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REVIEW 1 major objections 4 minor 83 references

Analog classical simulation of closed quantum systems

T0 review · 1 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper claims that the Schrödinger equation of any closed quantum system can be rewritten as a real second-order linear ODE, so that an analog classical computer—or even a spring-mass device—can simulate the full quantum dynamics.

desk verdict Solid ODE mapping and GPAC framing, but the spring-mass realization rests on an unproven truncation that demonstrably fails for sign-flipped Hamiltonians—worth refereeing because the core math is sound. read the letter →

arxiv 2502.06311 v1 pith:YIHOOV4J submitted 2025-02-10 quant-ph

Ka-Wa Yip This is my paper · ORCID
classification quant-ph PACS 03.65.-w
keywords analogclassicalsimulationSchrödingerequationreal-valuedODEspring-masssystemsecond-orderlineardynamicalQAOAfast-forwardingHamiltonian
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

The paper claims that the Schrödinger equation for a closed quantum system can be differentiated once and then decomplexified into a real second-order linear ODE, $|\ddot{\varphi}(t)\rangle = -K(t)|\varphi(t)\rangle$, whose solutions are exactly the real and imaginary parts of the original probability amplitudes. Because for a time-independent Hamiltonian the matrix $K$ is real symmetric and positive semidefinite, it can be read as the stiffness matrix of a classical second-order dynamical system such as a one-dimensional spring-mass network. This would mean that an analog classical computer can reveal the full time-dependent wavefunction without the repeated measurements and state collapses that quantum simulators need. It also implies that analog classical devices can in principle execute quantum algorithms, and the paper demonstrates the idea on an eight-qubit QAOA instance.

What carries the argument

The load-bearing object is the matrix $K(t) = i\dot{H}(t)+H(t)^2$ and its real $2N \times 2N$ block representation $K(t) = \begin{pmatrix} \mathrm{Re}\,K(t) & -\mathrm{Im}\,K(t) \\ \mathrm{Im}\,K(t) & \mathrm{Re}\,K(t) \end{pmatrix}$. The paper's central move is to differentiate the Schrödinger equation, which removes the factor $i$ from $H^2$ and produces a matrix that is Hermitian (indeed positive semidefinite) whenever $H$ is time-independent; decomplexification then makes this matrix real symmetric, so it qualifies mechanically as a stiffness matrix. The initial-value data are transferred through $|\dot{\varphi}(0)\rangle = -J(0)|\varphi(0)\rangle$, where $J(0)$ is the real representation of $iH(0)$, and the spring-mass construction encodes the entries of $K$ as spring constants, with a translation-and-truncation step used when $K$ has positive off-diagonal entries.

What would settle it

Take a four-qubit Hamiltonian with positive off-diagonals in H², apply the translation-and-truncation rule with increasing α, and check whether the truncated stiffness matrix K'' remains positive semidefinite and whether the spring-mass trajectory stays within a fixed error of $e^{{-iHt}}$|ψ(0)⟩; a single instance with a negative eigenvalue or an error that does not shrink with α would falsify the claim that any time-independent Hamiltonian can be simulated this way.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Schrödinger equation $i|\dot{\psi}(t)\rangle = H(t)|\psi(t)\rangle$, after one differentiation, is equivalent to the real-valued second-order equation $|\ddot{\varphi}(t)\rangle = -K(t)|\varphi(t)\rangle$, where $|\varphi(t)\rangle$ stacks the real and imaginary parts of $|\psi(t)\rangle$ and $K(t)$ is the real block representation of $K(t) = i\dot{H}(t)+H(t)^2$. The equivalence holds when the initial velocity is set from the Schrödinger equation at $t=0$, namely $|\dot{\varphi}(0)\rangle = -J(0)|\varphi(0)\rangle$. For a time-independent Hamiltonian, $K = H^2$ is Hermitian and positive semidefinite, so $K$ is real symmetric and positive semidefinite; Theorem 1 then says this matrix can serve directly as the stiffness matrix of a second-order linear dynamical system with unit mass matrix and no damping. The paper calls this analog classical simulation and shows that a spring-mass realization, an eight-qubit QAOA run on a general-purpose analog computer, and a twofold speedup by scaling spring constants all follow from the same mapping.

Load-bearing premise

The whole result rests on the assumption that the real ODE obtained by differentiating and decomplexifying the Schrödinger equation—with K built from H²—can be realized as a genuine spring-mass or second-order dynamical system, and in particular that zeroing out the wrong-sign off-diagonal entries of K (the translation-and-truncation step) keeps the approximation faithful.

