pith. machine review for the scientific record. sign in

arxiv: 2605.12656 · v1 · submitted 2026-05-12 · 🪐 quant-ph · cs.CC· cs.DS

Recognition: no theorem link

Optimal Bounds, Barriers, and Extensions for Non-Hermitian Bivariate Quantum Signal Processing

Joshua M. Courtney
Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:17 UTC · model grok-4.3

classification 🪐 quant-ph cs.CCcs.DS
keywords non-Hermitian Hamiltonian simulationquantum signal processingquery complexitybivariate polynomialsoptimization landscapeangle findingblock encodingwalk-operator model
0
0 comments X

The pith

Bivariate quantum signal processing establishes a tight query complexity of Θ(β_I T + log(1/ε)/log log(1/ε)) for anti-Hermitian Hamiltonian simulation.

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

This paper determines the optimal number of queries needed to simulate non-Hermitian systems using multivariate quantum signal processing. It proves that the anti-Hermitian query complexity is tightly bounded by a term linear in the integral of the imaginary part of the Hamiltonian plus a slowly growing logarithmic factor in the error. The work shows that fast-forwarding to square-root scaling is impossible in the model, though state-dependent linear improvements remain achievable. It further demonstrates that the optimization landscape contains spurious local minima, yet a warm-start basin guarantees convergence of the two-stage algorithm, while block-peeling reduces classical angle-finding cost to quadratic time.

Core claim

In the bivariate quantum signal processing framework applied to non-Hermitian Hamiltonians, the anti-Hermitian query complexity equals Θ(β_I T + log(1/ε)/log log(1/ε)). This bound is tight, derived via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert W inversion. Fast-forwarding to O(√(β_I T)) is ruled out, though an effective-β linear improvement to O(β_eff T + log term) is possible. The optimization landscape admits spurious minima, but the two-stage algorithm converges from a warm start; CRC-exploiting block peeling lowers angle-finding from O(d^3) to O(d^2) operations. A constant-ratio condition extends to non-identical operators, enabling time-dependentnon

What carries the argument

The walk-operator oracle model together with the bivariate polynomial representation of the signal operators, whose coefficient decay is controlled by Chebyshev bounds and modified Bessel asymptotics to produce the query-complexity expression.

If this is right

  • Fast-forwarding to square-root scaling O(√(β_I T)) is impossible in the bivariate polynomial model.
  • A state-dependent improvement achieving O(β_eff T + log(1/ε)/log log(1/ε)) queries is possible.
  • The two-stage algorithm converges to a global solution despite the presence of spurious local minima.
  • Time-dependent non-Hermitian simulation is achievable with query complexity O(∫(α_R(s) + β_I(s)) ds + log(1/ε)/log log(1/ε)).
  • Block-encoding overhead remains e^{-2 β_I T} across all function classes inside the walk-operator model.

Where Pith is reading between the lines

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

  • The same asymptotic techniques may extend to other non-unitary quantum channels that admit a walk-operator representation.
  • Overcoming the remaining bitorus-domain barrier would allow direct-access constructions to reach the intrinsic e^{-2 ω T} overhead for non-commuting Hamiltonians.
  • Block-peeling angle-finding reductions could apply to higher-variate M-QSP problems once analogous coefficient bounds are available.

Load-bearing premise

The bivariate polynomial model and walk-operator oracle assumptions continue to apply without modification in the non-Hermitian setting, including the constant-ratio condition for non-identical signal operators.

What would settle it

A concrete protocol that simulates a non-Hermitian Hamiltonian with query count scaling below Θ(β_I T) for arbitrarily small fixed error ε, while remaining inside the walk-operator oracle model, would falsify the claimed tightness.

Figures

Figures reproduced from arXiv: 2605.12656 by Joshua M. Courtney.

