Pith. sign in

REVIEW 5 minor 81 references

Poisson-Compiled Quantum Singular Value Transformation for Power-Exponential Dissipation

T0 review · 0 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read This paper establishes that a shifted-signal oracle reduces the QSVT polynomial degree for $e^{-TH^\alpha}$ from roughly $(T\|H\|^\alpha/\epsilon)^{1/\alpha}$ to $(T\|H\|^\alpha/\epsilon)^{1/(2\alpha)}$, with matching lower bounds in the…

desk verdict Careful, technically strong paper: the quadratic lift for the shifted signal is a genuine new idea with tight degree bounds, and the main caveat is the extra oracle assumption, not the math. read the letter →

arxiv 2608.04263 v1 pith:CAGVNLOP submitted 2026-08-04 quant-ph

Chao Wang , Xi-Ning Zhuang , Menghan Dou , Zhao-Yun Chen , Guo-Ping Guo This is my paper · ORCID
classification quant-ph MSC 68Q1241A1065F6065Y20 PACS 03.67.Ac
keywords quantumsingularvaluetransformationPoissonsummationmatrixfunctionapproximationdissipativedynamicsblockencodinglinearcombinationofHamiltoniansimulationamplitude-phaseseparationfractionaldiffusion
verification ladder T0 review T1 audit T2 compute T3 formal

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 studies quantum circuits that implement the contraction $e^{-T H^\alpha}$ for a Hermitian positive semidefinite matrix $H$ and power $\alpha>0$, the matrix function behind heat, biharmonic, and fractional diffusion. It uses Poisson summation as a classical compiler: the Fourier samples of the target are summed into one Chebyshev polynomial, and that polynomial is applied via QSVT. The central claim is access-dependent: under the standard block encoding of $H/\|H\|$, polynomial degree must scale like $(T\|H\|^\alpha/\epsilon)^{1/\alpha}$ for non-even $\alpha$, while under the shifted signal $2H/\|H\|-I$ the same contraction is achieved with degree roughly $(T\|H\|^\alpha/\epsilon)^{1/(2\alpha)}$ and every positive integer $\alpha$ enters the entire-function regime. This quadratic lift changes the fixed-scale approximation error from $\Theta(d^{-\alpha})$ to $\Theta(d^{-2\alpha})$ for noninteger powers. If the paper is right, it gives tight query bounds for preparing dissipative states and a reusable circuit construction for controlled dissipative families.

What carries the argument

Poisson-compiled QSVT: the exact Poisson-summation identity $h\sum_{k\in\mathbb{Z}} f_{\alpha,T}(kh)e^{-2\pi ikh x}=\sum_{n\in\mathbb{Z}} e^{-T\|H\|^\alpha |x+n/h|^\alpha}$ generates a finite cosine sum whose samples are converted by the Jacobi–Anger expansion into Chebyshev coefficients, yielding one even polynomial $P_d$. The fixed rescaling $P_d(x)/(2(1+\epsilon/2))$ makes the polynomial QSVT-admissible with constant normalization, and a parity decomposition with one extra ancilla handles the mixed-parity polynomial in the shifted variable. The quadratic lift is the substitution $y=\sqrt{(1+s)/2}$, which maps the lifted target $e^{-T\|H\|^\alpha[(1+s)/2]^\alpha}$ exactly to $e^{-T(\sqrt{\|H\|})^{2\alpha}|y|^{2\alpha}}$; this is the mechanism that doubles the approximation exponent. The classical preprocessing cost of kernel samples and Bessel sums is separated from the QSVT query count.

What would settle it

Numerically compute the best uniform error $E_d$ for the scalar target $f(s)=e^{-T\|H\|^\alpha[(1+s)/2]^\alpha}$ with noninteger $\alpha$ (for example $\alpha=1/2$, $T\|H\|^\alpha=1$). If $E_d$ decays as $d^{-\alpha}$ rather than $d^{-2\alpha}$, or if the minimal degree grows faster than $(T\|H\|^\alpha)^{1/(2\alpha)}$ at fixed error, the quadratic-lift claim fails; the paper's own Eq. (D3) reduces this to an even-polynomial approximation of $e^{-T\|H\|^\alpha |y|^{2\alpha}}$, so the scalar check is decisive.