Editorial extensions

If this is right

  • The full probability-amplitude vector of a closed quantum system can be read out continuously in real time from an analog device, something digital simulation can only approximate and quantum simulation cannot do without collapse.
  • Analog classical computers running Algorithm 1 can perform quantum algorithms; the paper concretely simulates an eight-qubit QAOA Max-Cut instance by repeated applications of the algorithm.
  • Scaling all spring constants by $p^2$ and all initial velocities by $p$ produces a linear $p$-fold speedup of the simulated quantum dynamics relative to clock time.
  • For real symmetric Hamiltonians, the simulation decouples into two identical oscillator subsystems, one encoding the real and one the imaginary part of the wavefunction.
  • Time-dependent Hamiltonians require knowledge of $\dot{H}(t)$ but are in principle also simulable, since $K(t)$ remains well defined for arbitrary Hermitian $H(t)$.

Reading between the lines

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

  • If the mapping is exact for all Hermitian Hamiltonians, the computational boundary between classical analog and quantum computation is not set by entanglement or state-vector size alone, but by how hard it is to construct $K = H^2$ and to achieve the required physical precision in an analog device.
  • A testable extension would use the same second-order form to simulate open systems or dissipative dynamics by adding the damping term $B\dot{x}$, since the framework already sits inside the general second-order linear system.
  • The translation-and-truncation step is the point to probe: a non-stoquastic Hamiltonian with large positive off-diagonals could reveal whether zeroing those entries is a controlled approximation or a hidden restriction that limits the spring-mass device to stoquastic or nearly diagonal $K$.
  • Because the analog device outputs amplitudes rather than samples, variational algorithms such as QAOA could in principle be run with gradients evaluated directly from continuous trajectories, avoiding the sampling overhead of quantum expectation estimation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper proposes an analog classical simulation algorithm for closed quantum systems. It differentiates the Schrödinger equation to obtain the second-order ODE |ψ¨⟩ = −(iH˙+H²)|ψ⟩, decomplexifies it into a real second-order system |φ¨⟩ = −K|φ⟩ with K the real representation of iH˙+H², and packages the mapping as Algorithm 1. Theorem 1 shows that for time-independent H the real matrix K is symmetric positive semidefinite, so the evolution can be viewed as a second-order linear dynamical system with M=I, D=0 and S=K; Corollary 1.1 gives a decoupled two-block form when H is real symmetric. The paper then considers one-dimensional spring-mass realizations, introducing an unconventional axis-flip encoding and a 'translation and truncation' approximation (Sec. VI B 3, Eq. 20) to obtain nonnegative spring constants, and illustrates the scheme on a four-qubit transverse-field example (Fig. 3), a scaled spring system with 2× speedup (Sec. VII), and an eight-qubit QAOA instance (Sec. VIII).

Significance. The algebraic derivation is self-contained and correct as far as Theorem 1 and Corollary 1.1 are concerned; these results give an exact, basis-independent reduction of Schrödinger dynamics to real second-order linear ODEs. The numerical examples are consistency checks rather than fits, and the code is made available. The central load-bearing caveat is that the spring-mass realization of a general real-symmetric H depends on the uncontrolled truncation in Sec. VI B 3; until that step is supplied with an error bound or a sign-adaptive construction, the headline claim that such systems 'may be solved by a simple analog mechanical device' remains unproven. If the truncation issue is resolved, the work would be a useful contribution to analog models of quantum simulation.

