacting fermions, where it yields polynomial or exponential speedups over classical methods for lattice systems with long-range interactions.

Core claim

The algorithm computes an entry of the solution matrix with query complexity ~O(ν L t / ε) for unitary or dissipative dynamics, where ν depends on problem parameters and L is a time integral of upper bounds on the norms of evolution operators. For unitary dynamics ν L is linear in t; for dissipative dynamics it can be constant. A matching lower bound of Ω(ν L t / ε) establishes optimality up to logarithmic factors. The approach is applied end-to-end to simulate dissipative dynamics of non-interacting fermions on lattices, with comparisons showing polynomial quantum speedups that become exponential for long-range interactions.

What carries the argument

Query-access quantum algorithm that computes individual matrix entries by integrating time-dependent upper bounds on the norms of the evolution operators to control complexity without exponential amplitude penalties.

If this is right

  • Simulation of open quantum systems becomes efficient on quantum hardware for unitary and dissipative dynamics whose norm integrals remain controlled.
  • Polynomial speedups appear for lattice systems and grow to exponential for models with long-range interactions.
  • The method extends directly to other quantum and classical linear systems that admit similar operator-norm bounds.
  • Classical algorithms for the same matrix differential equations are outperformed on lattice instances with growing system size.

Where Pith is reading between the lines

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

  • If efficient classical or quantum routines exist to compute the integrated norm L for a given model, the algorithm applies to a wide range of physical and engineered systems beyond the fermion example.
  • Hybrid quantum-classical implementations could combine this entry-wise solver with variational methods for systems where full-state simulation remains hard.
  • The same norm-integral technique may extend to time-dependent or mildly nonlinear differential equations provided comparable operator bounds can be derived.

Load-bearing premise

Upper bounds on the norms of the evolution operators can be integrated over time to produce a useful L that stays small or efficiently computable for the target systems.

What would settle it

A concrete matrix differential equation instance (such as a specific open-fermion lattice model) where the measured number of queries needed to achieve error ε either exceeds O(ν L t / ε) or violates the Ω(ν L t / ε) lower bound.

read the original abstract

We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity $\widetilde{\mathcal{O}}(\nu \mathcal{L} t/\epsilon)$, where the constant $\nu$ depends on the problem parameters, $\mathcal{L}$ involves a time integral of upper bounds on the norms of evolution operators, and $\epsilon$ is the error. In particular, $\nu \mathcal{L}$ is linear in $t$ for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of $\Omega(\nu \mathcal{L} t/\epsilon)$ for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.

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.

Referee Report

2 major / 2 minor

Summary. The paper presents a quantum algorithm for linear matrix differential equations that computes an entry of the solution matrix with query complexity ~O(ν L t / ε) for unitary or dissipative dynamics, where ν depends on problem parameters and L is a time integral of upper bounds on evolution operator norms. It claims ν L is linear in t for unitary cases but can be constant for dissipative dynamics, provides a matching Ω(ν L t / ε) lower bound proving near-optimality up to logs, contrasts with prior exponential-time methods due to small amplitudes, and demonstrates an end-to-end application to dissipative simulation of non-interacting fermions on lattices with evidence of polynomial (or exponential in general) quantum speedups over classical algorithms.

Significance. If the result holds, particularly the handling of dissipative dynamics with bounded L and the concrete speedups for lattice fermions, this would be a notable contribution to quantum algorithms for open quantum systems and linear ODEs, offering a nearly optimal query complexity that avoids the exponential overhead of state-encoding approaches and enables practical simulations where classical methods scale poorly with long-range interactions.

major comments (2)
  1. [Abstract and complexity analysis] Abstract and main complexity claim: the headline efficiency for dissipative dynamics (ν L constant, independent of t) rests on upper bounds for the norms of the relevant evolution operators (e.g., vectorized Lindblad propagator) admitting an integrable L that remains small and efficiently computable without hidden costs. The manuscript does not explicitly construct or bound these norms for general open systems, so it is unclear whether obtaining them incurs a cost that would negate the stated polynomial speedups for lattice fermions.
  2. [Lower bound argument] Lower-bound section: while a matching Ω(ν L t / ε) lower bound is stated, the high-level description leaves the explicit construction and proof details unverified; confirming that the lower-bound argument applies directly to the same encoding and query model used in the upper bound is load-bearing for the near-optimality claim.
minor comments (2)
  1. [Abstract] The dependence of ν on problem parameters is described only at a high level; a more precise definition or table relating ν to specific quantities (e.g., dissipation rates, system size) would improve clarity.
  2. [Applications section] In the fermion simulation application, the extension from non-interacting to interacting cases is mentioned only briefly; a short paragraph clarifying the additional assumptions required would help readers assess the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and have revised the manuscript to improve the presentation of the complexity claims and the lower-bound argument.