Watch

Extended reading notes

Core claim

The central discovery is an exact quadratic lift for QSVT with shifted signal access. With the standard signal $x$ from $H/\|H\|$, parity forces an even polynomial approximation of $e^{-T\|H\|^\alpha |x|^\alpha}$, a target that is entire only when $\alpha$ is an even positive integer. Replacing the signal by $S=2H/\|H\|-I$ makes the scalar target $e^{-T\|H\|^\alpha[(1+s)/2]^\alpha}$, and the change of variables $x=2y^2-1$ identifies a degree-$d$ polynomial in $s$ with an even degree-$2d$ polynomial in $y$ approximating $e^{-T\|H\|^\alpha |y|^{2\alpha}}$. Consequently every positive integer $\alpha$ becomes an entire-function approximation problem, and noninteger $\alpha$ has its interior singularity mapped to an endpoint, doubling the approximation exponent. Theorem II.3 and Table I state the resulting degree bounds: $d=O((T\|H\|^\alpha+\log(1/\epsilon))^{1/(2\alpha)}\log^{1-1/(2\alpha)}(1/\epsilon))$ for every positive integer, and $d=O(\sqrt{\|H\|}(T/\epsilon)^{1/(2\alpha)})$ for noninteger $\alpha$, with matching lower bounds in the fixed-error ($T\|H\|^\alpha\to\infty$) and fixed-scale ($\epsilon\to0$) limits.

Load-bearing premise

The improved degree bounds require a unit-normalized block encoding of the shifted signal $2H/\|H\|-I$; a generic block encoding of $H/\|H\|$ does not provide it, and for general $H$ this shifted access is an additional structural oracle assumption.

Editorial extensions

If this is right

  • For the heat equation ($\alpha=1$), shifted access replaces $O(u_r^2 \beta_L \kappa T/\epsilon)$ with $O(u_r\sqrt{\beta_L \kappa T \log(u_r/\epsilon)})$ in the regime $\beta_L \kappa T\ge \log(u_r/\epsilon)$, reproducing square-root dissipative dependence directly from one constant-normalization QSVT circuit.
  • For fractional diffusion with noninteger $\alpha$, the shifted model attains the optimal fixed-scale exponent $\epsilon^{-1/(2\alpha)}$ proved in Appendix D, while standard access gives $\epsilon^{-1/\alpha}$.
  • For the time-independent non-Hermitian generator $L+iG$, the Weyl–Poisson identity gives an exact operator-valued alias decomposition of the optimal LCHS quadrature that remains compatible with sinh–sinh discretization, though it does not change the optimal LCHS query order.
  • In amplitude–phase separation, one Poisson-compiled controlled semigroup family supplies every dissipative Dyson factor; with half-root access to $(H/\beta_H)^{1/(2\alpha)}$, the per-invocation degree scales as $(T\beta_H)^{1/(2\alpha)}$, recovering square-root scaling for $\alpha=1$ and giving beyond-square-root scaling for $\alpha>1$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the quadratic lift is one instance of composing QSVT with an even change of variables; higher-degree polynomial maps could in principle push noninteger powers to even larger approximation exponents, at the cost of oracle complexity this paper does not analyze.
  • Inference: because the Fourier sample count enters only classical preprocessing, the query bounds suggest that fault-tolerant cost estimates for this approach should concentrate on classical kernel evaluation and QSP phase synthesis rather than on ancilla branching; the paper does not provide optimized bit complexities for these steps.
  • Inference: if unit-normalized shifted access can be synthesized from sparse-matrix oracles with constant overhead, as the paper explicitly verifies for nearest-neighbor Laplacians, the improved degree bounds would carry over to practical quantum PDE solvers; the paper leaves this synthesis for general sparse $H$ as an implementation requirement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper presents a Poisson-summation-based classical compilation of the quantum signal processing polynomial for the contractive matrix function exp(-T H^α), where H=H†⪰0 and α>0, under two access models: a standard block encoding of H/||H|| and a unit-normalized shifted signal S=2H/||H||-I. The main theoretical results are Theorem II.3 and Table I, which give polynomial degree bounds for the QSVT implementation: for standard access, even-integer α has degree O((T||H||^α+log(1/ε))^{1/α} log^{1-1/α}(1/ε)) and non-even α has O(||H||(T/ε)^{1/α}); for shifted access, every positive integer α has degree O((T||H||^α+log(1/ε))^{1/(2α)} log^{1-1/(2α)}(1/ε)) and noninteger α has O(√||H||(T/ε)^{1/(2α)}). These bounds are shown to be tight in the separate fixed-error and fixed-precision limits. The shifted-signal improvement relies on an exact quadratic lift (Prop. II.2) that identifies a degree-d polynomial in s with an even degree-2d polynomial in y, thereby doubling the algebraic singularity exponent. The paper also proves end-to-end state-preparation costs involving the u_r overhead, applies the construction to heat/biharmonic/fractional dissipation, derives a noncommutative Weyl–Poisson identity for time-independent non-Hermitian LCHS simulation, and gives a Poisson-compiled realization of the controlled dissipative family used in amplitude–phase separation.