major comments (1)
  1. [Sec. VI B 3 (Eq. 20) and footnote 8] The spring-mass realization of an arbitrary real-symmetric Hamiltonian is not established because the 'translation and truncation' step is an uncontrolled approximation. The text asserts that as α grows, positive off-diagonal entries of K′ may be truncated 'without affecting the dynamics,' but no error bound is given and footnote 8 only assumes positive semidefiniteness of K′′, not accuracy. The sign-dependence is concrete: for H = −Σ_{i=1}^4 σx_i and |ψ(0)⟩=|0000⟩, H′ = H − αI gives K′ = (Σσx_i + αI)², whose off-diagonal entries are all positive (+2α for one-bit flips, +2 for two-bit flips); Trunc removes all of them, leaving K′′ = (α²+4)I. Using H′ in Eq. 12 as in the Fig. 3 recipe gives u0(t)=cos(ωt), v0(t)=(α/ω)sin(ωt) with ω=√(α²+4), hence |c0(t)|² = cos²(ωt)+α²/(α²+4)sin²(ωt) → 1 as α→∞, whereas the exact Schrödinger value is cos⁸(t) (≈0.007 at t=1). Thus the truncation error is order-one for a valid real-symmetric H and grows with α, directly contradicting the claim that larger α improves the approximation. The paper needs either a sign-adaptive construction that retains the dominant couplings, a rigorous error bound with conditions on H, or a clear restriction of the spring-mass claim to a class of Hamiltonians for which the truncation is controlled.
minor comments (4)
  1. [Global (typos)] There are several typographical errors: 'similiar' in the proof of Theorem 1, 'seqeuence' in the caption of Fig. 5, 'Rigoriously' in Sec. IX, and 'a L0-matrix' in Sec. VI B 1. Please proofread.
  2. [Fig. 3] The four dotted curves for α ∈ {8,10,20,100} are not labeled or distinguished in the figure, so the reader cannot assess the approximation quality per α or the convergence with α. Please add a legend or labels and, ideally, an error panel.
  3. [Sec. I / Sec. V] The statement that 'the global stability of the real ODEs can be proven' is not followed by a proof. For time-dependent H, K(t) is non-Hermitian (Table I) and may have negative eigenvalues, so any stability statement should be specified (e.g., invariance of the Schrödinger subspace) and either proved or qualified.
  4. [Sec. IX / Fig. 7] The runtime comparison between fastexpm and matrix squaring is implementation-specific; the asymptotic cost of both is exponential in n, as the text acknowledges. The discussion should be framed as a heuristic benchmark rather than a general complexity advantage.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the core derivation is a self-contained algebraic mapping, and the flagged truncation is a stated approximation, not a fit disguised as a prediction.

full rationale

The paper's central claim is a direct mathematical transformation: differentiating the Schrödinger equation (Eq. 2) yields the second-order ODE |ψ¨⟩ = −K|ψ⟩ with K = iḢ + H² (Eqs. 5–6), and the decomplexification step (Eqs. 8–12) converts this into real ODEs exactly. Theorem 1 then identifies the real symmetric positive-semidefinite stiffness matrix S with K, which is an algebraic identity rather than a fitted or self-referential construction. The numerical examples, including the QAOA demonstration, are consistency checks of this mapping against exact Schrödinger evolution, not predictions tuned to data. The only potentially weak step is the truncation approximation in Sec. VI B 3 (Eq. 20 and footnote 8), where positive off-diagonals of K′ are set to zero; the paper explicitly flags this as an assumption and provides no error bound. That is a correctness or validation gap, not circularity: the approximation is compared with the exact dynamics rather than used to define them. The paper contains no load-bearing self-citations and no imported uniqueness or ansatz results from the author's prior work. The derivation is self-contained and the central claim does not reduce to its inputs by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper does not introduce new physical entities; it relies on standard mathematical transformations and on the GPAC model of analog computation. The main extra assumptions are the realizability of a PSD matrix as a stiffness matrix and the validity of the truncation approximation.

free parameters (2)
  • Translation parameter α = 8, 10, 20, 100 (example-specific)
    Used in Sec. VI B 3 to shift H to H − αI so that positive off-diagonals of K' can be truncated; larger α improves the approximation but no error bound is given.
  • Speedup factor p = 2 in Sec. VII
    Scales spring constants by p² and initial velocities by p to achieve a p-fold time contraction; this is a design choice, not fitted, but it is a free parameter of the fast-forwarding claim.
assumptions (6)
  • standard math The IVP equivalence in Appendix B: solutions of the second-order ODE with Schrödinger-compatible initial conditions match the Schrödinger solutions.
    Follows from differentiating the Schrödinger equation and checking the initial condition; a standard ODE argument.
  • standard math For time-independent Hermitian H, K=H² is Hermitian positive semidefinite, and its real representation is real symmetric positive semidefinite.
    Uses spectral properties of Hermitian matrices; proofs sketched in Appendices F and G.
  • domain assumption A real symmetric positive semidefinite matrix can serve as the stiffness matrix S in the second-order linear dynamical system Eq. 13.
    True for the abstract linear system, but not for a passive spring-mass network with only positive springs, which requires S to be a Z-matrix (off-diagonals nonpositive). The paper relies on this for Theorem 1.
  • ad hoc to paper Truncation of positive off-diagonal entries preserves positive semidefiniteness of K'' and the approximation error is small.
    Explicitly assumed in Sec. VI B 3 footnote 8; no proof or error bound is supplied.
  • domain assumption The GPAC model can solve the real ODE system Eq. 1 for polynomial ODEs.
    Standard result in analog computability (Shannon, Bournez et al.), cited by the paper.
  • domain assumption Global stability of the real ODEs for time-dependent H follows from Hermiticity of H.
    Asserted in the Introduction but not proven; for time-dependent H, K is non-Hermitian and the second-order system admits spurious solutions, so this is not immediate.