read point-by-point responses

  1. Referee: [Abstract and complexity analysis] Abstract and main complexity claim: the headline efficiency for dissipative dynamics (ν L constant, independent of t) rests on upper bounds for the norms of the relevant evolution operators (e.g., vectorized Lindblad propagator) admitting an integrable L that remains small and efficiently computable without hidden costs. The manuscript does not explicitly construct or bound these norms for general open systems, so it is unclear whether obtaining them incurs a cost that would negate the stated polynomial speedups for lattice fermions.

    Authors: We agree that an explicit construction of the relevant operator-norm bounds strengthens the presentation. In the revised manuscript we have added a dedicated subsection (Section 3.2) that constructs upper bounds on the norms of the vectorized Lindblad propagator for general open systems under standard assumptions (bounded Lindblad operators and Hamiltonian). For the concrete lattice-fermion application we explicitly compute these bounds and show that the resulting L is O(1) independent of t; the classical pre-processing cost to obtain the bounds is only polylogarithmic in the system size and does not cancel the polynomial quantum speedup. We have also clarified that the stated complexity is expressed in terms of L, which is assumed to be provided by an efficient classical routine or oracle when the problem instance is specified. revision: yes

  2. Referee: [Lower bound argument] Lower-bound section: while a matching Ω(ν L t / ε) lower bound is stated, the high-level description leaves the explicit construction and proof details unverified; confirming that the lower-bound argument applies directly to the same encoding and query model used in the upper bound is load-bearing for the near-optimality claim.

    Authors: We acknowledge that the lower-bound argument was presented at a high level. In the revision we have expanded Section 5 with a self-contained proof that includes the explicit hard-instance construction (a suitable block-encoding of a time-dependent matrix whose solution entry encodes a known quantum query-hard problem) and verifies that the reduction applies to precisely the same query model and output encoding used in the upper bound (i.e., access to the matrix oracles and extraction of a single matrix entry). The revised proof confirms the Ω(ν L t / ε) lower bound holds in this model, establishing near-optimality up to logarithmic factors. revision: yes

Circularity Check

0 steps flagged

No significant circularity; complexity and lower bound derived independently from problem parameters

full rationale

The paper derives an upper-bound query complexity of ~O(ν ℒ t / ε) for solving linear matrix ODEs via a quantum algorithm that avoids exponentially small amplitudes, then separately establishes a matching Ω(ν ℒ t / ε) lower bound. Neither quantity is obtained by fitting to data, redefining the target in terms of itself, or relying on a self-citation chain whose validity is presupposed. The parameter ℒ is defined as an integral of operator-norm bounds that are external to the algorithm's runtime; the claim that ν ℒ can be constant for dissipative dynamics is an assumption about the target system class rather than a definitional tautology. The derivation therefore remains self-contained against external benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard quantum-oracle access and norm-bounding assumptions common to quantum simulation literature; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Quantum oracles or unitaries implementing the linear operators of the differential equation are available
    Standard assumption for quantum algorithms that simulate dynamics via query access.
  • domain assumption Upper bounds on the norms of the evolution operators exist and can be integrated over time to define L
    This quantity directly determines the stated query complexity and efficiency claims.

pith-pipeline@v0.9.0 · 5778 in / 1415 out tokens · 66904 ms · 2026-05-20T17:49:21.114521+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum Koopman Algorithms

    quant-ph 2026-05 unverdicted novelty 6.0

    Quantum Koopman Algorithms define an observable-space quantum framework for simulating linear quantum and nonlinear classical dynamics with polylog gate costs in some cases.

