Pith. sign in

REVIEW 4 major objections 4 minor 7 cited by

For dissipative linear ODEs, this paper constructs quantum solvers whose query complexity to the coefficient matrix is O(log^3(1/ε)) — independent of the evolution time T — while keeping initial-state preparation at constant cost.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

For strictly dissipative linear ODEs, quantum solvers based on time-marching or LCHS achieve query complexity O(polylog(1/ε)) that is independent of the evolution time T.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Effective-window truncation is a genuine new mechanism and the final-state T-independence likely holds, but the history-state claims are undermined by Q factors that reintroduce T dependence. the 4 major comments →

arxiv 2508.15170 v1 pith:KU65PVYG submitted 2025-08-21 quant-ph

Quantum Differential Equation Solvers with Low State Preparation Cost: Eliminating the Time Dependence in Dissipative Equations

Gengzhi Yang , Akwum Onwunta , Dong An This is my paper
classification quant-ph MSC 81P68 PACS 03.67.Ac
keywords dissipative ODEsquantum simulationfast-forwardingtime-marching methodlinear combination of Hamiltonian simulationhistory state preparationquery complexitystate preparation cost
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper claims that a broad class of linear differential equations—those whose dynamics dissipate energy at a uniform rate η—can be simulated on a quantum computer with a query cost that does not grow with the evolution time T at all. The authors build two solvers, one based on time-marching and one on linear combinations of Hamiltonian simulations, and show that each can prepare an ε-accurate copy of the solution, either at the final time or as a history state, using O(log^3(1/ε)) queries to the coefficient matrix. The idea is to simulate only the last O((1/η)log(1/ε)) units of time, because earlier forcing and initial data decay exponentially and cannot influence the final answer beyond tolerance ε. If correct, this removes the main barrier to long-time simulation of dissipative systems: previous best algorithms scaled at best as √T for final states, so the improvement is more than exponential. The speedup rests on a genuine assumption—the dissipation rate η must be known in advance and the dissipation must satisfy a strict uniform condition.

Core claim

At the paper's center is the observation that for dissipative ODEs, the time-evolution operator satisfies ||T exp(∫_{t0}^{t1} A(s)ds)|| ≤ e^{-η(t1-t0)}. Consequently, once the physical time T exceeds T0 = (1/η)log(1/ε), everything that happened before T−T0 is exponentially suppressed. The authors exploit this by discretizing only the window [T−T0, T] for final states, or the first M0 = T0/h steps for homogeneous history states, using Gaussian quadrature, and by implementing the resulting short-time propagators with the time-marching method or with LCHS. Because T0 depends only on η and ε, the query complexity to the coefficient matrix becomes O(log^3(1/ε)) for time-marching and O(log^{3+o(1)

What carries the argument

The load-bearing object is the effective simulation window T0 = (1/η)log(1/ε), chosen so that the truncation error from omitting the interval [0, T−T0] is at most ε, by the decay bound ||T exp(∫_{t0}^{t1} A)|| ≤ e^{-η(t1-t0)}. On this short window the algorithms run the time-marching propagators (truncated Dyson series with uniform amplitude amplification, where the amplification parameter δ is set to Θ(ηh) instead of Θ(1/M)) or LCHS's integral representation of the propagator as a weighted sum of Hamiltonian simulations; a final LCU layer combines the Gaussian-quadrature nodes. This converts a long-time simulation problem into a short-time one whose length is independent of T, which is exac

Load-bearing premise

The scheme stands on knowing a uniform dissipation rate η ahead of time, since the truncation window is set by it and estimating η is QMA-hard in general; it also assumes strict uniform dissipativity A+A†≤−2η<0 and that the history-state normalization factor does not grow with T, without which the T-independent bound fails.

What would settle it

Implement the algorithm on the scalar equation u' = −ηu + b(t) with a fixed nonzero forcing b(t), constant η, and ε fixed, then count queries to the A-block-encoding for T ranging from 10 to 10^6 while T ≥ (1/η)log(1/ε). If the query count grows with T rather than remaining bounded independent of T, the central T-independence claim fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Final-state preparation for dissipative inhomogeneous ODEs drops from O(√T) queries to O(log^3(1/ε)) (time-marching) or O(log^{3+o(1)}(1/ε)) (LCHS), independent of T.
  • History-state preparation for homogeneous dissipative ODEs becomes T-independent at O(log^3(1/ε)) queries, filling a gap where evolution-based solvers had no history-state result.
  • Inhomogeneous dissipative history states can be prepared with O(log^3(1/ε)) queries despite containing O(T) time steps, by constructing the time-step superpositions with a stronger global HAM-T input model.
  • Semi-dissipative ODEs gain a new history-state preparation algorithm—O(T^2 log^2(T/ε)) with time-marching and O(T log^{2+o(1)}(T/ε)) with LCHS—while keeping state preparation cost low.
  • For applications such as non-Hermitian quantum dynamics and reaction-diffusion processes, the query count to the coefficient matrix no longer depends on the simulation horizon, so long-time simulation costs no more than short-time simulation.