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Cite this review

Pith. "Pith review of Analog classical simulation of closed quantum systems." pith.science (2026). https://pith.science/paper/YIHOOV4J

@misc{pith2026250206311,
  author       = {Pith},
  title        = {Pith review of: Analog classical simulation of closed quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YIHOOV4J}},
  note         = {Machine review of arXiv:2502.06311}
}
read the original abstract

We develop an analog classical simulation algorithm of noiseless quantum dynamics. By formulating the Schr\"{o}dinger equation into a linear system of real-valued ordinary differential equations (ODEs), the probability amplitudes of a complex state vector can be encoded in the continuous physical variables of an analog computer. Our algorithm reveals the full dynamics of complex probability amplitudes. Such real-time simulation is impossible in quantum simulation approaches without collapsing the state vector, and it is relatively computationally expensive for digital classical computers. For a real symmetric time-independent Hamiltonian, the ODEs may be solved by a simple analog mechanical device such as a one-dimensional spring-mass system. Since the underlying dynamics of quantum computers is governed by the Schr\"{o}dinger equation, our findings imply that analog computers can also perform quantum algorithms. We illustrate how to simulate the Schr\"{o}dinger equation in such a paradigm, with an application to quantum approximate optimization algorithm. This may pave the way to emulate quantum algorithms with physical computing devices, including analog, continuous-time circuits.

Figures

Figures reproduced from arXiv: 2502.06311 by the authors.

Figure 1
Figure 1. FIG. 1. (a) An illustration of a fully-connected one-dimensional spring-mass system with 4 masses and 10 springs. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The highlighted matrix elements switch signs [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Solid line represents the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (a) shows the time evolution of Re{c0(t)}, Re{c1(t)} and Im{c7(t)} in a closed quantum sys￾tem. The values are obtained by solving the Schr¨odinger equation with the given values of transverse field and coupling strengths [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The time evolution of the first matrix element [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) The time evolution of objective value [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Computation ratio [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Two different approaches of calculating [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

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Reference graph

Works this paper leans on

83 extracted references · 73 canonical work pages

  1. [1]

    Therefore, a sufficient condition for K(t) to be Hermitian is that the Hamiltonian is time-independent; i.e., H(t) = H for all t

    Hermitian K At time t, the matrix K(t) = i ˙H(t) + H(t)2 is Hermitian if and only if ˙H(t) = 0. Therefore, a sufficient condition for K(t) to be Hermitian is that the Hamiltonian is time-independent; i.e., H(t) = H for all t. 5 TABLE I. A table illustrating the relationship between the Hamiltonian H(t), K(t) and K(t). H(t) K(t) K(t) Time-dependent non-Her...

  2. [2]

    This occurs in the case of time-reversal invariant Hamiltonian

    Real symmetric K If H is real symmetric, then K = H2 is also real symmetric. This occurs in the case of time-reversal invariant Hamiltonian. There are additional sce- narios where K can be real symmetric. For exam- ple, when Re {H} Im{H} is real symmetric. This condition is trivially met when H is either itself real symmetric or purely imaginary. These po...

  3. [3]

    For more on the classification of matrix families, see for ex- ample Refs

    The matrix K is a Z-matrix We introduce two matrix definitions. For more on the classification of matrix families, see for ex- ample Refs. [40–42]. Definition 1. A Z-matrix is a real matrix whose off-diagonal entries are nonpositive. Definition 2. A ZM-matrix is a matrix whose ev- ery power is a Z-matrix. Hence, if H is a ZM-matrix, then its square — K — ...