Reference graph

Works this paper leans on

107 extracted references · 107 canonical work pages · cited by 1 Pith paper · 5 internal anchors

  1. [1]

    , journal=

    Trefethen, Lloyd N. , journal=. Is. 2008 , publisher=

    work page 2008

  2. [2]

    1987 , publisher=

    Introduction to Numerical Analysis , author=. 1987 , publisher=

    work page 1987

  3. [4]

    Maarten Stroeks and Daan Lenterman and Barbara Terhal and Yaroslav Herasymenko , year=

    work page

  4. [7]

    2405.12714 , archivePrefix=

    Hsuan-Cheng Wu and Jingyao Wang and Xiantao Li , year=. 2405.12714 , archivePrefix=

    work page arXiv

  5. [8]

    doi:10.1109/focs61266.2024.00070 , booktitle=

    Low, Guang Hao and Su, Yuan , year=. doi:10.1109/focs61266.2024.00070 , booktitle=

    work page doi:10.1109/focs61266.2024.00070 2024

  6. [9]

    Efficient quantum algorithm for dissipative nonlinear differential equations , volume=

    Liu, Jin-Peng and Kolden, Herman. Efficient quantum algorithm for dissipative nonlinear differential equations , volume=. Proceedings of the National Academy of Sciences , publisher=. doi:10.1073/pnas.2026805118 , number=

    work page doi:10.1073/pnas.2026805118

  7. [12]

    2021 Proceedings of the Conference on Control and its Applications (CT) , chapter =

    Arash Amini and Qiyu Sun and Nader Motee , title =. 2021 Proceedings of the Conference on Control and its Applications (CT) , chapter =. doi:10.1137/1.9781611976847.1 , URL =. https://epubs.siam.org/doi/pdf/10.1137/1.9781611976847.1 , abstract =

    work page doi:10.1137/1.9781611976847.1 2021

  8. [13]

    and Budi

    Brunton, Steven L. and Budi. SIAM Review , volume =. 2022 , doi =

    work page 2022

  9. [19]

    Physical Review Letters , volume=

    Quantum algorithm for linear systems of equations , author=. Physical Review Letters , volume=. 2009 , doi=

    work page 2009

  10. [20]

    Proceedings of the 51st annual ACM SIGACT symposium on theory of computing , pages=

    Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics , author=. Proceedings of the 51st annual ACM SIGACT symposium on theory of computing , pages=

    work page

  11. [21]

    arXiv preprint arXiv:2411.00434 , url=

    Quantum linear algebra for disordered electrons , author=. arXiv preprint arXiv:2411.00434 , url=

    work page arXiv

  12. [22]

    2018 , doi=

    Hached, Mustapha and Jbilou, Khalide , journal=. 2018 , doi=

    work page 2018

  13. [23]

    PRX Quantum , volume=

    Focus beyond quadratic speedups for error-corrected quantum advantage , author=. PRX Quantum , volume=. 2021 , doi=

    work page 2021

  14. [24]

    Reviews of Modern Physics , volume=

    The kernel polynomial method , author=. Reviews of Modern Physics , volume=. 2006 , doi=

    work page 2006

  15. [25]

    Physical Review Research , volume=

    Solving free fermion problems on a quantum computer , author=. Physical Review Research , volume=. 2025 , publisher=

    work page 2025

  16. [26]

    Physical Review Letters , volume=

    Universal computation by quantum walk , author=. Physical Review Letters , volume=. 2009 , doi=

    work page 2009

  17. [27]

    Low, Guang Hao and Chuang, Isaac L. , doi =. Hamiltonian simulation by qubitization , volume =. Quantum , pages =

    work page

  18. [28]

    npj Quantum Information , volume=

    Quantum algorithms for matrix geometric means , author=. npj Quantum Information , volume=. 2025 , doi=

    work page 2025

  19. [29]

    Solving the matrix differential

    Nguyen, Thang and Gajic, Zoran , journal=. Solving the matrix differential. 2009 , doi=

    work page 2009

  20. [30]

    Lyapunov equation in open quantum systems and non-

    Purkayastha, Archak , journal=. Lyapunov equation in open quantum systems and non-. 2022 , doi=

    work page 2022

  21. [31]

    2019 , doi=

    Applied stochastic differential equations , author=. 2019 , doi=

    work page 2019

  22. [32]

    Physical Review X , volume=

    Exponential quantum speedup in simulating coupled classical oscillators , author=. Physical Review X , volume=. 2023 , doi=

    work page 2023

  23. [33]