Significance. The central contribution is the quadratic-lift reduction: for noninteger powers, the shifted-signal access changes the fixed-scale approximation error from Θ(d^{-α}) to Θ(d^{-2α}), and every positive integer power becomes entire, yielding the corresponding degree improvements in Eqs. (25)–(26). The tightness claims are properly scoped: lower bounds are one-parameter asymptotic statements for the indicated limits, and the standard-access non-even lower bound is restricted to ordinary single-sequence QSVT. The paper is careful to state that the shifted signal in Eq. (4) is an independent structural assumption, not implied by a generic H/||H|| block encoding, and it provides a concrete nearest-neighbor Laplacian realization. The appendices contain detailed proofs of the Poisson residual bounds, the Bernstein-ellipse degree estimates, the lower bounds, the Weyl–Poisson identity, and the APS query counts. The non-Hermitian LCHS section is explicit that the Weyl–Poisson reformulation does not improve the optimal LCHS query order, and the APS claims are stated under the additional half-root access assumption.

minor comments (5)
  1. [Abstract and Sec. II.A] The abstract states the improved 1/(2α) exponents without the qualifier 'under the unit-normalized shifted-signal access of Eq. (4)'; the body is explicit that this is an independent structural assumption, and the abstract would benefit from the same qualifier to avoid over-reading.
  2. [Sec. II.A, Eq. (17)] The phrase 'normalization-two block encoding' is slightly terse; since B itself satisfies ||B||≤1/2, calling B a normalization-two block encoding of exp(-T H^α) is consistent, but a parenthetical explanation in the text would improve readability.
  3. [Sec. III.B, Eq. (48)] The Weyl–Poisson identity in Eq. (48) is exact and norm-convergent as shown in Appendix E; a sentence immediately after the display stating that all sums converge in operator norm for finite-dimensional systems would prevent a possible misreading of the equality.
  4. [Appendix B, around Eq. (B6)] The bounding of the higher-order local expansion terms as a 'convergent geometric majorant' could be expanded into one or two explicit inequalities, since this is the step that turns the local singularity analysis into the Chebyshev coefficient decay.
  5. [Sec. III.A, Eq. (38)] The notation '-Ann/(2D)' could be misread as 'A_nn' with a subscript; ensure the subscript is clear in the final typeset version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: shifted-signal oracle is an explicit scope condition; the central degree bounds are proven from scratch against external approximation theory.

full rationale

The central derivation is self-contained. Theorem II.1 constructs the polynomial in Eq. (14) from the exact Poisson identity Eq. (10), which is proven in Appendix A, and the projection error is bounded by the standard Lebesgue constant plus classical Bernstein/Watson estimates; no fitted parameters or target-dependent assumptions enter. Proposition II.2 is an exact change of variables: for y = sqrt((1+s)/2), the identity e^{-T||H||^alpha|y|^{2alpha}} = e^{-T||H||^alpha((1+s)/2)^alpha} holds, so the shifted-signal degree bounds are obtained by applying the same theorem with exponent 2alpha and normalization sqrt(||H||), not by assuming the conclusion. The only special assumption, the unit-normalized shifted oracle in Eq. (4), is explicitly stated as an independent access condition: "Eq. (4) is a unit-normalized block encoding and is an independent access assumption." A scope restriction is not a circular reduction, and the paper supplies a concrete nearest-neighbor Laplacian realization. The self-citation to [25] is contextual and non-load-bearing: "A broader Poisson-summation framework for quantum matrix transformations was introduced previously [25]. Here we focus on power-exponential dissipation and pursue a different implementation." The lower bounds are taken from external approximation theory (Bernstein, [46-48], optimal Gaussian degree [41]) and are carefully scoped to the stated parity and access classes. The Weyl-Poisson identity is proved directly in Appendix E, and the paper explicitly disclaims any query-order improvement from it. No step reduces, by the paper's own equations or by a self-citation chain, to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities are needed. The central claims rest on standard Fourier and approximation theory, the QSVT framework, and explicit oracle assumptions. All these assumptions are stated and scoped by the authors.