  4. [4]

    Therefore, for time-independent H, the Hermitian matrix K is necessarily positive semidefinite

    Positive semidefinite K Since the eigenvalues of H(t) are real, the eigen- values of H(t)2 are nonnegative. Therefore, for time-independent H, the Hermitian matrix K is necessarily positive semidefinite. Positive semidef- initeness of K also implies that it has nonnegative diagonal entries. Such matrices with nonpositive off-diagonal and nonnegative diago...

    work page

  5. [5]

    A simple proof is given in Appendix F

    Real symmetric K If K is Hermitian, then K is real symmetric. A simple proof is given in Appendix F

    work page

  6. [6]

    Block diagonal K If K is real, then K is block diagonal

    work page

  7. [7]

    The matrix K is a block diagonal Z-matrix If K is a Z-matrix, then K is a block diagonal Z-matrix

    work page

  8. [8]

    A simple proof is given in Ap- pendix G

    Positive semidefinite K If K is positive semidefinite, then K is posi- tive semidefinite. A simple proof is given in Ap- pendix G. H(t). Adding an arbitrary infinitesimal ϵI to H(t) per- turbs the whole eigenspectrum by + ϵ, effectively remov- ing all zero eigenvalues. 6 VI. REAL SECOND-ORDER LINEAR DYNAMICAL SYSTEM Recall that the second-order real-value...

    work page

Show all 83 references
  1. [9]

    For example, for a 1-D system, the positives are either all to the right or all to the left

    Conventional encoding: every mass shares the same positive axis direction In the conventional encoding scheme, every mass shares the same positive axis direction. For example, for a 1-D system, the positives are either all to the right or all to the left. In this way, the off-...

  2. [10]

    Unconventional encoding and the change of signs The general observation is, if the default posi- tive axis of mass i is “flipped”, the signs of these particular off-diagonal matrix elements Sij have to switch as follows, to preserve the dynamics de- scribed in Eq. 16. Sij → (−...

  3. [11]

    translation and truncation

    Translation and truncation Consider again the conventional encoding scheme in Sec. VI B 1, where every mass shares the same positive direction. Given a Hamiltonian H of which the K has many positive off-diagonals, one can define a translated Hamiltonian H′ = H − αI, from which...

  4. [12]

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

    2014

  5. [13]

    Wiebe, D

    N. Wiebe, D. W. Berry, P. Høyer, and B. C. Sanders, Simulating quantum dynamics on a quantum computer, Journal of Physics A: Math- ematical and Theoretical 44, 445308 (2011)

    2011

  6. [14]

    Babbush, D

    R. Babbush, D. W. Berry, J. R. McClean, and H. Neven, Quantum simulation of chemistry with sublinear scaling in basis size, npj Quantum In- formation 5, 92 (2019)

    2019

  7. [15]

    J. I. Cirac and P. Zoller, Goals and opportuni- ties in quantum simulation, Nature physics 8, 264 (2012)

    2012

  8. [16]

    D. A. Lidar and H. Wang, Calculating the ther- mal rate constant with exponential speedup on a quantum computer, Physical Review E 59, 2429 (1999)

    1999

  9. [17]

    Raeisi, N

    S. Raeisi, N. Wiebe, and B. C. Sanders, Quantum- circuit design for efficient simulations of many- body quantum dynamics, New Journal of Physics 14, 103017 (2012)

    2012

  10. [18]

    Preskill, Quantum computing in the nisq era and beyond, Quantum 2, 79 (2018)

    J. Preskill, Quantum computing in the nisq era and beyond, Quantum 2, 79 (2018)

    2018

  11. [19]

    Lloyd, Universal quantum simulators, Science 273, 1073 (1996)

    S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996)

    1996

  12. [20]

    A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of trotter error with commutator scaling, Physical Review X 11, 011020 (2021)

    2021

  13. [21]

    Aharonov and A

    D. Aharonov and A. Ta-Shma, Adiabatic quan- tum state generation and statistical zero knowl- edge, in Proceedings of the thirty-fifth annual ACM symposium on Theory of computing (2003) pp. 20–29

    2003

  14. [22]

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating hamiltonian dy- namics with a truncated taylor series, Physical review letters 114, 090502 (2015)

    2015

  15. [23]

    G. H. Low and I. L. Chuang, Hamiltonian simu- lation by qubitization, Quantum 3, 163 (2019)

    2019

  16. [24]

    Aharonov and L

    D. Aharonov and L. Zhou, Hamiltonian Spar- sification and Gap-Simulation, in 10th Innova- tions in Theoretical Computer Science Confer- ence (ITCS 2019), Leibniz International Proceed- ings in Informatics (LIPIcs), Vol. 124, edited by A. Blum (Schloss Dagstuhl–Leibniz-Zentrum ...