    Nature Physics , volume=

    Read the fine print , author=. Nature Physics , volume=. 2015 , doi=

    work page 2015

  24. [34]

    SIAM Journal on Computing , volume=

    Quantum algorithm for systems of linear equations with exponentially improved dependence on precision , author=. SIAM Journal on Computing , volume=. 2017 , doi=

    work page 2017

  25. [35]

    Proceedings of the National Academy of Sciences , volume=

    Efficient quantum algorithm for dissipative nonlinear differential equations , author=. Proceedings of the National Academy of Sciences , volume=. 2021 , doi=

    work page 2021

  26. [36]

    Quantum , volume=

    Improved quantum algorithms for linear and nonlinear differential equations , author=. Quantum , volume=. 2023 , DOI=

    work page 2023

  27. [37]

    Journal of Physics A: Mathematical and Theoretical , volume=

    High-order quantum algorithm for solving linear differential equations , author=. Journal of Physics A: Mathematical and Theoretical , volume=. 2014 , doi=

    work page 2014

  28. [38]

    Physical Review B , volume=

    Towards simulation of the dynamics of materials on quantum computers , author=. Physical Review B , volume=. 2020 , doi=

    work page 2020

  29. [39]

    Universal quantum simulators , volume =

    Seth Lloyd , doi =. Universal quantum simulators , volume =. Science , number =

    work page

  30. [41]

    arXiv preprint arXiv:2508.02822 , url=

    Quantum algorithm for linear matrix equations , author=. arXiv preprint arXiv:2508.02822 , url=

    work page arXiv

  31. [42]

    arXiv preprint arXiv:2508.19238 , url=

    Optimal quantum simulation of linear non-unitary dynamics , author=. arXiv preprint arXiv:2508.19238 , url=

    work page arXiv

  32. [43]

    Low, Guang Hao and Chuang, Isaac L , journal=. Optimal. 2017 , doi=

    work page 2017

  33. [44]

    Efficient quantum algorithms for simulating sparse

    Berry, Dominic W and Ahokas, Graeme and Cleve, Richard and Sanders, Barry C , journal=. Efficient quantum algorithms for simulating sparse. 2007 , doi=

    work page 2007

  34. [45]

    2025 , eprint=

    Quantum algorithms for general nonlinear dynamics based on the Carleman embedding , author=. 2025 , eprint=

    work page 2025

  35. [46]

    2024 , journal=

    The discrete adiabatic quantum linear system solver has lower constant factors than the randomized adiabatic solver , author=. 2024 , journal=

    work page 2024

  36. [47]

    2025 , eprint=

    Quantum simulation of a noisy classical nonlinear dynamics , author=. 2025 , eprint=

    work page 2025

  37. [48]

    Efficient Quantum Algorithms for Simulating Lindblad Evolution

    Efficient quantum algorithms for simulating Lindblad evolution , author=. arXiv preprint arXiv:1612.09512 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  38. [49]

    Physical Review A , volume=

    Gate-count estimates for performing quantum chemistry on small quantum computers , author=. Physical Review A , volume=. 2014 , doi=

    work page 2014

  39. [50]

    Chemical basis of

    Babbush, Ryan and McClean, Jarrod and Wecker, Dave and Aspuru-Guzik, Al. Chemical basis of. Physical Review A , volume=. 2015 , doi=

    work page 2015

  40. [51]

    Physical Review X , volume=

    Fast quantum simulation of electronic structure by spectral amplification , author=. Physical Review X , volume=. 2025 , doi=

    work page 2025

  41. [52]

    2019 , doi=

    Behr, Maximilian and Benner, Peter and Heiland, Jan , journal=. 2019 , doi=

    work page 2019

  42. [53]

    arXiv preprint arXiv:2305.04908 , url=

    Tight bounds for quantum phase estimation and related problems , author=. arXiv preprint arXiv:2305.04908 , url=

    work page arXiv

  43. [54]

    2007 , publisher=

    Theoretical Numerical Analysis: A Functional Analysis Framework , author=. 2007 , publisher=

    work page 2007

  44. [55]

    2025 , eprint=

    Quantum algorithm for linear matrix equations , author=. 2025 , eprint=

    work page 2025

  45. [56]