Figure 1
Figure 1. Figure 1: FIG. 1. Empirical distribution of the non-commuting direct-access advantage factor [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
read the original abstract

Multivariate quantum signal processing (M-QSP) has recently been shown to be applicable for non-Hermitian Hamiltonian simulation, opening several problems regarding the optimization landscape, angle-finding, and constant-factor analysis. We resolve several of these problems here. We find the anti-Hermitian query complexity $d_I = \Theta(\betaI T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ to be tight, established via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert~$W$ inversion. Fast-forwarding to $d_I = \mathcal{O}(\sqrt{\betaI T})$ is impossible in the bivariate polynomial model, though a linear state-dependent improvement to $d_I = \mathcal{O} \beta_{\mathrm{eff}} T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ is achievable. The optimization landscape of M-QSP admits spurious local minima, but a warm-start basin guarantee ensures the two-stage algorithm converges. CRC-exploiting block peeling reduces angle-finding from $\mathcal{O}(d^3)$ to $\mathcal{O}(d^2)$ classical operations, and optimized error allocation yields a leading constant of approximately~$2$ relative to the information-theoretic lower bound. A constant-ratio condition extends to non-identical signal operators, enabling time-dependent non-Hermitian simulation with query complexity $\mathcal{O}(\int_0^T(\alphaR(s) + \betaI(s))\,ds + \log(1/\varepsilon)/\log\log(1/\varepsilon))$. Block-encoding overhead $e^{-2\betaI T}$ holds across all function classes within the walk-operator oracle model, and dilational methods (Schr\"odingerization) achieve the walk-operator barrier. A precisely characterized direct-access construction achieves the intrinsic barrier $e^{-2\omega T}$ (with $\omega < \betaI$ for non-commuting Hamiltonians) on a restricted domain, though extension to the full bitorus remains open.

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

3 major / 2 minor

Summary. The paper claims to resolve open problems in multivariate quantum signal processing (M-QSP) for non-Hermitian Hamiltonian simulation. It establishes the tight anti-Hermitian query complexity d_I = Θ(β_I T + log(1/ε)/log log(1/ε)) via Chebyshev coefficient bounds, modified Bessel asymptotics, and Lambert W inversion; shows fast-forwarding to O(√(β_I T)) is impossible in the bivariate polynomial model while linear state-dependent improvements are possible; proves a warm-start basin guarantee for the two-stage angle-finding algorithm despite spurious local minima; reduces classical angle-finding cost to O(d²) via CRC-exploiting block peeling; achieves a leading constant ≈2 relative to the information-theoretic lower bound; extends the constant-ratio condition to non-identical signal operators for time-dependent simulation with complexity O(∫(α_R(s)+β_I(s)) ds + log(1/ε)/log log(1/ε)); and characterizes barriers including the walk-operator block-encoding overhead e^{-2 β_I T} and a direct-access construction achieving e^{-2 ω T} (ω < β_I) on a restricted domain.

Significance. If the central tightness and extension results hold, the work supplies the first optimal query-complexity characterization for non-Hermitian M-QSP, together with concrete algorithmic improvements (block peeling, optimized error allocation, warm-start convergence) that are immediately usable. The explicit identification of model barriers (walk-operator and direct-access) delineates the intrinsic limits of the framework and supplies falsifiable predictions for future implementations. These contributions would materially advance the design of quantum algorithms for open-system and non-Hermitian dynamics.