    2019

  17. [25]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al., Probing many-body dynamics on a 51-atom quantum simulator, Na- ture 551, 579 (2017)

    2017

  18. [26]

    Zhang, G

    J. Zhang, G. Pagano, P. W. Hess, A. Kypriani- dis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe, Observation of a many- body dynamical phase transition with a 53-qubit quantum simulator, Nature 551, 601 (2017)

    2017

  19. [27]

    A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Na- ture 607, 667 (2022)

    2022

  20. [28]

    O. Bournez, Computability, Complexity and Pro- gramming with Ordinary Differential Equations (Invited Talk), in 37th International Sympo- sium on Theoretical Aspects of Computer Sci- ence (STACS 2020) , Leibniz International Pro- ceedings in Informatics (LIPIcs), Vol. 154, edi...

    2020

  21. [29]

    Bournez and A

    O. Bournez and A. Pouly, A survey on analog models of computation, in Handbook of Com- putability and Complexity in Analysis (Springer,

  22. [30]

    Bournez and M

    O. Bournez and M. L. Campagnolo, A survey on continuous time computations, in New computa- tional paradigms: Changing conceptions of what is computable (Springer, 2008) pp. 383–423

    2008

  23. [31]

    C. E. Shannon, Mathematical theory of the dif- ferential analyzer, Journal of Mathematics and Physics 20, 337 (1941)

    1941

  24. [32]

    Babbush, D

    R. Babbush, D. W. Berry, R. Kothari, R. D. Somma, and N. Wiebe, Exponential quantum speedup in simulating coupled classical oscillators, Phys. Rev. X 13, 041041 (2023)

    2023

  25. [33]

    L. K. Grover and A. M. Sengupta, Classical analog of quantum search, Physical Review A 65, 032319 (2002)

    2002

  26. [34]

    A. A. Patel, A coupled oscillator model for grover’s quantum database search algorithm, 14 arXiv preprint arXiv:0711.4733 (2007)

    2007 arXiv

  27. [35]

    Hasler, Opportunities in physical computing driven by analog realization, in 2016 IEEE in- ternational conference on rebooting computing (ICRC) (IEEE, 2016) pp

    J. Hasler, Opportunities in physical computing driven by analog realization, in 2016 IEEE in- ternational conference on rebooting computing (ICRC) (IEEE, 2016) pp. 1–8

    2016

  28. [36]

    Yon Neumann, The computer and the brain, New Haven, Conn.: Yale Uni (1958)

    J. Yon Neumann, The computer and the brain, New Haven, Conn.: Yale Uni (1958)

    1958

  29. [37]

    Mead, Neuromorphic electronic systems, Pro- ceedings of the IEEE 78, 1629 (1990)

    C. Mead, Neuromorphic electronic systems, Pro- ceedings of the IEEE 78, 1629 (1990)

    1990

  30. [38]

    Mead and M

    C. Mead and M. Ismail, Analog VLSI implemen- tation of neural systems , Vol. 80 (Springer Science & Business Media, 1989)

    1989

  31. [39]

    G. M. Gladwell, Inverse problems in vibration (Springer, 1986)

    1986

  32. [40]

    H. T. Siegelmann and S. Fishman, Analog compu- tation with dynamical systems, Physica D: Non- linear Phenomena 120, 214 (1998)

    1998

  33. [41]

    Bournez, M

    O. Bournez, M. L. Campagnolo, D. S. Gra¸ ca, and E. Hainry, The general purpose analog com- puter and computable analysis are two equiva- lent paradigms of analog computation, in Interna- tional Conference on Theory and Applications of Models of Computation (Springer, 2006) pp...

    2006

  34. [42]

    Thomson, Vi

    W. Thomson, Vi. mechanical integration of the general linear differential equation of any or- der with variable coefficients, Proceedings of the Royal Society of London 24, 271 (1876)

  35. [43]

    Bournez, D

    O. Bournez, D. Gra¸ ca, and A. Pouly, Computing with polynomial ordinary differential equations, Journal of Complexity 36, 106 (2016)

    2016

  36. [44]

    Schr¨ odinger,Collected papers on wave mechan- ics, Vol

    E. Schr¨ odinger,Collected papers on wave mechan- ics, Vol. 302 (American Mathematical Society, 2003)

    2003

  37. [45]

    R. L. Chen, Derivation of the real form of schr¨ odinger’s equation for a nonconservative sys- tem and the unique relation between Re( ψ) and Im(ψ), Journal of mathematical physics 30, 83 (1989)

    1989

  38. [46]

    R. L. Chen, Concerning the gauge invariance and the apparent nonlocality of the real form of Schr¨ odinger’s equation, Journal of mathematical physics 32, 464 (1991)