    Trudy Matematicheskogo Instituta imeni VA Steklova , volume=

    On Newton's method , author=. Trudy Matematicheskogo Instituta imeni VA Steklova , volume=. 1949 , publisher=

    work page 1949

  46. [58]

    Physical Review Letters , volume=

    Quantum simulation with sum-of-squares spectral amplification , author=. Physical Review Letters , volume=. 2026 , publisher=

    work page 2026

  47. [59]

    SIAM Review , volume=

    Computational methods for linear matrix equations , author=. SIAM Review , volume=. 2016 , doi=

    work page 2016

  48. [60]

    A restarted

    Eiermann, Michael and Ernst, Oliver G , journal=. A restarted. 2006 , doi=

    work page 2006

  49. [63]

    Physical Review A , volume=

    Optimal quantum measurements of expectation values of observables , author=. Physical Review A , volume=. 2007 , doi=

    work page 2007

  50. [65]

    and Chuang, Isaac L

    Nielsen, Michael A. and Chuang, Isaac L. , year=. Quantum Computation and Quantum Information: 10th Anniversary Edition , DOI=

    work page

  51. [66]

    Quantum state preparation with optimal

    David Gosset and Robin Kothari and Kewen Wu , year=. Quantum state preparation with optimal. 2411.04790 , archivePrefix=

    work page arXiv

  52. [71]

    Gengzhi Yang and Akwum Onwunta and Dong An , journal=

    work page

  53. [73]

    PRX Quantum , volume=

    Improved precision scaling for simulating coupled quantum-classical dynamics , author=. PRX Quantum , volume=. 2024 , publisher=

    work page 2024

  54. [74]

    author author Dominic W \ Berry , author Graeme \ Ahokas , author Richard \ Cleve , \ and\ author Barry C \ Sanders ,\ title title Efficient quantum algorithms for simulating sparse H amiltonians , \ 10.1007/s00220-006-0150-x journal journal Communications in Mathematical Physics \ volume 270 ,\ pages 359--371 ( year 2007 ) NoStop

    work page doi:10.1007/s00220-006-0150-x 2007

  55. [75]

    \ Berry , author Andrew M

    author author Dominic W. \ Berry , author Andrew M. \ Childs , author Richard \ Cleve , author Robin \ Kothari , \ and\ author Rolando D. \ Somma ,\ title title Simulating Hamiltonian dynamics with a truncated Taylor series , \ 10.1103/PhysRevLett.114.090502 journal journal Physical Review Letters \ volume 114 ,\ pages 090502 ( year 2015 ) NoStop

    work page doi:10.1103/physrevlett.114.090502 2015

  56. [76]

    author author Guang Hao \ Low \ and\ author Isaac L \ Chuang ,\ title title Optimal H amiltonian simulation by quantum signal processing , \ 10.1103/PhysRevLett.118.010501 journal journal Physical Review Letters \ volume 118 ,\ pages 010501 ( year 2017 ) NoStop

    work page doi:10.1103/physrevlett.118.010501 2017

  57. [77]

    author author Dave \ Wecker , author Bela \ Bauer , author Bryan K \ Clark , author Matthew B \ Hastings , \ and\ author Matthias \ Troyer ,\ title title Gate-count estimates for performing quantum chemistry on small quantum computers , \ 10.1103/PhysRevA.90.022305 journal journal Physical Review A \ volume 90 ,\ pages 022305 ( year 2014 ) NoStop

    work page doi:10.1103/physreva.90.022305 2014

  58. [78]

    author author Ryan \ Babbush , author Jarrod \ McClean , author Dave \ Wecker , author Al \'a n \ Aspuru-Guzik , \ and\ author Nathan \ Wiebe ,\ title title Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation , \ 10.1103/PhysRevA.91.022311 journal journal Physical Review A \ volume 91 ,\ pages 022311 ( year 2015 ) NoStop

    work page doi:10.1103/physreva.91.022311 2015

  59. [79]

    author author Guang Hao \ Low , author Robbie \ King , author Dominic W \ Berry , author Qiushi \ Han , author A Eugene \ DePrince III , author Alec F \ White , author Ryan \ Babbush , author Rolando D \ Somma , \ and\ author Nicholas C \ Rubin ,\ title title Fast quantum simulation of electronic structure by spectral amplification , \ 10.1103/pb2g-j9cw j...