Where Pith is reading between the lines

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

  • The same truncation argument may extend to the weaker Lyapunov condition P(t)A(t)+A†(t)P(t) ≤ −2η<0, where the effective window would be measured in the P-weighted norm; whether the query count remains polylogarithmic in 1/ε is a testable open question.
  • In reaction-diffusion problems, the lower bound η = π²a* ties the effective window to the smallest diffusion coefficient, suggesting that diffusion-dominated PDEs are simulable with cost scaling only in ε and spatial resolution, not in the time horizon T.
  • The QMA-hardness of estimating η means the practical payoff is largest when η is fixed by known physical parameters such as a measured decay rate or diffusion coefficient, rather than inferred from the dynamics.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper considers quantum algorithms for linear dissipative ODEs du/dt=A(t)u+b(t) with A(t)+A†(t)≤-2η<0. The central idea is to simulate only an effective time window T0=O((1/η)log(1/ε)), since dissipative propagators decay exponentially; this is used with both time-marching and LCHS to prepare final and history states with claimed query complexity O(log^3(1/ε)), independent of the physical time T, while keeping state-preparation cost low. The effective-window idea is clearly presented and the appendix gives extensive complexity analyses, but several load-bearing normalization factors Q are defined inconsistently with the truncation, so the headline claims are not established as written.

Significance. If the T-independence and polylogarithmic ε-dependence are correct, this would be a substantial improvement over prior dissipative ODE solvers (e.g., O(√T) final-state preparation) and would strengthen the case that dissipativity enables fast-forwarding. The paper contains no fitted parameters and the arguments are analytic; the application to non-Hermitian dynamics and reaction-diffusion is concrete. However, the complexity claims are currently undermined by unresolved Q-dependencies that may reintroduce T or ε, and the theorem statements hide important parameters. The core idea is promising, but a major revision is needed before the claims can be relied upon.

major comments (4)
  1. [Appendix C.2.2, Eqs. (85), (91)] Q in Eq. (85) is defined with M=T/h. Under dissipativity, ∥u(kh)∥≤e^{-ηkh}∥u(0)∥, so ∑_{k=0}^{M-1}∥u(kh)∥² is O(∥u(0)∥²/(1-e^{-2ηh})); hence 1/Q=O(1/√M) and Q=Ω(√T). Thus Eq. (91) as written carries an implicit √T and does not support the T-independent history-state claim. Replacing M by the effective M0 removes the √T, but gives Q=O(√(log(1/ε))), making the state-preparation term Q·Cost(O_init) dependent on ε and the matrix-query term O~(log^{7/2}(1/ε)), not O(log^3(1/ε)). This needs to be corrected and the statements in Result 2/Table 1 adjusted accordingly.
  2. [Appendix C.2.4, Eqs. (116), (118)] For the inhomogeneous dissipative history state, Q is defined with denominator ∥u(0)∥+∫_0^{jh}|b(t)|dt. With a constant nonzero source, u(jh) approaches a steady state while the integral grows like jh∥b∥∞; hence 1/Q=O(1/√M) and Q=Ω(√T). Consequently Eq. (118) does not establish the claimed T-independent O((log(1/ε))^3) complexity. The select oracle in Eq. (101) only integrates over the last r steps, so the denominator should be over the effective window [max(0,j-r)h,jh]; if that replacement is made, Q can be bounded independently of T, but the derivation must be rewritten.
  3. [Appendix C.1.3, Eqs. (68), (74); also D.1.3, Eq. (159)] The final-state complexity keeps Q=(∥b∥_{L1}+∥u(0)∥)/∥u(T)∥, where ∥b∥_{L1} is the L1 norm over the full [0,T]. For a constant source, ∥b∥_{L1}=Θ(T∥b∥∞) and ∥u(T)∥≈∥b∥∞/η, so Q=Θ(ηT), reintroducing linear T-dependence. The truncation to [T-T0,T] should replace Q by a quantity involving only the effective window, but this is not done in the displayed derivation. As written, Eq. (74) and the corresponding LCHS result do not support the T-independent final-state claim.
  4. [Result 2 and Tables 1/2] The main theorem statements suppress α_A, Q, η, and the state-preparation normalization factors. This is not a cosmetic issue: the appendix equations (74), (91), (118), (159), (172), (187) all contain Q factors that, as defined, can depend on T or ε and can dominate the claimed complexity. The paper should either state explicit bounds under which Q=O(1) or a bounded polylog factor, or restate the claims with the full dependencies. The present formulation overstates what the analysis proves.
minor comments (4)
  1. [Result 3] The two complexity options are both attributed to 'time-marching method'; the second, O(T log^{2+o(1)}(T/ε)), should refer to LCHS. Please fix.
  2. [Throughout] There are numerous typos: 'genenal', 'leval', 'redefinied', 'facor', 'quries', 'scles', 'exhcange', 'excahnge'. A thorough proofreading pass is needed.
  3. [Eq. (14) and Eq. (99)] The displayed state-preparation cost formulas are typeset in a garbled way; the square-root/division structure is hard to read. Please use clearer notation for the normalization factor and specify which state-preparation oracle is being counted.
  4. [Figure 1] The text refers to 'Figure 1' but no figure is included in the submitted text; please ensure the figure is present in the final version.

Circularity Check

0 steps flagged

No circularity found; the main T-independent claim has an internal proof gap (Eq. 85 vs 91) that is a correctness concern, not a circular reduction.

full rationale

The derivation is self-contained with respect to circularity. The fast-forwarding premise is the dissipative decay bound ||T e^{∫ A}|| ≤ e^{-ηΔt}, quoted as Lemma 1 from the authors' earlier [34]. This bound is an elementary one-line consequence of A+A† ≤ -2η (d/dt||u||² ≤ -2η||u||²), so the citation is independent support, not a self-referential premise. The LCHS representation and quadrature results are imported from [19] (which shares an author) but are standalone theorems with stated assumptions and are not the target T-independence claim. No parameter is fitted to data, and no 'prediction' is a renamed input: the algorithm's cost is derived from explicit block-encoding oracles and the chosen effective window T0 = O(η^{-1} log(1/ε)). The paper honestly records limitations (need for a priori η; QMA-hardness of estimating it; stronger-than-Lyapunov dissipativity). One genuine proof gap exists but is not circular: in Appendix C.2.2, Q is defined in Eq. (85) over all M = T/h steps, which for a homogeneous dissipative ODE gives Q = Ω(√T) because Σ_k ||u(kh)||² is bounded; Eq. (91) then carries this Q into the stated complexity, so the claimed T-independent O(log³(1/ε)) history-state bound does not follow as written unless Q is redefined over the effective window M0. That is an internal inconsistency / missing support, not an equivalence between a result and its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central complexity claims rest on the strict dissipativity assumption with known η, the exponential-decay lemma from [34], the LCHS and quadrature bounds from [19], and standard block-encoding and amplification methods. No free parameters are fit to data, and no new physical entities are introduced. The main unstated background assumptions are the smoothness of A(t), b(t) and the known-η requirement.

axioms (6)
  • domain assumption Lemma 1 from [34]: for A+A† ≤ -2η<0, ∥T e^{∫_{t0}^{t1} A}∥ ≤ e^{-η(t1-t0)}.
    The exponential-decay bound is the basis for truncating the simulation to the effective window [T-T0,T]. It is cited from the authors' prior work [34] and is a direct consequence of the dissipativity condition.
  • standard math LCHS representation of the propagator as a weighted integral of Hamiltonian simulations (Eq. 3) and the associated discretization bounds from [19].
    Used to express the truncated propagators T e^{∫_t^T A} as linear combinations of unitaries. The bounds on K, h1, Q are from [19] and are assumed valid.
  • domain assumption Gaussian quadrature error bound (Eq. 21) requires the integrand f(t)=T e^{∫_t^T A} b(t) to be sufficiently smooth with bounded high-order derivatives.
    The numerical integration error bound in Appendix B, inherited from [19], assumes smoothness of A(t) and b(t). The main text does not state this smoothness requirement explicitly.
  • standard math Time-marching block-encoding construction from [17,36] with truncated Dyson series and uniform amplitude amplification.
    The one-step propagator block-encodings Pℓ rely on these results, quoted in Appendix C. The possible quadratically underestimated error in [17] is addressed by the paper's own re-derivation.
  • domain assumption Block-encoding input models (HAM-TA,q, HAM-TA) and state preparation oracles for u(0) and b(t) exist with unit cost per query.
    The query complexity is measured in these oracles; the stronger global HAM-TA model needed for history states is assumed constructible with O(1) queries from the basic time-dependent block-encoding.
  • domain assumption A priori knowledge of η with constant multiplicative error.
    The truncation time T0 and the uniform amplitude amplification parameter δ depend on η. The paper notes estimating η is QMA-hard in general (Section 3.2).

reviewed 2026-08-05 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Differential Equation Solvers with Low State Preparation Cost: Eliminating the Time Dependence in Dissipative Equations." pith.science (2026). https://pith.science/paper/KU65PVYG

@misc{pith2026250815170,
  author       = {Pith},
  title        = {Pith review of: Quantum Differential Equation Solvers with Low State Preparation Cost: Eliminating the Time Dependence in Dissipative Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KU65PVYG}},
  note         = {Machine review of arXiv:2508.15170}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Linear dissipative differential equation is a fundamental model for a large number of physical systems, such as quantum dynamics with non-Hermitian Hamiltonian, open quantum system dynamics, diffusion process and damped system. In this work, we propose efficient quantum algorithms for simulating linear dissipative differential equations. The key idea of our algorithms is to perform the simulation only over an effective time period when the dynamics has not significantly dissipated yet, rather than over the entire physical evolution period. We conduct detailed analysis on the complexity of our algorithms and show that, while maintaining low state preparation cost, our algorithms can completely eliminate the time dependence. This is a more than exponential improvement compared to the previous state-of-the-art quantum algorithms.