    1991

  39. [47]

    Callender, Quantum mechanics: Keeping it real?, The British Journal for the Philosophy of Science (2020)

    C. Callender, Quantum mechanics: Keeping it real?, The British Journal for the Philosophy of Science (2020)

    2020

  40. [48]

    C. N. Yang, Square root of minus one, complex phases and erwin schr¨ odinger, in Schr¨ odinger: Centenary Celebration of a Polymath , edited by C. W. Kilmister (Cambridge University Press,

  41. [49]

    Hestenes, The kinematic origin of complex wave functions, Physics and Probability: Essays in Honor of Edwin T

    D. Hestenes, The kinematic origin of complex wave functions, Physics and Probability: Essays in Honor of Edwin T. Jaynes, Cambridge U. Press, Cambridge , 153 (1993)

    1993

  42. [50]

    R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, 1985)

    1985

  43. [51]

    Azimzadeh, A fast and stable test to check if a weakly diagonally dominant matrix is an M - matrix, Math

    P. Azimzadeh, A fast and stable test to check if a weakly diagonally dominant matrix is an M - matrix, Math. Comput. 88, 783 (2017)

    2017

  44. [52]

    Friedland, D

    S. Friedland, D. Hershkowitz, and H. Schnei- der, Matrices whose powers are m-matrices or z- matrices, Transactions of the American Mathe- matical Society 300, 343 (1987)

    1987

  45. [53]

    Berman and R

    A. Berman and R. J. Plemmons, Nonnegative ma- trices in the mathematical sciences (SIAM, 1994)

    1994

  46. [54]

    Bravyi, D

    S. Bravyi, D. P. DiVincenzo, R. I. Oliveira, and B. M. Terhal, The complexity of stoquastic lo- cal hamiltonian problems (2007), arXiv:quant- ph/0606140 [quant-ph]

    2007

  47. [55]

    Marvian, D

    M. Marvian, D. A. Lidar, and I. Hen, On the com- putational complexity of curing non-stoquastic hamiltonians, Nature communications 10, 1571 (2019)

    2019

  48. [56]

    Klassen and B

    J. Klassen and B. M. Terhal, Two-local qubit hamiltonians: when are they stoquastic?, Quan- tum 3, 139 (2019)

    2019

  49. [57]

    Choi, Essentiality of the non-stoquastic hamiltonians and driver graph design in quan- tum optimization annealing, arXiv preprint arXiv:2105.02110 (2021)

    V. Choi, Essentiality of the non-stoquastic hamiltonians and driver graph design in quan- tum optimization annealing, arXiv preprint arXiv:2105.02110 (2021)

    2021 arXiv

  50. [58]

    D. T. Kawano, M. Morzfeld, and F. Ma, The de- coupling of second-order linear systems with a sin- gular mass matrix, Journal of Sound and Vibra- tion 332, 6829 (2013)

    2013

  51. [59]

    D. T. Kawano, R. G. Salsa, and F. Ma, Decou- pling of second-order linear systems by isospectral transformation, Zeitschrift f¨ ur angewandte Math- ematik und Physik 69, 1 (2018)

    2018

  52. [60]

    D. L. Logan, A first course in the finite element method (PWS Engineering Boston, Boston, 1986)

    1986

  53. [61]

    K. Y. Tsang, R. E. Mirollo, S. H. Strogatz, and K. Wiesenfeld, Dynamics of a globally coupled os- cillator array, Physica D: Nonlinear Phenomena 48, 102 (1991)

    1991

  54. [62]

    K. M. Stiefel and G. B. Ermentrout, Neurons as oscillators, Journal of neurophysiology 116, 2950 (2016)

    2016

  55. [63]

    Gerstner, W

    W. Gerstner, W. M. Kistler, R. Naud, and L. Paninski, Neuronal dynamics: From single neu- rons to networks and models of cognition (Cam- bridge University Press, 2014)

    2014

  56. [64]

    Brown, P

    E. Brown, P. Holmes, and J. Moehlis, Glob- ally coupled oscillator networks, Perspectives and Problems in Nolinear Science: A Celebratory Vol- ume in Honor of Lawrence Sirovich , 183 (2003)

    2003

  57. [65]

    K. Zhao, L. Cheng, S. Li, and A. Liao, A new up- dating method for the damped mass-spring sys- tems, Applied Mathematical Modelling 62, 119 (2018)

    2018

  58. [66]