    work page doi:10.1103/pb2g-j9cw 2025

  60. [80]

    author author Seth \ Lloyd ,\ title title Universal quantum simulators , \ 10.1126/science.273.5278.1073 journal journal Science \ volume 273 ,\ pages 1073--1078 ( year 1996 ) NoStop

    work page doi:10.1126/science.273.5278.1073 1996

  61. [81]

    author author R. D. \ Somma , author G. Ortiz , author J. E. \ Gubernatis , author E. Knill , \ and\ author R. Laflamme ,\ title title Simulating physical phenomena by quantum networks , \ 10.1103/PhysRevA.65.042323 journal journal Physical Review A \ volume 65 ,\ pages 042323 ( year 2002 ) NoStop

    work page doi:10.1103/physreva.65.042323 2002

  62. [82]

    author author Lindsay \ Bassman Oftelie , author Kuang \ Liu , author Aravind \ Krishnamoorthy , author Thomas \ Linker , author Yifan \ Geng , author Daniel \ Shebib , author Shogo \ Fukushima , author Fuyuki \ Shimojo , author Rajiv K \ Kalia , author Aiichiro \ Nakano , et al. ,\ title title Towards simulation of the dynamics of materials on quantum co...

    work page doi:10.1103/physrevb.101.184305 2020

  63. [83]

    author author Dominic W \ Berry ,\ title title High-order quantum algorithm for solving linear differential equations , \ 10.1088/1751-8113/47/10/105301 journal journal Journal of Physics A: Mathematical and Theoretical \ volume 47 ,\ pages 105301 ( year 2014 ) NoStop

    work page doi:10.1088/1751-8113/47/10/105301 2014

  64. [84]

    \ Berry , author Andrew M

    author author Dominic W. \ Berry , author Andrew M. \ Childs , author Aaron \ Ostrander , \ and\ author Guoming \ Wang ,\ title title Quantum algorithm for linear differential equations with exponentially improved dependence on precision , \ 10.1007/s00220-017-3002-y journal journal Communications in Mathematical Physics \ volume 356 ,\ pages 1057–1081 ( ...

    work page doi:10.1007/s00220-017-3002-y 2017

  65. [85]

    author author Hari \ Krovi ,\ title title Improved quantum algorithms for linear and nonlinear differential equations , \ 10.22331/q-2023-02-02-913 journal journal Quantum \ volume 7 ,\ pages 913 ( year 2023 ) NoStop

    work page doi:10.22331/q-2023-02-02-913 2023

  66. [86]

    author author Jin-Peng \ Liu , author Herman ie \ Kolden , author Hari K \ Krovi , author Nuno F \ Loureiro , author Konstantina \ Trivisa , \ and\ author Andrew M \ Childs ,\ title title Efficient quantum algorithm for dissipative nonlinear differential equations , \ 10.1073/pnas.202680511 journal journal Proceedings of the National Academy of Sciences \...

    work page doi:10.1073/pnas.202680511 2021

  67. [87]

    author author Shi \ Jin , author Nana \ Liu , \ and\ author Yue \ Yu ,\ title title Quantum simulation of partial differential equations via Schr\"odingerization , \ 10.1103/PhysRevLett.133.230602 journal journal Physical Review Letters \ volume 133 ,\ pages 230602 ( year 2024 ) NoStop

    work page doi:10.1103/physrevlett.133.230602 2024

  68. [88]

    author author Dong \ An , author Jin-Peng \ Liu , \ and\ author Lin \ Lin ,\ title title Linear Combination of Hamiltonian Simulation for Nonunitary Dynamics with Optimal State Preparation Cost , \ 10.1103/physrevlett.131.150603 journal journal Physical Review Letters \ volume 131 ,\ pages 150603 ( year 2023 ) NoStop

    work page doi:10.1103/physrevlett.131.150603 2023

  69. [89]

    author author Guang Hao \ Low \ and\ author Rolando D \ Somma ,\ title title Optimal quantum simulation of linear non-unitary dynamics , \ https://doi.org/10.48550/arXiv.2508.19238 journal journal arXiv preprint arXiv:2508.19238 \ ( year 2025 ) NoStop

    work page doi:10.48550/arxiv.2508.19238 2025

  70. [90]

    author author Aram W \ Harrow , author Avinatan \ Hassidim , \ and\ author Seth \ Lloyd ,\ title title Quantum algorithm for linear systems of equations , \ 10.1103/PhysRevLett.103.150502 journal journal Physical Review Letters \ volume 103 ,\ pages 150502 ( year 2009 ) NoStop

    work page doi:10.1103/physrevlett.103.150502 2009

  71. [91]

    author author Andrew M \ Childs , author Robin \ Kothari , \ and\ author Rolando D \ Somma ,\ title title Quantum algorithm for systems of linear equations with exponentially improved dependence on precision , \ 10.1137/16M1087072 journal journal SIAM Journal on Computing \ volume 46 ,\ pages 1920--1950 ( year 2017 ) NoStop

    work page internal anchor Pith review doi:10.1137/16m1087072 1920

  72. [92]

    author author Scott \ Aaronson ,\ title title Read the fine print , \ 10.1038/nphys3272 journal journal Nature Physics \ volume 11 ,\ pages 291--293 ( year 2015 ) NoStop

    work page doi:10.1038/nphys3272 2015

  73. [93]

    author author Thang \ Nguyen \ and\ author Zoran \ Gajic ,\ title title Solving the matrix differential R iccati equation: a L yapunov equation approach , \ 10.1109/TAC.2009.2033841 journal journal IEEE Transactions on Automatic Control \ volume 55 ,\ pages 191--194 ( year 2009 ) NoStop

    work page doi:10.1109/tac.2009.2033841 2009

  74. [94]

    volume 10 \ ( publisher Cambridge University Press ,\ year 2019 ) NoStop

    author author Simo \ S \"a rkk \"a \ and\ author Arno \ Solin ,\ 10.1017/9781108186735 title Applied stochastic differential equations ,\ Vol. volume 10 \ ( publisher Cambridge University Press ,\ year 2019 ) NoStop

    work page doi:10.1017/9781108186735 2019

  75. [95]

    author author Archak \ Purkayastha ,\ title title Lyapunov equation in open quantum systems and non- H ermitian physics , \ 10.1103/PhysRevA.105.062204 journal journal Physical Review A \ volume 105 ,\ pages 062204 ( year 2022 ) NoStop

    work page doi:10.1103/physreva.105.062204 2022

  76. [96]

    author author Rolando D \ Somma , author Guang Hao \ Low , author Dominic W \ Berry , \ and\ author Ryan \ Babbush ,\ title title Quantum algorithm for linear matrix equations , \ https://doi.org/10.48550/arXiv.2508.02822 journal journal arXiv preprint arXiv:2508.02822 \ ( year 2025 a ) NoStop

    work page doi:10.48550/arxiv.2508.02822 2025

  77. [97]

    author author Ryan \ Babbush , author Dominic W \ Berry , author Robin \ Kothari , author Rolando D \ Somma , \ and\ author Nathan \ Wiebe ,\ title title Exponential quantum speedup in simulating coupled classical oscillators , \ 10.1103/PhysRevX.13.041041 journal journal Physical Review X \ volume 13 ,\ pages 041041 ( year 2023 ) NoStop

    work page doi:10.1103/physrevx.13.041041 2023

  78. [98]

    author author Andrew M \ Childs ,\ title title Universal computation by quantum walk , \ 10.1103/PhysRevLett.102.180501 journal journal Physical Review Letters \ volume 102 ,\ pages 180501 ( year 2009 ) NoStop

    work page doi:10.1103/physrevlett.102.180501 2009

  79. [99]

    \ Chuang ,\ title title Hamiltonian simulation by qubitization , \ 10.22331/q-2019-07-12-163 journal journal Quantum \ volume 3 ,\ pages 163 ( year 2019 ) NoStop

    author author Guang Hao \ Low \ and\ author Isaac L. \ Chuang ,\ title title Hamiltonian simulation by qubitization , \ 10.22331/q-2019-07-12-163 journal journal Quantum \ volume 3 ,\ pages 163 ( year 2019 ) NoStop

    work page doi:10.22331/q-2019-07-12-163 2019

  80. [100]

    author author András \ Gilyén , author Yuan \ Su , author Guang Hao \ Low , \ and\ author Nathan \ Wiebe ,\ title title Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics , \ in\ 10.1145/3313276.3316366 booktitle Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing ,\ series and ...

    work page doi:10.1145/3313276.3316366 2019

Showing first 80 references.