Figures

Figures reproduced from arXiv: 2508.15170 by Akwum Onwunta, Dong An, Gengzhi Yang.

Figure 1
Figure 1. Figure 1: Examples of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

discussion (0)

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

Forward citations

Cited by 7 Pith papers

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

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

    quant-ph 2026-05 conditional novelty 8.0

    Develops a quantum algorithm for linear matrix differential equations with query complexity O~(ν L t / ε) that is nearly optimal and yields polynomial to exponential speedups for open quantum system simulation.

  2. Circuit Depth Reduction of One-Ancilla Quantum Differential Equation Solver via Extrapolation

    quant-ph 2026-07 accept novelty 7.0

    Classical step-size extrapolation reduces the maximum single-run circuit depth of a one-ancilla quantum ODE solver from O(1/ε) to O(polylog(1/ε)) without adding ancillae.

  3. Transmutation based Quantum Simulation for Non-unitary Dynamics

    quant-ph 2026-01 conditional novelty 7.0

    The Kannai transform turns dissipative quantum simulation for A = L†L into a Gaussian-weighted sum of unitary wave propagators, giving query complexity Õ(√(‖A‖T log(1/ε))) and a κ^{3/2} linear-solver corollary.

  4. Quantum Algorithms for Nonlinear Differential Equations via Pivot-Shifted Carleman Linearization

    quant-ph 2026-05 unverdicted novelty 6.0

    Pivot-shifted Carleman linearization with Lyapunov transform enables logarithmic truncation and removes initial-condition lower bounds for quantum simulation of a broader class of nonlinear ODEs.

  5. Quantum Simulation of Non-Hermitian Special Functions and Dynamics via Contour-based Matrix Decomposition

    quant-ph 2025-11 conditional novelty 6.0

    CBMD decomposes non-Hermitian evolution operators into Hermitian LCU terms via a matrix residue theorem, matching known optimal query bounds and offering a route to polynomial matrix functions.

  6. Quantum Simulation of Non-Hermitian Special Functions and Dynamics via Contour-based Matrix Decomposition

    quant-ph 2025-11 unverdicted novelty 6.0

    CBMD decomposes non-Hermitian operators via contour residues to enable optimal-query quantum simulation of first-order dynamics and special functions such as Bessel and Airy evolutions without requiring diagonalizability.

  7. Circuit-Efficient Randomized Quantum Simulation of Non-Unitary Dynamics with Observable-Driven and Symmetry-Aware Designs

    quant-ph 2025-09 reject novelty 5.0

    A randomized compilation of LCHS for non-unitary dynamics, with an observable-driven variant and a symmetry-aware sampler, claims reduced ancilla and circuit depth at the cost of more repetitions.

Reference graph

Works this paper leans on

67 extracted references · 36 canonical work pages · cited by 6 Pith papers · 1 internal anchor

  1. [1]

    Quantum algorithm for linear systems of equations

    Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. “Quantum algorithm for linear systems of equations”. In: Physical review letters 103.15 (2009), p. 150502

    work page 2009

  2. [2]

    Variable time amplitude amplification and quantum algorithms for linear algebra problems

    Andris Ambainis. “Variable time amplitude amplification and quantum algorithms for linear algebra problems”. In: STACS’12 (29th Symposium on Theoretical Aspects of Computer Science) . Vol. 14. LIPIcs. 2012, pp. 636–647

    work page 2012

  3. [3]

    Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision

    Andrew M. Childs, Robin Kothari, and Rolando D. Somma. “Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision”. In: SIAM Journal on Computing 46.6 (Jan. 2017), pp. 1920–1950. issn: 1095-7111. doi: 10.1137/16m1087072 . url: http://dx.doi. org/10.1137/16M1087072

    doi:10.1137/16m1087072 2017

  4. [4]

    Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Computing

    Yi˘ git Suba¸ sı, Rolando D. Somma, and Davide Orsucci. “Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Computing”. In: Physical Review Letters 122.6 (Feb. 2019). issn: 1079-7114. doi: 10 . 1103 / physrevlett . 122 . 060504. url: http : / / dx . doi . org / 10 . 1103 / PhysRevLett.122.060504

    work page 2019

  5. [5]

    Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm

    Dong An and Lin Lin. “Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm”. In: ACM Transactions on Quantum Computing 3.2 (2022), pp. 1–28

    work page 2022

  6. [6]

    Optimal scaling quantum linear-systems solver via discrete adiabatic theorem

    Pedro CS Costa et al. “Optimal scaling quantum linear-systems solver via discrete adiabatic theorem”. In: PRX quantum 3.4 (2022), p. 040303

    work page 2022

  7. [7]

    A shortcut to an optimal quantum linear system solver

    Alexander M Dalzell. “A shortcut to an optimal quantum linear system solver”. In: arXiv preprint arXiv:2406.12086 (2024)

    Pith/arXiv arXiv 2024

  8. [8]

    Quantum eigenvalue processing

    Guang Hao Low and Yuan Su. “Quantum eigenvalue processing”. In: arXiv preprint arXiv:2401.06240 (2024)

    arXiv 2024

  9. [9]

    High-order quantum algorithm for solving linear differential equations

    Dominic W Berry. “High-order quantum algorithm for solving linear differential equations”. In: Journal of Physics A: Mathematical and Theoretical 47.10 (2014), p. 105301

    2014

  10. [10]

    Quantum algorithm for linear differential equations with exponentially im- proved dependence on precision

    Dominic W Berry et al. “Quantum algorithm for linear differential equations with exponentially im- proved dependence on precision”. In: Communications in Mathematical Physics 356 (2017), pp. 1057– 1081

    work page 2017

  11. [11]

    Quantum algorithm for time-dependent differential equations using Dyson series

    Dominic W Berry and Pedro CS Costa. “Quantum algorithm for time-dependent differential equations using Dyson series”. In: Quantum 8 (2024), p. 1369

    work page 2024

  12. [12]

    Efficient quantum algorithm for dissipative nonlinear differential equations

    Jin-Peng Liu et al. “Efficient quantum algorithm for dissipative nonlinear differential equations”. In: Proceedings of the National Academy of Sciences 118.35 (2021), e2026805118

    work page 2021

  13. [13]

    Quantum spectral methods for differential equations

    Andrew M Childs and Jin-Peng Liu. “Quantum spectral methods for differential equations”. In: Com- munications in Mathematical Physics 375.2 (2020), pp. 1427–1457

    work page 2020

  14. [14]

    High-precision quantum algorithms for partial differential equations

    Andrew M Childs, Jin-Peng Liu, and Aaron Ostrander. “High-precision quantum algorithms for partial differential equations”. In: Quantum 5 (2021), p. 574

    2021

  15. [15]

    Improved quantum algorithms for linear and nonlinear differential equations

    Hari Krovi. “Improved quantum algorithms for linear and nonlinear differential equations”. In: Quan- tum 7 (2023), p. 913. 8

    work page 2023

  16. [16]

    Quantum linear system algorithm with optimal queries to initial state preparation

    Guang Hao Low and Yuan Su. “Quantum linear system algorithm with optimal queries to initial state preparation”. In: arXiv preprint arXiv:2410.18178 (2024)

    arXiv 2024

  17. [17]

    Time-marching based quantum solvers for time-dependent linear differential equations

    Di Fang, Lin Lin, and Yu Tong. “Time-marching based quantum solvers for time-dependent linear differential equations”. In: Quantum 7 (2023), p. 955

    work page 2023

  18. [18]

    Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost

    Dong An, Jin-Peng Liu, and Lin Lin. “Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost”. In: Physical Review Letters 131.15 (2023), p. 150603

    work page 2023

  19. [19]

    Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters

    Dong An, Andrew M Childs, and Lin Lin. “Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters”. In: arXiv preprint arXiv:2312.03916 (2023)

    arXiv 2023

  20. [20]

    Quantum simulation of partial differential equations via schrodin- gerisation: technical details

    Shi Jin, Nana Liu, and Yue Yu. “Quantum simulation of partial differential equations via schrodin- gerisation: technical details”. In: arXiv preprint arXiv:2212.14703 (2022)

    Pith/arXiv arXiv 2022

  21. [21]

    On the Schr¨ odingerization method for linear non-unitary dynamics with optimal depen- dence on matrix queries

    Shi Jin et al. On the Schr¨ odingerization method for linear non-unitary dynamics with optimal depen- dence on matrix queries . 2025. arXiv: 2505.00370 [math.NA]. url: https://arxiv.org/abs/2505. 00370

    arXiv 2025

  22. [22]

    From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms

    Xiantao Li. “From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms”. In: arXiv preprint arXiv:2507.10285 (2025)

    arXiv 2025

  23. [23]

    Design nearly optimal quantum algorithm for linear differential equations via Lindbladians