    C. B. Churchill, D. W. Shahan, S. P. Smith, A. C. Keefe, and G. P. McKnight, Dynamically variable negative stiffness structures, Science advances 2, e1500778 (2016)

    2016

  59. [67]

    Zhao, Is the negative equivalent stiffness of a system possible?, Physics Education 51, 013002 (2016)

    M. Zhao, Is the negative equivalent stiffness of a system possible?, Physics Education 51, 013002 (2016)

    2016

  60. [68]

    Wang and R

    Y. Wang and R. Lakes, Extreme stiffness systems due to negative stiffness elements, American Jour- nal of Physics 72, 40 (2004)

    2004

  61. [69]

    L. Tu, D. Ning, S. Sun, W. Li, H. Huang, M. Dong, and H. Du, A novel negative stiffness magnetic spring design for vehicle seat suspension system, Mechatronics 68, 102370 (2020)

    2020

  62. [70]

    A. O. Oyelade, Z. Wang, and G. Hu, Dynamics of 1d mass–spring system with a negative stiff- ness spring realized by magnets: Theoretical and experimental study, Theoretical and Applied Me- chanics Letters 7, 17 (2017)

    2017

  63. [71]

    Farreras and A

    M. Farreras and A. Cervera-Lierta, Simulation of the 1d xy model on a quantum computer (2024), arXiv:2410.21143 [quant-ph]. 15

    2024 arXiv

  64. [72]

    Farhi, J

    E. Farhi, J. Goldstone, and S. Gutmann, A quan- tum approximate optimization algorithm (2014), arXiv:1411.4028 [quant-ph]

    2014 arXiv

  65. [73]

    Blanc and O

    M. Blanc and O. Bournez, Simulation of tur- ing machines with analytic discrete odes: Fp- time and fpspace over the reals characterised with discrete ordinary differential equations (2024), arXiv:2307.11747 [cs.CC]

    2024 arXiv

  66. [74]

    Mentink-Vigier, Fast exponential matrix for matlab (full/sparse), fastexpm, https://github

    F. Mentink-Vigier, Fast exponential matrix for matlab (full/sparse), fastexpm, https://github. com/fmentink/fastExpm/releases/tag/1.0, GitHub (2024), [Retrieved March 22, 2024.]

    2024

  67. [75]

    R. B. Laughlin, A different universe: Reinventing physics from the bottom down (Basic Books (AZ), 2005)

    2005

  68. [76]

    D. K. Ferry, R. Akis, M. J. Gilbert, and I. Kneze- vic, Do we need quantum for quantum comput- ing?, in Noise and Information in Nanoelectron- ics, Sensors, and Standards , Vol. 5115 (SPIE,

  69. [77]

    L. B. Kish, Hilbert space computing by analog circuits, in Noise and Information in Nanoelec- tronics, Sensors, and Standards , Vol. 5115 (SPIE,

  70. [78]

    Bournez, D

    O. Bournez, D. S. Gra¸ ca, and A. Pouly, Turing machines can be efficiently simulated by the gen- eral purpose analog computer, in Theory and Ap- plications of Models of Computation: 10th Inter- national Conference, TAMC 2013, Hong Kong, China, May 20-22, 2013. Proceedings 10 ...

    2013

  71. [79]

    Chang, M

    B. Chang, M. Chen, E. Haber, and E. H. Chi, An- tisymmetricrnn: A dynamical system view on re- current neural networks (2019), arXiv:1902.09689 [stat.ML]

    2019 arXiv

  72. [80]

    Abernot, N

    M. Abernot, N. Azemard, and A. Todri-Sanial, Oscillatory neural network learning for pattern recognition: an on-chip learning perspective and implementation, Frontiers in Neuroscience 17, 10.3389/fnins.2023.1196796 (2023)

    2023

  73. [81]

    F. A. Firestone, A new analogy between mechan- ical and electrical systems, The Journal of the Acoustical Society of America 4, 249 (1933). Appendix A: Encoding probability amplitudes as velocity: First-order ODE and damping To convert the Schr¨ odinger equation into a system ...

    1933

  74. [82]

    If further Im n ˙H(t) o = 0 in Eq

    Symmetric Re {H(t)} Im{H(t)}. If further Im n ˙H(t) o = 0 in Eq. E1, the matrix K(t) is real symmetric: K(t) = (Re{H(t)}2 − Im{H(t)}2)T = Re{H(t)}T Re{H(t)}T − Im{H(t)}T Im{H(t)}T = Re{H(t)} Re{H(t)} −(− Im{H(t)})(− Im{H(t)}) = