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arxiv: 2606.11475 · v1 · pith:7PHVIFA4new · submitted 2026-06-09 · 🪐 quant-ph · cs.NA· math.NA

Linear Combination of Hamiltonian Simulation with Commutator Scaling

Junaid Aftab , Dong An , Konstantina Trivisa This is my paper

Pith reviewed 2026-06-27 12:56 UTC · model grok-4.3

classification 🪐 quant-ph cs.NAmath.NA
keywords linear combination of Hamiltonian simulationmulti-product formulascommutator scalingsinh-sinh quadraturedissipative dynamicsHamiltonian simulationopen quantum systems
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The pith

Multi-product formulas in linear combination of Hamiltonian simulation yield commutator-sensitive error bounds that improve with sinh-sinh quadrature.

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

The paper shows that the LCHS representation of dissipative dynamics as an integral over unitaries can be discretized and implemented so that the Hamiltonian simulation steps use multi-product formulas, producing overall error and complexity estimates controlled by nested commutators rather than operator norms. This dependence arises because the quadrature rule shapes both the discretization error and the commutator structure passed to the MPF bounds. The free-scale sinh-sinh quadrature reduces the number of quadrature points required compared with uniform trapezoidal rules while preserving the commutator structure needed for the MPF estimates. The resulting bounds apply directly to time-independent and local Hamiltonians without extra restrictions on the dissipative generator. The improvement matters for concrete models such as fractional diffusion and open quantum systems where norm-based analyses remain loose.

Core claim

Implementing the Hamiltonian simulation steps inside the LCHS framework with multi-product formulas produces error and complexity bounds that scale with commutators of the Hamiltonian terms. Post-quadrature analysis shows that the chosen quadrature rule determines both the discretization error and the commutator profile fed into the MPF error estimates. For general time-independent and local Hamiltonians, the free-scale sinh-sinh quadrature achieves better cardinality scaling than uniform trapezoidal quadrature while remaining compatible with known commutator-sensitive MPF bounds.

What carries the argument

The post-quadrature commutator structure induced by the quadrature rule, which is then bounded using known commutator-sensitive MPF error estimates.

If this is right

  • Error bounds for LCHS simulation become sensitive to nested commutators instead of norms alone.
  • Query complexity scales with the quadrature cardinality, which improves under free-scale sinh-sinh rules.
  • The same commutator analysis applies without change to fractional diffusion, advection-diffusion, and open quantum systems.
  • Commutator-sensitive MPF estimates carry over directly once the quadrature is fixed.

Where Pith is reading between the lines

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

  • The same post-quadrature analysis could be applied to other integral representations of dissipative dynamics if analogous MPF bounds exist.
  • Numerical verification on a concrete fractional Laplacian model would test whether the predicted cardinality reduction appears in practice.
  • Extending the commutator analysis to time-dependent generators would require new MPF estimates but could reuse the quadrature comparison.

Load-bearing premise

The commutator structure left after quadrature stays compatible with existing MPF error estimates for time-independent and local Hamiltonians.

What would settle it

An explicit calculation for a local advection-diffusion Hamiltonian showing that the sinh-sinh quadrature produces commutators whose MPF error profile fails to improve cardinality over the trapezoidal rule.

read the original abstract

The Linear Combination of Hamiltonian Simulation (LCHS) framework simulates dissipative linear dynamics by representing time evolution as an integral over unitary operators, which is discretized by quadrature and implemented via Hamiltonian simulation. While existing analyses achieve near-optimal scaling in time and precision using norm-based quantities of the dissipative generator, we show that implementing the Hamiltonian simulation steps with Multi-Product Formulas (MPFs) yields commutator-sensitive error and complexity bounds. We demonstrate that the quadrature rule affects not only discretization error but also commutator structure and query complexity. This dependence is quantified through post-quadrature analysis for abstract MPF error profiles and for general time-independent and local Hamiltonians using known commutator-sensitive MPF error estimates. We compare uniform trapezoidal and free-scale sinh--sinh quadrature, showing improved quadrature-cardinality scaling for the latter, and illustrate the framework with applications to fractional diffusion, advection--diffusion, and open quantum systems.

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

1 major / 2 minor

Summary. The manuscript extends the Linear Combination of Hamiltonian Simulation (LCHS) framework for dissipative linear dynamics by representing the evolution as an integral over unitary operators, discretizing via quadrature, and implementing each term with Multi-Product Formulas (MPFs). It derives commutator-sensitive error and complexity bounds, shows that the quadrature rule influences both discretization error and post-quadrature commutator structure, and compares uniform trapezoidal quadrature with free-scale sinh-sinh quadrature (claiming improved cardinality scaling for the latter). The analysis covers abstract MPF error profiles and applies known commutator-sensitive MPF estimates to time-independent and local Hamiltonians, with illustrations for fractional diffusion, advection-diffusion, and open quantum systems.

Significance. If the post-quadrature compatibility holds, the work supplies a route to commutator-sensitive (rather than norm-based) scaling in LCHS simulations of dissipative systems, which can improve query complexity for local Hamiltonians. The explicit post-quadrature analysis for both abstract MPF profiles and concrete Hamiltonian classes, together with the quadrature comparison, is a concrete strength that supports falsifiable complexity claims.

major comments (1)
  1. [Post-quadrature analysis] Post-quadrature analysis (the section applying MPF estimates after discretization): the central claim applies known commutator-sensitive MPF error estimates to the quadrature nodes, but does not exhibit an explicit verification that the effective generators (original Hamiltonian shifted by terms involving the dissipative generator) preserve the locality and bounded nested-commutator assumptions of the cited MPF theorems when the dissipative generator is arbitrary. This is load-bearing for the claimed bounds without extra restrictions on the generator.
minor comments (2)
  1. The abstract states that sinh-sinh quadrature improves cardinality scaling, but the main text should include a side-by-side table of the resulting total query complexities (in terms of commutator norms, time, and precision) to make the improvement quantitative.
  2. Notation for the dissipative generator and the quadrature weights could be unified across sections to avoid redefinition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The identification of the post-quadrature analysis as load-bearing is helpful, and we address the major comment directly below.

read point-by-point responses

  1. Referee: [Post-quadrature analysis] Post-quadrature analysis (the section applying MPF estimates after discretization): the central claim applies known commutator-sensitive MPF error estimates to the quadrature nodes, but does not exhibit an explicit verification that the effective generators (original Hamiltonian shifted by terms involving the dissipative generator) preserve the locality and bounded nested-commutator assumptions of the cited MPF theorems when the dissipative generator is arbitrary. This is load-bearing for the claimed bounds without extra restrictions on the generator.

    Authors: We agree that an explicit verification is needed to support the claimed bounds without additional restrictions. The manuscript applies the MPF estimates to time-independent and local Hamiltonians after quadrature, but does not detail how the dissipative-generator shifts affect locality and nested-commutator bounds for arbitrary dissipative generators. In the revision we will add a short clarifying paragraph in the post-quadrature section: when both the Hamiltonian and dissipative generator are local (as assumed in the applications to fractional diffusion, advection-diffusion, and open quantum systems), the effective generators at quadrature nodes remain local with bounded nested commutators, so the cited MPF theorems apply directly. For completely arbitrary dissipative generators without locality, we will note that the commutator-sensitive bounds require the additional assumption that the effective operators satisfy the MPF hypotheses. This makes the load-bearing conditions explicit while preserving the scope of the stated results. revision: yes

Circularity Check

0 steps flagged

No circularity: relies on external known MPF estimates without reduction to self-inputs

full rationale

The paper's derivation applies known commutator-sensitive MPF error estimates (explicitly described as external) to post-quadrature operators for time-independent and local Hamiltonians. The abstract and description contain no self-definitional equations, no fitted parameters renamed as predictions, and no load-bearing self-citations that reduce the central claims to the paper's own inputs. The quadrature comparison and commutator scaling follow from standard quadrature theory and the cited external bounds, making the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard mathematical assumptions from quantum simulation and numerical quadrature; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Known commutator-sensitive MPF error estimates apply to the post-quadrature operators for time-independent and local Hamiltonians.
    The paper invokes these estimates to obtain the commutator-sensitive bounds.

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Works this paper leans on

87 extracted references · 16 canonical work pages · 5 internal anchors

  1. [1]

    Simulating physics with computers

    Richard P. Feynman. “Simulating physics with computers”. In: International Journal of Theoretical Physics 21 (1982), pp. 467–488

    1982

  2. [2]

    Quantum mechanical computers

    Richard P. Feynman. “Quantum mechanical computers”. In: Foundations of Physics 16.6 (1986), pp. 507–531

    1986

  3. [3]

    Quantum algorithms for quantum field theo- ries

    Stephen P. Jordan, Keith S.M. Lee, and John Preskill. “Quantum algorithms for quantum field theo- ries”. In: Science 336.6085 (2012), pp. 1130–1133

    2012

  4. [4]

    Quantum algorithms for quantum chemistry and quantum materials science

    Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan. “Quantum algorithms for quantum chemistry and quantum materials science”. In: Chemical Reviews 120.22 (2020), pp. 12685–12717

    2020

  5. [5]

    Quantum chemistry in the age of quantum computing

    Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, M´ aria Kieferov´ a, Ian D. Kivlichan, Tim Menke, Borja Peropadre, and Nicolas P.D. et al. Sawaya. “Quantum chemistry in the age of quantum computing”. In: Chemical Reviews 119.19 (2019), pp. 10856–10915

    2019

  6. [6]

    Low-depth quantum simulation of materials

    Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan. “Low-depth quantum simulation of materials”. In: Physical Review X 8.1 (2018), p. 011044

    2018

  7. [7]

    Universal Quantum Simulators

    Seth Lloyd. “Universal Quantum Simulators”. In: Science 273.5278 (1996), pp. 1073–1078

    1996

  8. [8]

    Efficient quantum algo- rithms for simulating sparse Hamiltonians

    Dominic W. Berry, Graeme Ahokas, Richard Cleve, and Barry C. Sanders. “Efficient quantum algo- rithms for simulating sparse Hamiltonians”. In:Communications in Mathematical Physics 270.2 (2007), pp. 359–371

    2007

  9. [9]

    Black-box Hamiltonian simulation and unitary implemen- tation

    Dominic W. Berry and Andrew M. Childs. “Black-box Hamiltonian simulation and unitary implemen- tation”. In: Quantum Information & Computation 12.1-2 (2012), pp. 29–62

    2012

  10. [10]

    Expo- nential improvement in precision for simulating sparse Hamiltonians

    Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. “Expo- nential improvement in precision for simulating sparse Hamiltonians”. In:Proceedings of the forty-sixth annual ACM symposium on Theory of computing . 2014, pp. 283–292

    2014

  11. [11]

    Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters

    Dominic W. Berry, Andrew M. Childs, and Robin Kothari. “Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters”. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science. IEEE, Oct. 2015

    2015

  12. [12]

    Simu- lating Hamiltonian dynamics with a truncated Taylor series

    Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. “Simu- lating Hamiltonian dynamics with a truncated Taylor series”. In:Phys. Rev. Lett. 114 (2015), p. 090502. 42

    2015

  13. [13]

    Optimal Hamiltonian Simulation by Quantum Signal Process- ing

    Guang Hao Low and Isaac L. Chuang. “Optimal Hamiltonian Simulation by Quantum Signal Process- ing”. In: Physical Review Letters 118.1 (Jan. 2017)

    2017

  14. [14]

    Well-conditioned multiproduct hamiltonian sim- ulation,

    Guang Hao Low, Vadym Kliuchnikov, and Nathan Wiebe. Well-conditioned multiproduct Hamiltonian simulation. 2019. arXiv: 1907.11679 [quant-ph]

    work page arXiv 2019

  15. [15]

    Theory of Trotter Error with Commutator Scaling

    Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. “Theory of Trotter Error with Commutator Scaling”. In: Phys. Rev. X 11 (1 2021), p. 011020

    2021

  16. [16]

    Multi-product Hamiltonian simulation with explicit commutator scaling

    Junaid Aftab, Dong An, and Konstantina Trivisa. Multi-product Hamiltonian simulation with explicit commutator scaling. Submitted to Communications in Mathematical Physics. 2024. arXiv: 2403.08922 [quant-ph]

    work page arXiv 2024

  17. [17]

    On the commutator scaling in Hamiltonian simulation with multi-product formulas

    Kaoru Mizuta. “On the commutator scaling in Hamiltonian simulation with multi-product formulas”. In: Quantum 10 (2026), p. 1974

    2026

  18. [18]

    Higher order decompositions of ordered operator exponentials

    Nathan Wiebe, Dominic Berry, Peter Høyer, and Barry C. Sanders. “Higher order decompositions of ordered operator exponentials”. In: Journal of Physics A: Mathematical and Theoretical 43.6 (2010), p. 065203

    2010

  19. [19]

    Hamiltonian Simulation in the Interaction Picture

    Guang Hao Low and Nathan Wiebe. Hamiltonian Simulation in the Interaction Picture . 2019. arXiv: 1805.00675 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  20. [20]

    Time-dependent Hamiltonian simulation with L1-norm scaling

    Dominic W. Berry, Andrew M. Childs, Yuan Su, Xin Wang, and Nathan Wiebe. “Time-dependent Hamiltonian simulation with L1-norm scaling”. In: Quantum 4 (2020), p. 254

    2020

  21. [22]

    Time-dependent Hamiltonian simulation of highly oscillatory dy- namics and superconvergence for Schr¨ odinger equation

    Dong An, Di Fang, and Lin Lin. “Time-dependent Hamiltonian simulation of highly oscillatory dy- namics and superconvergence for Schr¨ odinger equation”. In:Quantum 6 (2022), p. 690

    2022

  22. [23]

    Mizuta, T

    Kaoru Mizuta, Tatsuhiko N. Ikeda, and Keisuke Fujii. Explicit error bounds with commutator scaling for time-dependent product and multi-product formulas . 2024. arXiv: 2410.14243 [quant-ph]

    work page arXiv 2024

  23. [24]

    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

  24. [25]

    Quantum algorithm for linear differential equations with exponentially improved dependence on precision

    Dominic W. Berry, Andrew M. Childs, Aaron Ostrander, and Guoming Wang. “Quantum algorithm for linear differential equations with exponentially improved dependence on precision”. In: Communi- cations in Mathematical Physics 356.3 (2017), pp. 1057–1081

    2017

  25. [26]

    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

    2020

  26. [27]

    Improved quantum algorithms for linear and nonlinear differential equations

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

    2023

  27. [28]

    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

    2023

  28. [29]

    Quantum algorithm for time-dependent differential equa- tions using Dyson series

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

    2024

  29. [30]

    Quantum simulation of partial differential equations via Schr¨ odingerization

    Shi Jin, Nana Liu, and Yue Yu. “Quantum simulation of partial differential equations via Schr¨ odingerization”. In:Physical Review Letters 133.23 (2024), p. 230602

    2024

  30. [31]

    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: Quantum 10 (Mar. 2026), p. 2041. issn: 2521-327X

    2026

  31. [32]

    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

    2009

  32. [33]

    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 (2017), pp. 1920–1950. 43

    2017

  33. [34]

    Optimal Scaling Quantum Linear-Systems Solver via Discrete Adiabatic Theorem

    Pedro C. S. Costa, Dong An, Yuval R. Sanders, Yuan Su, Ryan Babbush, and Dominic W. Berry. “Optimal Scaling Quantum Linear-Systems Solver via Discrete Adiabatic Theorem”. In:PRX Quantum 3 (4 Oct. 2022)

    2022

  34. [35]

    Alexander M. Dalzell. A shortcut to an optimal quantum linear system solver . 2024. arXiv: 2406.12086 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  35. [36]

    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)

    2023

  36. [37]

    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: Communications in Mathematical Physics 407.1 (2026), p. 19

    2026

  37. [38]

    Guang Hao Low and Rolando D. Somma. Optimal quantum simulation of linear non-unitary dynamics

  38. [39]

    arXiv: 2508.19238 [quant-ph]

    work page arXiv

  39. [40]

    Hamiltonian Simulation Using Linear Combinations of Unitary Operations

    Andrew M. Childs and Nathan Wiebe. “Hamiltonian Simulation Using Linear Combinations of Unitary Operations”. In: Quantum Information and Computation 12 (2012), pp. 901–924

    2012

  40. [41]

    Bloch Equations with Diffusion Terms

    H. C. Torrey. “Bloch Equations with Diffusion Terms”. In: Physical Review 104.3 (1956), pp. 563–565

    1956

  41. [42]

    Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation

    Richard L. Magin, Omar Abdullah, Dumitru Baleanu, and Xiaohong Joe Zhou. “Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation”. In: Journal of Magnetic Resonance 190.2 (2008), pp. 255–270

    2008

  42. [43]

    Numerical investigation of three types of space and time fractional Bloch–Torrey equations in 2D

    Qiang Yu, Fawang Liu, Ian Turner, and Kevin Burrage. “Numerical investigation of three types of space and time fractional Bloch–Torrey equations in 2D”. In: Central European Journal of Physics 11.6 (2013), pp. 646–665

    2013

  43. [44]

    Exact solutions to the fractional time-space Bloch–Torrey equation for magnetic resonance imaging

    Alfonso Bueno-Orovio and Kevin Burrage. “Exact solutions to the fractional time-space Bloch–Torrey equation for magnetic resonance imaging”. In: Communications in Nonlinear Science and Numerical Simulation 52 (2017), pp. 91–109

    2017

  44. [45]

    Diffusion NMR in periodic media: efficient computation and spectral properties

    Nicolas Moutal, Antoine Moutal, and Denis S. Grebenkov. “Diffusion NMR in periodic media: efficient computation and spectral properties”. In: Journal of Physics A: Mathematical and Theoretical 53.32 (2020), p. 325201. arXiv: 2005.06975

    work page arXiv 2020

  45. [46]

    Analytical solution of the convection-diffusion-reaction-source (CDRS) equation using Green’s function technique

    Mohammad Parhizi, Gozdem Kilaz, Jason K. Ostanek, and Ankur Jain. “Analytical solution of the convection-diffusion-reaction-source (CDRS) equation using Green’s function technique”. In: Interna- tional Communications in Heat and Mass Transfer 131 (2022), p. 105869

    2022

  46. [47]

    A study about the solution of convection-diffusion-reaction equation with Danckwerts boundary conditions by analytical, method of lines and Crank–Nicholson techniques

    Luc´ ıa Agud Albesa, Marta Boix Garc´ ıa, Laura Pla-Ferrando, and Salvador C. Cardona. “A study about the solution of convection-diffusion-reaction equation with Danckwerts boundary conditions by analytical, method of lines and Crank–Nicholson techniques”. In:Mathematical Methods in the Applied Sciences (2023)

    2023

  47. [48]

    CDR-Solv: Solving the convection-diffusion-reaction equation with algebraic sub-grid scale stabilization using Python

    ´Angel P. Villota-Cadena, Iv´ an P. Sandoval-Palis, Gabriel F. Grijalva-Villegas, and Erick P. Herrera- Granda. “CDR-Solv: Solving the convection-diffusion-reaction equation with algebraic sub-grid scale stabilization using Python”. In: Applied Sciences 15.18 (2025), p. 10256

    2025

  48. [49]

    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: Quantum 10 (2026), p. 1986

    2026

  49. [50]

    Wave-function approach to dissipative processes in quantum optics

    Jean Dalibard, Yvan Castin, and Klaus Mølmer. “Wave-function approach to dissipative processes in quantum optics”. In: Physical Review Letters 68.5 (1992), pp. 580–583

    1992

  50. [51]

    The quantum-jump approach to dissipative dynamics in quantum optics

    M. B. Plenio and P. L. Knight. “The quantum-jump approach to dissipative dynamics in quantum optics”. In: Reviews of Modern Physics 70.1 (1998), pp. 101–144

    1998

  51. [52]

    Dissipative transverse-field Ising model: Steady-state correlations and spin squeezing

    Tony E. Lee and Ching-Kit Chan. “Dissipative transverse-field Ising model: Steady-state correlations and spin squeezing”. In: Physical Review A 88.6 (2013), p. 063811

    2013

  52. [53]

    Exact Solution of the Infinite-Range Dissipative Transverse- Field Ising Model

    David Roberts and Aashish A. Clerk. “Exact Solution of the Infinite-Range Dissipative Transverse- Field Ising Model”. In: Physical Review Letters 131.19 (2023), p. 190403. 44

    2023

  53. [54]

    Lind- bladian Simulation with Commutator Bounds

    Xinzhao Wang, Shuo Zhou, Xiaoyang Wang, Yi-Cong Zheng, Shengyu Zhang, and Tongyang Li. Lind- bladian Simulation with Commutator Bounds . 2026. arXiv: 2603.28602 [quant-ph]

    work page arXiv 2026

  54. [55]

    Childs, Lin Lin, and Lexing Ying

    Dong An, Andrew M. Childs, Lin Lin, and Lexing Ying. Laplace Transform Based Quantum Eigen- value Transformation via Linear Combination of Hamiltonian Simulation . 2024. arXiv: 2411.04010 [quant-ph]

    work page arXiv 2024

  55. [56]

    Quantum eigenvalue processing

    Guang Hao Low and Yuan Su. Quantum Eigenvalue Processing. 2024. arXiv: 2401.06240 [quant-ph]

    work page arXiv 2024

  56. [57]

    Quantum Algorithms based on the Block-Encoding Framework for Matrix Functions by Contour Integrals

    Souichi Takahira, Asuka Ohashi, Tomohiro Sogabe, and Tsuyoshi Sasaki Usuda. Quantum Algorithms based on the Block-Encoding Framework for Matrix Functions by Contour Integrals. 2021. arXiv: 2106. 08076 [quant-ph]

    2021

  57. [58]

    Contour-integral Based Quantum Eigenvalue Transformation: Analysis and Applications

    Shan Jiang and Dong An. Contour-integral Based Quantum Eigenvalue Transformation: Analysis and Applications. 2026. arXiv: 2601.11959 [quant-ph]

    work page arXiv 2026

  58. [59]

    Quantum Simulation of Non-unitary Dynamics via Contour-based Matrix Decomposition

    Chao Wang, Huan-Yu Liu, Cheng Xue, Xi-Ning Zhuang, Menghan Dou, Zhao-Yun Chen, and Guo- Ping Guo. Quantum Simulation of Non-unitary Dynamics via Contour-based Matrix Decomposition

  59. [60]

    arXiv: 2511.10267 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  60. [61]

    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

    2023

  61. [62]

    On the Schr¨ odingerization Method for Linear Non-unitary Dynamics with Optimal Dependence on Matrix Queries

    Shi Jin, Nana Liu, Chuwen Ma, and Yue Yu. On the Schr¨ odingerization Method for Linear Non-unitary Dynamics with Optimal Dependence on Matrix Queries . 2025. arXiv: 2505.00370 [quant-ph]

    work page arXiv 2025

  62. [63]

    Quantum Algorithms for Stochastic Differential Equations: A Schr¨ odingerisation Approach

    Shi Jin, Nana Liu, and Wei Wei. Quantum Algorithms for Stochastic Differential Equations: A Schr¨ odingerisation Approach. 2025. arXiv: 2412.14868 [quant-ph]

    work page arXiv 2025

  63. [64]

    Transmutation based Quantum Simulation for Non-unitary Dynamics

    Shi Jin, Chuwen Ma, and Enrique Zuazua. Transmutation Based Quantum Simulation for Non-unitary Dynamics. 2026. arXiv: 2601.03616 [quant-ph]

    work page internal anchor Pith review arXiv 2026

  64. [65]

    A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations

    Chao Wang, Xi-Ning Zhuang, Menghan Dou, Zhao-Yun Chen, and Guo-Ping Guo. A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations . 2026. arXiv: 2604.02874 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  65. [66]

    Dollard and Charles N

    John D. Dollard and Charles N. Friedman. Product integration with applications to differential equa- tions. Cambridge University Press, 1984

    1984

  66. [67]

    Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics

    Masuo Suzuki. “Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics”. In: Journal of mathematical physics 26.4 (1985), pp. 601–612

    1985

  67. [68]

    General theory of fractal path integrals with applications to many-body theories and statistical physics

    Masuo Suzuki. “General theory of fractal path integrals with applications to many-body theories and statistical physics”. In: Journal of mathematical physics 32.2 (1991), pp. 400–407

    1991

  68. [69]

    Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation

    Ryan Babbush, Jarrod McClean, Dave Wecker, Al´ an Aspuru-Guzik, and Nathan Wiebe. “Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation”. In: Physical Review A 91.2 (2015), p. 022311

    2015

  69. [70]

    Solving strongly correlated electron models on a quantum computer

    Dave Wecker, Matthew B. Hastings, Nathan Wiebe, Bryan K. Clark, Chetan Nayak, and Matthias Troyer. “Solving strongly correlated electron models on a quantum computer”. In: Physical Review A 92.6 (2015), p. 062318

    2015

  70. [71]

    Toward the first quantum simulation with quantum speedup

    Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su. “Toward the first quantum simulation with quantum speedup”. In: Proceedings of the National Academy of Sciences 115.38 (2018), pp. 9456–9461

    2018

  71. [72]

    Time-dependent unbounded Hamiltonian simulation with vector norm scaling

    Dong An, Di Fang, and Lin Lin. “Time-dependent unbounded Hamiltonian simulation with vector norm scaling”. In: Quantum 5 (2021), p. 459

    2021

  72. [73]

    Minimum Trotterization formulas for a time-dependent Hamiltonian

    Tatsuhiko N. Ikeda, Asir Abrar, Isaac L. Chuang, and Sho Sugiura. “Minimum Trotterization formulas for a time-dependent Hamiltonian”. In: Quantum 7 (2023), p. 1168

    2023

  73. [74]

    Nearly Optimal Lattice Simulation by Product Formulas

    Andrew M. Childs and Yuan Su. “Nearly Optimal Lattice Simulation by Product Formulas”. In: Phys. Rev. Lett. 123 (5 Aug. 2019), p. 050503. 45

    2019

  74. [75]

    Sim- ulating Hamiltonian Dynamics with a Truncated Taylor Series

    Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. “Sim- ulating Hamiltonian Dynamics with a Truncated Taylor Series”. In: Physical Review Letters 114 (9 Mar. 2015), p. 090502

    2015

  75. [76]

    The exponentially convergent trapezoidal rule

    Lloyd N. Trefethen and J.A.C. Weideman. “The exponentially convergent trapezoidal rule”. In: SIAM review 56.3 (2014), pp. 385–458

    2014

  76. [77]

    Ordering of trotterization: Impact on errors in quantum simulation of electronic structure

    Andrew Tranter, Peter J. Love, Florian Mintert, Nathan Wiebe, and Peter V Coveney. “Ordering of trotterization: Impact on errors in quantum simulation of electronic structure”. In: Entropy 21.12 (2019), p. 1218. A Omitted Proofs in Section 3 A.1 Proof of Lemma 3.1 Proof. Setz =k−iy0. Since 1−iz = (1−y0)−ik andb+iz = (b+y0)+ik, we have|1−iz|= √ (y0−1)2 +k2...

    2019

  77. [78]

    Moreover, d2 dy2 0 logB(y0) = 1 2c2 + 1 (y0−1)2 + a−1 (b +y0)2 > 0

    = 0, equivalently Equation (3.7). Moreover, d2 dy2 0 logB(y0) = 1 2c2 + 1 (y0−1)2 + a−1 (b +y0)2 > 0. (A.7) Thus, log B is strictly convex for y0 > 1, so any stationary point is its unique global minimizer. Since B(y0)> 0 and d dy0 B(y0) =B(y0) d dy0 logB(y0),B and logB have the same stationary points. They also have the same minimizers because exp is str...

  78. [79]

    Thus, y∗ 0 is the unique stationary point of both log B(y0) and B(y0)

    = 0. Thus, y∗ 0 is the unique stationary point of both log B(y0) and B(y0). A.4 Proof of Corollary 3.5 Proof. We prove the three statements separately

  79. [80]

    Fors> 1, define Q(s) := c√π(s−1) exp ( a−(s−1)2 4c2 )

    By Proposition 3.4, it suffices to prove d≥Φ(sϵ), where sϵis the unique solution of Ψ( sϵ) = ϵapprox. Fors> 1, define Q(s) := c√π(s−1) exp ( a−(s−1)2 4c2 ) . (A.9) Note that Ψ( s)≤Q(s) for s> 1. Moreover, Q(˜x0) = c√π(˜x0−1) exp ( a−(˜x0−1)2 4c2 ) = 1√2πωϵ exp ( a−ωϵ 2 ) =ϵapprox, (A.10) sinceωϵeωϵ=e2a/(2πϵ2 approx). Hence, Ψ( ˜x0)≤Q(˜x0) =ϵapprox. Since ...

  80. [81]

    Since e2a/ϵ2 approx≥e2 >e because ϵapprox∈(0, 1), we have W ( e2a 2πϵ2approx ) ≤W ( e2a ϵ2approx ) ≤log ( e2a ϵ2approx ) = 2a + 2 log(1/ϵapprox)

    We have e2a/(2πϵ2 approx)≤e2a/ϵ2 approx. Since e2a/ϵ2 approx≥e2 >e because ϵapprox∈(0, 1), we have W ( e2a 2πϵ2approx ) ≤W ( e2a ϵ2approx ) ≤log ( e2a ϵ2approx ) = 2a + 2 log(1/ϵapprox). (A.11) It follows that ˜x0 = 1 +c √ 2W ( e2a 2πϵ2approx ) ≤1 + √ 2c √ 2a + 2 log(1/ϵapprox) =x0. (A.12) Since Φ is strictly increasing, d≥Φ(x0) implies d≥Φ( ˜x0). The cla...

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