major comments (3)
  1. [Abstract and §3] Abstract and §3 (query-complexity derivation): the claim that d_I = Θ(β_I T + log(1/ε)/log log(1/ε)) remains tight for non-commuting [H_R, H_I] ≠ 0 rests on the unverified assertion that the leading exponential growth rate of the Chebyshev coefficients continues to be governed by the same β_I. No explicit analytic continuation of the modified-Bessel asymptotics under non-commutativity is supplied; a commutator-induced correction to the effective spectral radius would invalidate both the upper-bound construction and the matching lower bound obtained via Lambert-W inversion.
  2. [Abstract] Abstract: the statement that 'a constant-ratio condition extends to non-identical signal operators' is load-bearing for the time-dependent simulation result O(∫(α_R(s)+β_I(s)) ds + …). The manuscript supplies no verification that |α_R(s)/β_I(s)| = const survives when the two signal operators fail to commute, nor does it quantify the resulting perturbation to the bivariate polynomial model.
  3. [§4] §4 (optimization landscape): the warm-start basin guarantee is asserted to ensure convergence of the two-stage algorithm, yet no quantitative bound on basin size, success probability, or dependence on the spurious-minima landscape is provided. This gap affects the practical utility of the claimed O(d²) angle-finding procedure.
minor comments (2)
  1. Notation: βI, αR, and β_eff appear without consistent subscripting or definition on first use; a single nomenclature table would improve readability.
  2. [Abstract] The abstract states the leading constant is 'approximately 2'; the precise error-allocation argument that produces this value should be stated explicitly rather than summarized.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments point-by-point below, providing clarifications and indicating the revisions made to strengthen the presentation and proofs.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (query-complexity derivation): the claim that d_I = Θ(β_I T + log(1/ε)/log log(1/ε)) remains tight for non-commuting [H_R, H_I] ≠ 0 rests on the unverified assertion that the leading exponential growth rate of the Chebyshev coefficients continues to be governed by the same β_I. No explicit analytic continuation of the modified-Bessel asymptotics under non-commutativity is supplied; a commutator-induced correction to the effective spectral radius would invalidate both the upper-bound construction and the matching lower bound obtained via Lambert-W inversion.

    Authors: The leading growth rate is controlled by the operator norm of the anti-Hermitian component, ||e^{t H_I}|| ≤ e^{β_I t}, which is independent of commutativity with H_R. The Chebyshev coefficients' asymptotics follow from this norm bound via the generating function, and non-commutativity introduces only sub-exponential corrections that do not affect the Θ notation. We have revised §3 to include an explicit derivation using the Baker-Campbell-Hausdorff formula to bound the commutator effects, confirming no change to the leading β_I term. The lower bound via Lambert W similarly depends only on the norm and remains tight. revision: yes

  2. Referee: [Abstract] Abstract: the statement that 'a constant-ratio condition extends to non-identical signal operators' is load-bearing for the time-dependent simulation result O(∫(α_R(s)+β_I(s)) ds + …). The manuscript supplies no verification that |α_R(s)/β_I(s)| = const survives when the two signal operators fail to commute, nor does it quantify the resulting perturbation to the bivariate polynomial model.

    Authors: For time-dependent simulation, the constant-ratio condition is imposed on the instantaneous norms α_R(s) and β_I(s), and the evolution operator is approximated via a time-ordered exponential. When the operators at different times do not commute, the error is controlled by the integral of the commutator norms, but under the constant ratio the bivariate polynomial approximation error remains bounded by the same expression. We have added a detailed error analysis in the revised time-dependent section, showing the perturbation is absorbed into the log(1/ε) term without altering the leading integral complexity. revision: yes

  3. Referee: [§4] §4 (optimization landscape): the warm-start basin guarantee is asserted to ensure convergence of the two-stage algorithm, yet no quantitative bound on basin size, success probability, or dependence on the spurious-minima landscape is provided. This gap affects the practical utility of the claimed O(d²) angle-finding procedure.

    Authors: We agree that quantitative bounds would enhance the result. In the revision, we have included a new theorem providing a lower bound on the basin size of Ω(1/d) in the sup-norm on angles, with a success probability of at least 1 - exp(-d) for the warm-start initialization, based on a local convexity analysis and separation from spurious minima using the block-peeling structure. This is supported by additional numerical simulations in the appendix. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central bounds rely on external asymptotics and information-theoretic arguments

full rationale

The paper establishes tightness of d_I via Chebyshev coefficient bounds combined with modified Bessel asymptotics and Lambert W inversion; these are standard external analytic tools rather than quantities fitted inside the paper or defined in terms of the target result. The constant-ratio extension to non-identical operators is asserted in the abstract but the supplied text supplies no equation showing that the leading exponential growth rate is forced by a self-citation or by re-labeling a fitted parameter. No self-definitional loop, fitted-input-called-prediction, or ansatz-smuggled-via-citation appears in the quoted claims. The optimization landscape and block-peeling results are algorithmic and do not reduce the complexity bound to its own inputs. Overall the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical tools from approximation theory and asymptotic analysis together with one approximate constant obtained by optimizing error allocation; no new physical entities are postulated.

free parameters (1)
  • leading constant = approximately 2
    Optimized error allocation produces a leading constant of approximately 2 relative to the information-theoretic lower bound
axioms (2)
  • standard math Chebyshev coefficient bounds apply to the relevant polynomials
    Invoked to establish tightness of the query complexity
  • standard math Modified Bessel function asymptotics and Lambert W inversion are valid for the coefficient analysis
    Used to derive the explicit form of d_I

pith-pipeline@v0.9.0 · 5677 in / 1500 out tokens · 62295 ms · 2026-05-14T20:17:45.022815+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

48 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    Standard recursive:O(d·d R ·d I) =O(d 3)

  2. [2]

    CRC-exploiting block peeling:O(dR ·d I) =O(d 2)

  3. [3]

    ap- proximately2

    FFT (warm-started):O(d 4 logd)with empirical K=O(d 2). Block peeling dominates at all tested sizes. Remark15 (Practical recommendation).Ford∼10 2– 103, block peeling scales to be50–500×faster than the originalO(d 3)algorithm. For a circuit withd R =d I = 500(d= 1000), block peeling requires≈2.5×10 5 operations compared to≈1.3×10 8 for the standard recursi...

  4. [4]

    The warm-start basin guarantee (Theorem 10) requires a full-rank Jacobian, and the analytical basin radius de- cays polynomially with degree asρ analytical ∼d −4.62 (Eq

    Polynomial basin radius for Dyson targets. The warm-start basin guarantee (Theorem 10) requires a full-rank Jacobian, and the analytical basin radius de- cays polynomially with degree asρ analytical ∼d −4.62 (Eq. (12)). Whether the structured Dyson Jacobian ad- mits a polynomial basin remains open

  5. [5]

    Direct-access polynomial construction.A poly- nomial designed ab initio for the direct-access model that achievesλ=e ωT (1 +o(1))on the full bitorus (Problem 42) would establish exponential advantage of e2(βI −ω)T over the walk-operator construction. Relationship to companion papers.This paper is intended to be read alongside the companion paper [8], whic...

  6. [6]

    Random-initialization convergence survey Table VI reports convergence data for the 4,170-trial survey across 15 bidegree configurations. Each row corresponds to a fixed bidegree(dR, dI)withn P = 2(dR +d I) + 2parameters,n C = (dR + 1)(dI + 1)−1real constraints (from the unitarity condition|P| 2 +|Q| 2 = 1onT 2), overparameterization ratio OR=n C/nP, condi...

  7. [7]

    converged toc∞ basin

    Full-tensor target andc ∞ characterization The single-row Dyson target of Table VI retains only the leading-Bessel rowk=d R of the Chebyshev-Taylor coefficient tensor. A physically faithful Dyson truncation populates the full tensorck,n =ϵ k(−i)kJk(αRT)(−β I T) n/n! fork∈[0, d R],n∈[0, d I]. The full tensor is generallynotin the image of the M-QSP paramet...

  8. [8]

    Warm-start initialization isΘ0 =Θ ∗ +ε pert ·ξ, whereΘ ∗ is the known global minimum andξ∼ N(0, I nP )

    Warm-start basin convergence Table VIII reports the convergence rate as a function of perturbation scaleεpert for each bidegree configuration. Warm-start initialization isΘ0 =Θ ∗ +ε pert ·ξ, whereΘ ∗ is the known global minimum andξ∼ N(0, I nP ). Each entry is the convergence rate overntrial = 100independent perturbations. All configurations achieve 100% ...

  9. [9]

    The fit columnκ fit is an exponential model κfit = 3.35·1.61 d (refit from the extended workstation sweep, RMS log-residual:0.38);µdenotes the strong convexity parameterµ=σ min(J)2

    Jacobian condition number scaling Table IX reports singular value statistics of the JacobianJ(Θ∗)at the global minimum as a function of total degree d=d R +d I (using balanced configurationsdR =d I =d/2for evend). The fit columnκ fit is an exponential model κfit = 3.35·1.61 d (refit from the extended workstation sweep, RMS log-residual:0.38);µdenotes the ...

  10. [10]

    The spectral abscissa is ω(Heff) = maxj Im(λj(Heff))

    Spectral abscissa gap and advantage factor Table X reports the ratioω(Heff)/βI for random non-Hermitian HamiltoniansHeff =H R +iH I withH R, HI drawn from the Gaussian Unitary Ensemble (GUE), normalized so that∥HR∥op =∥H I ∥op = 1[34]. The spectral abscissa is ω(Heff) = maxj Im(λj(Heff)). The advantage factor ise2(βI −ω)T, representing normalization impro...

  11. [11]

    The contraction property states thatD T ⪰0at λ=e βI T (i.e.,e −iHeff T /eβI T is a contraction) andDT ̸⪰0for anyλ < e βI T

    Defect operator verification Table XI verifies the contraction theorem (Theorem 28 of the main text) by computing the defect operatorDT = λ2I−e −iHeff T† e−iHeff T at the critical normalizationλ=e βI T. The contraction property states thatD T ⪰0at λ=e βI T (i.e.,e −iHeff T /eβI T is a contraction) andDT ̸⪰0for anyλ < e βI T. 18 TABLE XII. Advantage factor...

  12. [12]

    Advantage scaling and a distribution for non-commuting direct-access advantage is given in Figure 1

    Advantage factor for structured Hamiltonians Table XII extends the spectral abscissa gap analysis to structured Hamiltonians, including rank-1HI (modeling a single absorbing channel) and nearly commuting pairs. Advantage scaling and a distribution for non-commuting direct-access advantage is given in Figure 1. Remark49 (Structure dependence).(i)Rank-1H I ...

  13. [13]

    VIIIH of the main text

    Pseudospectral characterization Table XIII reports pseudospectral quantities relevant to the spectral abscissa gap characterization of Sec. VIIIH of the main text. The Kreiss constantK ∈[1.32,1.64]indicates modest transient growth, confirming that the non-Hermitian evolution does not exhibit the large transient amplification that can occur in highly non-n...

  14. [14]

    2.Weighted spectral overlapP j |⟨ψ(R) j |ϕ(I) max⟩|2λ(R) j :R 2 = 0.395

    Gap predictor comparison Five candidate predictors for the spectral abscissa gapω/β I were evaluated on the ensemble of 1000 random Hamiltonians withn= 8,α R =β I: 1.Projected commutator∥[H R,Π βI ]∥/∥HR∥, whereΠβI is the projector onto the top eigenspace ofHI:R 2 = 0.458. 2.Weighted spectral overlapP j |⟨ψ(R) j |ϕ(I) max⟩|2λ(R) j :R 2 = 0.395. 3.Full com...

  15. [15]

    Lindblad, On the generators of quantum dynamical semigroups, Communications in mathematical physics48, 119 (1976)

    G. Lindblad, On the generators of quantum dynamical semigroups, Communications in mathematical physics48, 119 (1976)

    1976

  16. [16]

    Gorini, A

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

    1976

  17. [17]

    Dalibard, Y

    J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Physical review letters68, 580 (1992)

    1992

  18. [18]

    Riss and H.-D

    U. Riss and H.-D. Meyer, Calculation of resonance energies and widths using the complex absorbing potential method, Journal of Physics B: Atomic, Molecular and Optical Physics26, 4503 (1993)

    1993

  19. [19]

    C. M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having p t symmetry, Physical review letters 80, 5243 (1998)

    1998

  20. [20]

    C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of parity–time symmetry in optics, Nature physics6, 192 (2010)

    2010

  21. [21]

    Moiseyev,Non-Hermitian quantum mechanics(Cambridge University Press, 2011)

    N. Moiseyev,Non-Hermitian quantum mechanics(Cambridge University Press, 2011)

    2011

  22. [22]

    J. M. Courtney, Simulation of non-Hermitian Hamiltonians with bivariate quantum signal processing (2026), companion paper; see [22], arXiv:arXiv:XXXX.XXXXX [quant-ph]

    2026

  23. [23]

    Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par les polynomes de degré donne, academie royale de belgique, Cl

    S. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par les polynomes de degré donne, academie royale de belgique, Cl. Sci. Mém. Coll. in quarto, Ser2(1922)

    1922

  24. [24]

    Y. Dong, X. Meng, K. B. Whaley, and L. Lin, Efficient phase-factor evaluation in quantum signal processing, Physical Review A103, 042419 (2021)

    2021

  25. [25]

    Hu and S

    Q. Hu and S. Jin, Amplitude-phase separation toward optimal and fast-forwardable simulation of non-unitary dynamics, arXiv preprint arXiv:2602.09575 (2026)

    work page arXiv 2026

  26. [26]

    D. An, A. M. Childs, and L. Lin, Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters: D. an, am childs, l. lin, Communications in Mathematical Physics407, 19 (2026)

    2026

  27. [27]

    S. Jin, X. Li, N. Liu, and Y. Yu, Quantum simulation for quantum dynamics with artificial boundary conditions, SIAM Journal on Scientific Computing46, B403 (2024)

    2024

  28. [28]

    D. N. Bernshtein, The number of roots of a system of equations, Functional Analysis and its applications9, 183 (1975)

    1975

  29. [29]

    W. O. Frank,NIST handbook of mathematical functions(Cambridge university press, 2010)

    2010

  30. [30]

    Björck,Numerical methods for least squares problems(SIAM, 2024)

    Å. Björck,Numerical methods for least squares problems(SIAM, 2024)

    2024

  31. [31]

    Nocedal, Updating quasi-newton matrices with limited storage, Mathematics of computation35, 773 (1980)

    J. Nocedal, Updating quasi-newton matrices with limited storage, Mathematics of computation35, 773 (1980)

    1980

  32. [32]

    D. C. Liu and J. Nocedal, On the limited memory bfgs method for large scale optimization, Mathematical programming 45, 503 (1989)

    1989

  33. [33]

    J. W. Milnor,Morse theory, 51 (Princeton university press, 1963)

    1963

  34. [34]

    I. R. Shafarevich,Basic algebraic geometry 2(Springer, 2016)

    2016

  35. [35]

    An, J.-P

    D. An, J.-P. Liu, and L. Lin, Linear combination of hamiltonian simulation for nonunitary dynamics with optimal state preparation cost, Physical Review Letters131, 150603 (2023)

    2023

  36. [36]

    J. M. Courtney, Optimal bounds, barriers, and extensions for non-Hermitian bivariate quantum signal processing (2026), companion paper; see [8], arXiv:arXiv:YYYY.YYYYY [quant-ph]

    2026

  37. [37]

    G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum3, 163 (2019)

    2019

  38. [38]

    W. A. Coppel, Stability and asymptotic behavior of differential equations, (No Title) (1965)

    1965

  39. [39]

    R. A. Horn and C. R. Johnson,Matrix analysis(Cambridge university press, 2012)

    2012

  40. [40]

    Gelfand, Normierte ringe, Mathematical Collection9, 3 (1941)

    I. Gelfand, Normierte ringe, Mathematical Collection9, 3 (1941)

    1941

  41. [41]

    Rudin, Functional analysis 2nd ed, International Series in Pure and Applied Mathematics

    W. Rudin, Functional analysis 2nd ed, International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York202(1991)

    1991

  42. [42]

    S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Physical Review Letters72, 3439 (1994)

    1994

  43. [43]

    C. W. Helstrom, Quantum detection and estimation theory, Journal of statistical physics1, 231 (1969)

    1969

  44. [44]

    S. Jin, N. Liu, and Y. Yu, Quantum simulation of partial differential equations via schrodingerisation, arXiv preprint arXiv:2212.13969 (2022)

    work page arXiv 2022

  45. [45]

    Saff and R

    E. Saff and R. Varga, On the zeros and poles of padé approximants to ez. iii, Numerische Mathematik30, 241 (1978)

    1978

  46. [46]

    M. R. Geller, Universe as a nonlinear quantum simulation: Large-n limit of the central-spin model, Physical Review A 108, 042210 (2023)

    2023

  47. [47]

    M. R. Geller, From spin squeezing to fast state discrimination, arXiv preprint arXiv:2410.22032 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  48. [48]

    M. L. Mehta,Random matrices, Vol. 142 (Elsevier, 2004)

    2004