    Zhong-Xia Shang et al. Design nearly optimal quantum algorithm for linear differential equations via Lindbladians. 2024. arXiv: 2410.19628 [quant-ph]. url: https://arxiv.org/abs/2410.19628

    arXiv 2024

  24. [24]

    Qubit-Efficient Quantum Algorithm for Linear Differential Equations

    Di Fang, David Lloyd George, and Yu Tong. Qubit-Efficient Quantum Algorithm for Linear Differential Equations. 2025. arXiv: 2507.16995 [quant-ph]. url: https://arxiv.org/abs/2507.16995

    Pith/arXiv arXiv 2025

  25. [25]

    Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equa- tions in mathematical finance

    Dong An et al. “Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equa- tions in mathematical finance”. In: Quantum 5 (2021), p. 481

    work page 2021

  26. [26]

    Differentiable Quantum Computing for Large-scale Linear Control

    Connor Clayton et al. “Differentiable Quantum Computing for Large-scale Linear Control”. In: arXiv preprint arXiv:2411.01391 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  27. [27]

    Toward end-to-end quantum simulation for protein dynamics

    Zhenning Liu et al. “Toward end-to-end quantum simulation for protein dynamics”. In: arXiv preprint arXiv:2411.03972 (2024)

    Pith/arXiv arXiv 2024

  28. [28]

    Quantum simulation of partial differential equations: Applications and detailed analysis

    Shi Jin, Nana Liu, and Yue Yu. “Quantum simulation of partial differential equations: Applications and detailed analysis”. In: Physical Review A 108.3 (2023), p. 032603

    work page 2023

  29. [29]

    Towards provably efficient quantum algorithms for large-scale machine-learning models

    Junyu Liu et al. “Towards provably efficient quantum algorithms for large-scale machine-learning models”. In: Nature Communications 15.1 (2024), p. 434

    work page 2024

  30. [30]

    Exponential improvement in precision for simulating sparse Hamiltonians

    Dominic W Berry et al. “Exponential improvement in precision for simulating sparse Hamiltonians”. In: Proceedings of the forty-sixth annual ACM symposium on Theory of computing . 2014, pp. 283–292

    work page 2014

  31. [31]

    Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

    Andr´ as Gily´ en et al. “Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics”. In: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing . 2019, pp. 193–204

    work page 2019

  32. [32]

    Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series

    M´ aria Kieferov´ a, Artur Scherer, and Dominic W Berry. “Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series”. In: Physical Review A 99.4 (2019), p. 042314

    work page 2019

  33. [33]

    The cost of solving linear differential equations on a quantum computer: fast- forwarding to explicit resource counts

    David Jennings et al. “The cost of solving linear differential equations on a quantum computer: fast- forwarding to explicit resource counts”. In: Quantum 8 (2024), p. 1553

    work page 2024

  34. [34]

    Fast-forwarding quantum algorithms for linear dis- sipative differential equations

    Dong An, Akwum Onwunta, and Gengzhi Yang. “Fast-forwarding quantum algorithms for linear dis- sipative differential equations”. In: arXiv preprint arXiv:2410.13189 (2024)

    arXiv 2024

  35. [35]

    Simulating Hamiltonian Dynamics with a Truncated Taylor Series

    Dominic W. Berry et al. “Simulating Hamiltonian Dynamics with a Truncated Taylor Series”. In: Physical Review Letters 114.9 (Mar. 2015). issn: 1079-7114. doi: 10.1103/physrevlett.114.090502. url: http://dx.doi.org/10.1103/PhysRevLett.114.090502

    doi:10.1103/physrevlett.114.090502 2015

  36. [36]

    Hamiltonian simulation in the interaction picture

    Guang Hao Low and Nathan Wiebe. “Hamiltonian simulation in the interaction picture”. In: arXiv preprint arXiv:1805.00675 (2018)

    Pith/arXiv arXiv 2018

  37. [37]

    In: Quantum Information and Computation 12.11 & 12 (Nov. 2012). issn: 1533-7146. doi: 10.26421/ qic12.11-12. url: http://dx.doi.org/10.26421/QIC12.11-12. 9

    doi:10.26421/qic12.11-12 2012

  38. [38]

    Quantum NP-a survey

    Dorit Aharonov and Tomer Naveh. “Quantum NP-a survey”. In: arXiv preprint quant-ph/0210077 (2002)

    Pith/arXiv arXiv 2002

  39. [39]

    The power of quantum systems on a line

    Dorit Aharonov et al. “The power of quantum systems on a line”. In: Communications in mathematical physics 287.1 (2009), pp. 41–65

    work page 2009

  40. [40]

    The complexity of the local Hamiltonian problem

    Julia Kempe, Alexei Kitaev, and Oded Regev. “The complexity of the local Hamiltonian problem”. In: Siam journal on computing 35.5 (2006), pp. 1070–1097

    work page 2006

  41. [41]