assumptions (6)
  • standard math Poisson summation formula applies to the kernel f_{alpha,T} and to the matrix-valued Schwartz functions used in the Weyl-Poisson proof.
    Invoked in Eq. (10) and Eq. (48); Lemma A.1 and Appendix E justify absolute convergence and the operator-valued form.
  • standard math Jacobi-Anger expansion of cos(zx) into Chebyshev polynomials is valid.
    Used in Eq. (13) to convert Fourier modes into even Chebyshev coefficients for the compiled polynomial.
  • standard math QSVT completion theorem: real polynomials of fixed parity and modulus at most one on [-1,1] are implementable with degree-many signal queries.
    Assumed from [13] to turn the constructed polynomial into a circuit; also used for the normalization-one controlled family in Appendix F.
  • standard math Classical approximation lower bounds from Bernstein, Markov, and entire-function theory apply in the stated limits.
    Used in Appendix D to prove the matching degree lower bounds in Table I.
  • domain assumption The LCHS optimal kernel and its generalized LCHS theorem bound the kernel mismatch in Eq. (45).
    Borrowed from [32] for the non-Hermitian application; the paper does not reprove this theorem.
  • domain assumption Exact block-encoding access to U_H, U_S, U_A, and the half-root oracle of Eq. (60) is available when stated.
    These are explicit access assumptions, not derived consequences. The shifted-signal access is flagged as an additional structural assumption in Sec. II.A, and half-root access is assumed in Sec. III.C.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Poisson-Compiled Quantum Singular Value Transformation for Power-Exponential Dissipation." pith.science (2026). https://pith.science/paper/CAGVNLOP

@misc{pith2026260804263,
  author       = {Pith},
  title        = {Pith review of: Poisson-Compiled Quantum Singular Value Transformation for Power-Exponential Dissipation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CAGVNLOP}},
  note         = {Machine review of arXiv:2608.04263}
}
abstract

We study quantum implementations of the contraction $\exp(-T H^\alpha)$ for $H=H^\dagger\succeq0$ and $\alpha>0$. Poisson summation provides an exact target--alias--tail decomposition whose Fourier samples are compiled classically into a single Chebyshev polynomial, so the quantum circuit uses polynomial eigenvalue transformation rather than a frequency linear combination of unitaries. We compare block encodings of $H/\norm{H}$ and of the shifted signal $2H/\norm{H}-I$. Under ordinary single-sequence QSVT, parity forces the former to use an even extension, which is entire only for even positive integers. An exact quadratic lift for the shifted signal makes every positive integer entire and improves the fixed-scale approximation error for noninteger powers from $\Theta(d^{-\alpha})$ to $\Theta(d^{-2\alpha})$ within the stated access and parity classes. We derive matching degree bounds in the large-scale fixed-error and fixed-scale high-precision limits, including the output-normalization overhead $u_r$. Nearest-neighbor Laplacians give a unit-normalized shifted signal. We further establish a noncommutative Weyl--Poisson identity compatible with LCHS quadrature, and use the same polynomial construction to implement controlled dissipative families in amplitude--phase separation.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

81 extracted references · 45 canonical work pages

  1. [1]

    Linden, A

    N. Linden, A. Montanaro, and C. Shao, Quantum vs. classical algorithms for solving the heat equation, Communications in Mathematical Physics395, 601 (2022)

    2022

  2. [2]

    S. Jin, N. Liu, and Y . Yu, Quantum simulation of partial differ- ential equations: Applications and detailed analysis, Phys. Rev. A108, 032603 (2023)

    2023

  3. [3]

    Metzler and J

    R. Metzler and J. Klafter, The random walk’s guide to anoma- lous diffusion: a fractional dynamics approach, Physics Reports 339, 1 (2000)

    2000

  4. [4]

    Lischke, G

    A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, and G. E. Karniadakis, What is the fractional Laplacian? a comparative review with new results, Journal of Computational Physics404, 109009 (2020)

    work page 2020

  5. [5]

    Lloyd, Universal quantum simulators, Science273, 1073 (1996)

    S. Lloyd, Universal quantum simulators, Science273, 1073 (1996)

    1996

  6. [6]

    I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys.86, 153 (2014)

    2014

  7. [7]

    A. M. Childs, D. Maslov, Y . Nam, N. J. Ross, and Y . Su, Toward the first quantum simulation with quantum speedup, Proceed- ings of the National Academy of Sciences115, 9456 (2018)

    work page 2018

  8. [8]

    Altman, K

    E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Demler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriksson, K.- M. C. Fu, M. Greiner, K. R. Hazzard, R. G. Hulet, A. J. Koll´ar, B. L. Lev, M. D. Lukin, R. Ma, X. Mi, S. Misra, C. Monroe, K. Murch, Z. Nazario, K.-K. Ni, A. C. Potter, P. Roushan, M. Saffman, M. Schleier-Smith, I. Siddiqi, R. Simmonds, M. S...

    2021

Show all 81 references
  1. [9]

    A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quan- tum simulation, Nature607, 667 (2022)

    2022

  2. [10]

    D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, Efficient quantum algorithms for simulating sparse Hamiltonians, Com- munications in Mathematical Physics270, 359 (2007)

    2007

  3. [11]

    S ¨underhauf, E

    C. S ¨underhauf, E. Campbell, and J. Camps, Block-encoding structured matrices for data input in quantum computing, Quan- tum8, 1226 (2024)

    2024

  4. [12]

    G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett.118, 010501 (2017)

    2017

  5. [13]

    Gily ´en, Y

    A. Gily ´en, Y . Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics, inProceedings of the 51st An- nual ACM SIGACT Symposium on Theory of Computing, STOC 2019 (Association for Computing Machine...

    2019

  6. [14]

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating Hamiltonian dynamics with a truncated Taylor series, Phys. Rev. Lett.114, 090502 (2015)

    2015

  7. [15]

    A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett.103, 150502 (2009)

    2009

  8. [16]

    A. M. Childs, R. Kothari, and R. D. Somma, Quantum algo- rithm for systems of linear equations with exponentially im- proved dependence on precision, SIAM Journal on Computing 46, 1920 (2017)

    2017

  9. [17]

    G. H. Low and Y . Su, Quantum linear system algorithm with optimal queries to initial state preparation, Quantum10, 2041 (2026)

    2026

  10. [18]

    A. M. Childs and N. Wiebe, Hamiltonian simulation using lin- ear combinations of unitary operations, Quantum Information and Computation12, 901 (2012)

    2012

  11. [19]

    Kannai, Off diagonal short time asymptotics for fundamental solution of diffusion equation, Communications in Partial Dif- ferential Equations2, 781 (1977)

    Y . Kannai, Off diagonal short time asymptotics for fundamental solution of diffusion equation, Communications in Partial Dif- ferential Equations2, 781 (1977)

    1977

  12. [20]

    S. Jin, C. Ma, and E. Zuazua, Transmutation based quantum simulation for non-unitary dynamics (2026), arXiv:2601.03616 [quant-ph]

    2026 arXiv

  13. [21]

    Kharazi, A

    T. Kharazi, A. M. Alkadri, K. K. Mandadapu, and K. B. Wha- ley, A sublinear-time quantum algorithm for high-dimensional reaction rates (2026), arXiv:2601.15523 [quant-ph]

    2026 arXiv

  14. [22]

    S. D. Miller and W. Schmid, Summation formulas, from Pois- son and V oronoi to the present, inNoncommutative Harmonic Analysis: In Honor of Jacques Carmona, edited by P. Delorme and M. Vergne (Birkh ¨auser Boston, Boston, MA, 2004) pp. 419–440

    2004

  15. [23]

    E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, V ol. 32 (Princeton University Press, Princeton, NJ, 1971)

    1971

  16. [24]

    Katznelson,An introduction to harmonic analysis(Cam- bridge University Press, 2004)

    Y . Katznelson,An introduction to harmonic analysis(Cam- bridge University Press, 2004)

    2004

  17. [25]

    Wang, X.-N

    C. Wang, X.-N. Zhuang, M. Dou, Z.-Y . Chen, and G.-P. Guo, A unified Poisson summation framework for generalized quantum matrix transformations (2026), arXiv:2604.02874 [quant-ph]

    2026 arXiv

  18. [26]

    Ni and L

    H. Ni and L. Ying, Quantum eigenvalue transformation via lin- ear combination of Hamiltonian simulation: A Weyl calculus approach (2026), arXiv:2606.29848 [quant-ph]

    2026 arXiv

  19. [27]

    Weyl,The Theory of Groups and Quantum Mechanics (Dover Publications, New York, 1950) reprint of the 1931 En- glish translation

    H. Weyl,The Theory of Groups and Quantum Mechanics (Dover Publications, New York, 1950) reprint of the 1931 En- glish translation

    1950

  20. [28]

    R. F. V . Anderson, The Weyl functional calculus, Journal of Functional Analysis4, 240 (1969)

    1969

  21. [29]

    Jefferies,Spectral Properties of Noncommuting Operators, Lecture Notes in Mathematics, V ol

    B. Jefferies,Spectral Properties of Noncommuting Operators, Lecture Notes in Mathematics, V ol. 1843 (Springer, Berlin, 12 Heidelberg, 2004)

    2004

  22. [30]

    An, J.-P

    D. An, J.-P. Liu, and L. Lin, Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state prepara- tion cost, Phys. Rev. Lett.131, 150603 (2023)

    2023

  23. [31]

    D. An, A. M. Childs, and L. Lin, Quantum algorithm for lin- ear nonunitary dynamics with near-optimal dependence on all parameters, Communications in Mathematical Physics407, 19 (2026)

    2026

  24. [32]

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

    2025 arXiv

  25. [33]

    Aftab, D

    J. Aftab, D. An, and K. Trivisa, Linear combination of Hamiltonian simulation with commutator scaling (2026), arXiv:2606.11475 [quant-ph]

    2026 arXiv

  26. [34]

    Hu and S

    Q. Hu and S. Jin, Quantum simulation of non-unitary dynam- ics via Amplitude–Phase Separation (2026), arXiv:2602.09575 [quant-ph]

    2026 arXiv

  27. [35]

    S ¨underhauf, Generalized quantum singular value transfor- mation (2023), arXiv:2312.00723 [quant-ph]

    C. S ¨underhauf, Generalized quantum singular value transfor- mation (2023), arXiv:2312.00723 [quant-ph]

    2023 arXiv

  28. [36]

    DLMF,NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/, Release 1.2.7 of 2026-06-15 (2026), f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V . Saun- ders, H. S. Cohl, and M. A. McClain, eds

    2026

  29. [37]

    J. C. Mason and D. C. Handscomb,Chebyshev Polynomials (Chapman and Hall/CRC, Boca Raton, FL, 2003)

    2003

  30. [38]

    L. N. Trefethen,Approximation Theory and Approximation Practice, Extended Edition(Society for Industrial and Applied Mathematics, Philadelphia, 2019)

    2019

  31. [39]

    Haah, Product decomposition of periodic functions in quan- tum signal processing, Quantum3, 190 (2019)

    J. Haah, Product decomposition of periodic functions in quan- tum signal processing, Quantum3, 190 (2019)

    2019

  32. [40]

    Wong,Asymptotic Approximations of Integrals, Classics in Applied Mathematics, V ol

    R. Wong,Asymptotic Approximations of Integrals, Classics in Applied Mathematics, V ol. 34 (Society for Industrial and Ap- plied Mathematics, Philadelphia, 2001)

    2001

  33. [41]

    Aggarwal and J

    A. Aggarwal and J. Alman, Optimal-degree polynomial ap- proximations for exponentials and Gaussian kernel density estimation, in37th Computational Complexity Conference (CCC 2022), Leibniz International Proceedings in Informatics, V ol. 234 (Schloss Dagstuhl–Leibniz-Zentrum fu...

    2022

  34. [42]

    G. H. Low and I. L. Chuang, Hamiltonian simulation by uni- form spectral amplification (2017), arXiv:1707.05391 [quant- ph]

    2017 arXiv

  35. [43]

    R. A. DeV ore and G. G. Lorentz,Constructive Approxima- tion, Grundlehren der mathematischen Wissenschaften, V ol. 303 (Springer-Verlag, Berlin, 1993)

    1993

  36. [44]

    Ditzian and V

    Z. Ditzian and V . Totik,Moduli of Smoothness, Springer Series in Computational Mathematics, V ol. 9 (Springer, New York, 1987)

    1987

  37. [45]

    T. J. Yoder, G. H. Low, and I. L. Chuang, Fixed-point quantum search with an optimal number of queries, Phys. Rev. Lett.113, 210501 (2014)

    2014

  38. [46]

    S. N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degre donne, Mem- oires publies par la Classe des Sciences de l’Academie Royale de Belgique 2,4, 1 (1912)

    1912

  39. [47]

    A. F. Timan,Theory of Approximation of Functions of a Real Variable(Pergamon Press, Oxford, 1963)

    1963

  40. [48]

    Bernstein, Sur la meilleure approximation de|x|par des polynomes de degr´es donn´es, Acta Mathematica37, 1 (1914)

    S. Bernstein, Sur la meilleure approximation de|x|par des polynomes de degr´es donn´es, Acta Mathematica37, 1 (1914)

    1914

  41. [49]

    A. M. Childs and J.-P. Liu, Quantum spectral methods for dif- ferential equations, Communications in Mathematical Physics 375, 1427 (2020)

    2020

  42. [50]

    S. Jin, N. Liu, and Y . Yu, Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations, Journal of Computational Physics 471, 111641 (2022)

    2022

  43. [51]

    D. Fang, L. Lin, and Y . Tong, Time-marching based quantum solvers for time-dependent linear differential equations, Quan- tum7, 955 (2023)

    2023

  44. [52]

    D. W. Berry and P. C. S. Costa, Quantum algorithm for time- dependent differential equations using Dyson series, Quantum 8, 1369 (2024)

    2024

  45. [53]

    S. Jin, N. Liu, and Y . Yu, Quantum simulation of partial differ- ential equations via Schr ¨odingerization, Phys. Rev. Lett.133, 230602 (2024)

    2024

  46. [54]

    S. Jin, N. Liu, C. Ma, and Y . Yu, On the Schr ¨odingerization method for linear non-unitary dynamics with optimal depen- dence on matrix queries (2025), arXiv:2505.00370 [math.NA]

    2025

  47. [55]

    Gonzalez-Conde, A

    J. Gonzalez-Conde, A. Rodriguez-Rozas, E. Solano, and M. Sanz, Efficient Hamiltonian simulation for solving option price dynamics, Physical Review Research5, 10.1103/phys- revresearch.5.043220 (2023)

    2023 doi

  48. [56]

    W. W. Mullins, Theory of thermal grooving, Journal of Applied Physics28, 333 (1957)

    1957

  49. [57]

    Camps and R

    D. Camps and R. Van Beeumen, Fable: Fast approximate quan- tum circuits for block-encodings, in2022 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, 2022) pp. 104–113

    2022

  50. [58]

    Szegedy, Quantum speed-up of Markov chain based algo- rithms, in45th Annual IEEE Symposium on Foundations of Computer Science(2004) pp

    M. Szegedy, Quantum speed-up of Markov chain based algo- rithms, in45th Annual IEEE Symposium on Foundations of Computer Science(2004) pp. 32–41

    2004

  51. [59]

    V . M. Zolotarev,One-Dimensional Stable Distributions(Amer- ican Mathematical Society, Providence, RI, 1986)

    1986

  52. [60]

    D. Dong, Y . Li, and J. Xue, Products between block-encodings (2025), arXiv:2509.15779 [quant-ph]

    2025

  53. [61]

    Takahira, A

    S. Takahira, A. Ohashi, T. Sogabe, and T. S. Usuda, Quantum algorithm for matrix functions by Cauchy’s integral formula, Quantum Information and Computation20, 14 (2020)

    2020

  54. [62]

    Takahira, A

    S. Takahira, A. Ohashi, T. Sogabe, and T. S. Usuda, Quantum algorithms based on the block-encoding framework for matrix functions by contour integrals, Quantum Information and Com- putation22, 965 (2022)

    2022

  55. [63]

    Jiang and D

    S. Jiang and D. An, Contour-integral based quantum eigen- value transformation: analysis and applications (2026), arXiv:2601.11959 [quant-ph]

    2026

  56. [64]

    G. H. Low and Y . Su, Quantum eigenvalue processing, in2024 IEEE 65th Annual Symposium on Foundations of Computer Sci- ence (FOCS)(2024) pp. 1051–1062

    2024

  57. [65]

    D. An, A. M. Childs, L. Lin, and L. Ying, Laplace trans- form based quantum eigenvalue transformation via linear com- bination of Hamiltonian simulation (2024), arXiv:2411.04010 [quant-ph]

    2024 arXiv

  58. [66]

    H. H. S. Chan, D. M. Ramo, and N. Fitzpatrick, Simulating non- unitary dynamics using quantum signal processing with unitary block encoding (2023), arXiv:2303.06161 [quant-ph]

    2023 arXiv

  59. [67]

    Wang, H.-Y

    C. Wang, H.-Y . Liu, C. Xue, X.-N. Zhuang, M. Dou, Z.-Y . Chen, and G.-P. Guo, Quantum simulation of non-hermitian special functions and dynamics via contour-based matrix de- composition, Quantum Science and Technology11, 035027 (2026)

    2026

  60. [68]

    L. N. Trefethen and J. A. C. Weideman, The exponentially con- vergent trapezoidal rule, SIAM Review56, 385 (2014)

    2014

  61. [69]

    Mori, Discovery of the double exponential transformation and its developments, Publications of the Research Institute for Mathematical Sciences41, 897 (2005)

    M. Mori, Discovery of the double exponential transformation and its developments, Publications of the Research Institute for Mathematical Sciences41, 897 (2005)

    2005

  62. [70]

    Ooura, A double exponential formula for the Fourier trans- forms, Publications of the Research Institute for Mathematical 13 Sciences41, 971 (2005)

    T. Ooura, A double exponential formula for the Fourier trans- forms, Publications of the Research Institute for Mathematical 13 Sciences41, 971 (2005)

    2005

  63. [71]

    Sugihara, Optimality of the double exponential formula— functional analysis approach, Numerische Mathematik75, 379 (1997)

    M. Sugihara, Optimality of the double exponential formula— functional analysis approach, Numerische Mathematik75, 379 (1997)

    1997

  64. [72]

    Okayama, Double-exponential formula for infinite integrals of unilateral rapidly decreasing functions, JSIAM Letters14, 17 (2022)

    T. Okayama, Double-exponential formula for infinite integrals of unilateral rapidly decreasing functions, JSIAM Letters14, 17 (2022)

    2022

  65. [73]

    Weimer, A

    H. Weimer, A. Kshetrimayum, and R. Or ´us, Simulation meth- ods for open quantum many-body systems, Rev. Mod. Phys.93, 015008 (2021)

    2021

  66. [74]

    H.-Y . Liu, X. Lin, Z.-Y . Chen, C. Xue, T.-P. Sun, Q.-S. Li, X.-N. Zhuang, Y .-J. Wang, Y .-C. Wu, M. Gong, and G.-P. Guo, Simu- lation of open quantum systems on universal quantum comput- ers, Quantum9, 1765 (2025)

    2025

  67. [75]

    L. H. Delgado-Granados, T. J. Krogmeier, L. M. Sager-Smith, I. Avdic, Z. Hu, M. Sajjan, M. Abbasi, S. E. Smart, P. Narang, S. Kais, A. W. Schlimgen, K. Head-Marsden, and D. A. Mazz- iotti, Quantum algorithms and applications for open quantum systems, Chemical Reviews125, 1823 (2025)

    2025

  68. [76]

    An, J.-P

    D. An, J.-P. Liu, D. Wang, and Q. Zhao, Quantum differential equation solvers: Limitations and fast-forwarding, Communi- cations in Mathematical Physics406, 189 (2025). 14 Appendix A: Fourier-kernel estimates and Poisson truncation We first establish the Fourier decay and trun...

    2025

  69. [77]

    Indeed, the resulting finite-sum error is at mosth(2N+ 1)times the sample error, and Eq

    log(d+ 2)]. Indeed, the resulting finite-sum error is at mosth(2N+ 1)times the sample error, and Eq. (B1) supplies the remaining factor. Rounding the finald+ 1Chebyshev coefficients contributes at most the sum of their absolute rounding errors. Both are classical precision req...

  70. [78]

    The Chebyshev polynomial2x 2 −1has even parity and modulus at most one on[−1,1], so a degree-two QSVT sequence implements it exactly [13]

    Half-root access and controlled family Suppose first that an exact block encoding of R= H βH 1 2α (F1) is available. The Chebyshev polynomial2x 2 −1has even parity and modulus at most one on[−1,1], so a degree-two QSVT sequence implements it exactly [13]. SinceRis positive sem...

  71. [79]

    WritingU(T) = e −iT GV(T)and differentiating gives V ′(T) =−e iT GHe −iT GV(T), V(0) =I.(F5) The time-ordered solution is Eq

    Exact factorizations LetU(T) = e −T(H+iG) . WritingU(T) = e −iT GV(T)and differentiating gives V ′(T) =−e iT GHe −iT GV(T), V(0) =I.(F5) The time-ordered solution is Eq. (62). Alternatively, writingU(T) = e −T HW(T)gives W ′(T) =−i e T HGe−T HW(T), W(0) =I,(F6) which proves Eq...

  72. [80]

    Iterating Eq

    Dissipative Dyson products The amplitude-driven factorization can be used without separately implementing the possibly non-Hermitian similarity trans- forme sH Ge−sH. Iterating Eq. (F6) and multiplying bye −T Hfrom the left gives the exact series e−T(H+iG) = ∞X k=0 (−i)k Z 0≤s...

  73. [81]

    The APS circuit makesO(β GTlog(u r/ϵ)/log log(u r/ϵ))controlled dissipative-family calls per run

    Error and query allocation We now justify Corollary III.1. The APS circuit makesO(β GTlog(u r/ϵ)/log log(u r/ϵ))controlled dissipative-family calls per run. Telescoping over those calls and the subsequent amplitude-amplification steps shows that it is sufficient to choose the ...

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.