    Classical and quantum computation

    Alexei Yu Kitaev, Alexander Shen, and Mikhail N Vyalyi. Classical and quantum computation . 47. American Mathematical Soc., 2002

    work page 2002

  42. [42]

    The complexity of quantum spin systems on a two-dimensional square lattice

    Roberto Oliveira and Barbara M Terhal. “The complexity of quantum spin systems on a two-dimensional square lattice”. In: arXiv preprint quant-ph/0504050 (2005)

    Pith/arXiv arXiv 2005

  43. [43]

    Ground-state preparation and energy estimation on early fault- tolerant quantum computers via quantum eigenvalue transformation of unitary matrices

    Yulong Dong, Lin Lin, and Yu Tong. “Ground-state preparation and energy estimation on early fault- tolerant quantum computers via quantum eigenvalue transformation of unitary matrices”. In: PRX Quantum 3.4 (2022), p. 040305

    work page 2022

  44. [44]

    Near-optimal ground state preparation

    Lin Lin and Yu Tong. “Near-optimal ground state preparation”. In: Quantum 4 (2020), p. 372

    work page 2020

  45. [45]

    Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers

    Lin Lin and Yu Tong. “Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers”. In: PRX Quantum 3.1 (2022), p. 010318

    work page 2022

  46. [46]

    Single-ancilla ground state preparation via Lindbladians

    Zhiyan Ding, Chi-Fang Chen, and Lin Lin. “Single-ancilla ground state preparation via Lindbladians”. In: Physical Review Research 6.3 (2024), p. 033147

    work page 2024

  47. [47]

    Dissipative preparation of many-body quantum states: Towards practical quantum advan- tage

    Lin Lin. “Dissipative preparation of many-body quantum states: Towards practical quantum advan- tage”. In: arXiv preprint arXiv:2505.21308 (2025)

    arXiv 2025

  48. [48]

    End-to-End Efficient Quantum Thermal and Ground State Preparation Made Simple

    Zhiyan Ding et al. “End-to-End Efficient Quantum Thermal and Ground State Preparation Made Simple”. In: arXiv preprint arXiv:2508.05703 (2025)

    arXiv 2025

  49. [49]

    Rapid quantum ground state preparation via dissipative dynamics

    Yongtao Zhan et al. “Rapid quantum ground state preparation via dissipative dynamics”. In: arXiv preprint arXiv:2503.15827 (2025)

    arXiv 2025

  50. [50]

    Fast-forwarding of Hamiltonians and exponentially precise measure- ments

    Yosi Atia and Dorit Aharonov. “Fast-forwarding of Hamiltonians and exponentially precise measure- ments”. In: Nature communications 8.1 (2017), p. 1572

    work page 2017

  51. [51]

    Fast-forwarding quantum evolution

    Shouzhen Gu, Rolando D Somma, and Burak S ¸ahino˘ glu. “Fast-forwarding quantum evolution”. In: Quantum 5 (2021), p. 577

    work page 2021

  52. [52]

    A theory of quantum differential equation solvers: limitations and fast-forwarding

    Dong An et al. “A theory of quantum differential equation solvers: limitations and fast-forwarding”. In: arXiv preprint arXiv:2211.05246 (2022)

    Pith/arXiv arXiv 2022

  53. [53]

    Making sense of non-Hermitian Hamiltonians

    Carl M Bender. “Making sense of non-Hermitian Hamiltonians”. In: Reports on Progress in Physics 70.6 (May 2007), pp. 947–1018. issn: 1361-6633. doi: 10.1088/0034- 4885/70/6/r03 . url: http: //dx.doi.org/10.1088/0034-4885/70/6/R03

    doi:10.1088/0034- 2007

  54. [54]

    Classical field theory for a non-Hermitian Schr¨ odinger equation with position-dependent masses

    M. A. Rego-Monteiro and F. D. Nobre. “Classical field theory for a non-Hermitian Schr¨ odinger equation with position-dependent masses”. In: Phys. Rev. A 88 (3 Sept. 2013), p. 032105. doi: 10 . 1103 / PhysRevA.88.032105. url: https://link.aps.org/doi/10.1103/PhysRevA.88.032105

    work page doi:10.1103/physreva.88.032105 2013

  55. [55]

    Non-Hermitian Hamiltonian approach to quantum transport in disordered networks with sinks: Validity and effectiveness

    Giulio G. Giusteri, Francesco Mattiotti, and G. Luca Celardo. “Non-Hermitian Hamiltonian approach to quantum transport in disordered networks with sinks: Validity and effectiveness”. In:Physical Review B 91.9 (Mar. 2015). issn: 1550-235X. doi: 10.1103/physrevb.91.094301 . url: http://dx.doi. org/10.1103/PhysRevB.91.094301

    work page doi:10.1103/physrevb.91.094301 2015

  56. [56]

    Non-Hermitian physics and PT symmetry

    Ramy El-Ganainy et al. “Non-Hermitian physics and PT symmetry”. In: Nature Physics 14.1 (2018), pp. 11–19. doi: 10.1038/nphys4323. url: https://doi.org/10.1038/nphys4323

    doi:10.1038/nphys4323 2018

  57. [57]

    Topological Phases of Non-Hermitian Systems

    Zongping Gong et al. “Topological Phases of Non-Hermitian Systems”. In: Physical Review X 8.3 (Sept. 2018). issn: 2160-3308. doi: 10.1103/physrevx.8.031079 . url: http://dx.doi.org/10. 1103/PhysRevX.8.031079

    doi:10.1103/physrevx.8.031079 2018

  58. [58]

    Topological Origin of Non-Hermitian Skin Effects

    Nobuyuki Okuma et al. “Topological Origin of Non-Hermitian Skin Effects”. In: Phys. Rev. Lett. 124 (8 Feb. 2020), p. 086801. doi: 10.1103/PhysRevLett.124.086801. 10

    doi:10.1103/physrevlett.124.086801 2020

  59. [59]

    Non-Hermitian physics

    Yuto Ashida, Zongping Gong, and Masahito Ueda. “Non-Hermitian physics”. In: Advances in Physics 69.3 (July 2020), pp. 249–435. doi: 10.1080/00018732.2021.1876991 . url: https://doi.org/10. 1080/00018732.2021.1876991

    arXiv 2020

  60. [60]

    Continuous Phase Transition without Gap Closing in Non-Hermitian Quantum Many-Body Systems

    Norifumi Matsumoto et al. “Continuous Phase Transition without Gap Closing in Non-Hermitian Quantum Many-Body Systems”. In: Phys. Rev. Lett. 125 (26 Dec. 2020), p. 260601. doi: 10.1103/ PhysRevLett.125.260601

    work page 2020

  61. [61]

    Non-Hermitian topology and exceptional-point geome- tries

    Kun Ding, Chen Fang, and Guancong Ma. “Non-Hermitian topology and exceptional-point geome- tries”. In: Nature Reviews Physics 4.12 (Oct. 2022), pp. 745–760. issn: 2522-5820. doi: 10 . 1038 / s42254-022-00516-5 . url: http://dx.doi.org/10.1038/s42254-022-00516-5

    doi:10.1038/s42254-022-00516-5 2022

  62. [62]

    Topological Spin Excitations in Non-Hermitian Spin Chains with a Generalized Kernel Polynomial Algorithm

    Guangze Chen, Fei Song, and Jose L. Lado. “Topological Spin Excitations in Non-Hermitian Spin Chains with a Generalized Kernel Polynomial Algorithm”. In: Phys. Rev. Lett. 130 (10 Mar. 2023), p. 100401. doi: 10.1103/PhysRevLett.130.100401

    work page doi:10.1103/physrevlett.130.100401 2023

  63. [63]

    Exact Solution of the Bose-Hubbard Model with Unidirectional Hopping

    Mingchen Zheng et al. “Exact Solution of the Bose-Hubbard Model with Unidirectional Hopping”. In: Phys. Rev. Lett. 132 (8 Feb. 2024), p. 086502. doi: 10 . 1103 / PhysRevLett . 132 . 086502. url: https://link.aps.org/doi/10.1103/PhysRevLett.132.086502

    doi:10.1103/physrevlett.132.086502 2024

  64. [64]

    Non-Hermitian Fermi-Dirac Distribution in Persistent Current Transport

    Pei-Xin Shen et al. “Non-Hermitian Fermi-Dirac Distribution in Persistent Current Transport”. In: Phys. Rev. Lett. 133 (8 Aug. 2024), p. 086301. doi: 10.1103/PhysRevLett.133.086301. url: https: //link.aps.org/doi/10.1103/PhysRevLett.133.086301

    doi:10.1103/physrevlett.133.086301 2024

  65. [65]

    Lawrence C. Evans. Partial differential equations . Vol. 19. American Mathematical Soc., 2010

    work page 2010

  66. [66]

    Hundsdorfer and Jan G

    Willem H. Hundsdorfer and Jan G. Verwer. Numerical solution of time-dependent advection-diffusion- reaction equations. Vol. 33. Springer, 2003

    work page 2003

  67. [67]

    Growth, reaction, movement and diffusion from biology

    Benoıt Perthame. “Growth, reaction, movement and diffusion from biology”. In: Lecture Notes, Uni- versity Paris 6 (2012). 11 A Input models In this section, we discuss the input models in our algorithms with more details. Block-encodings. We say UA is an ( α, a, ϵ) block-encoding of a matrix A if UA is a unitary and ∥(α ⟨0a| ⊗I)UA(|0a⟩ ⊗I) − A∥ ≤ϵ. (15) S...

    work page 2012

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.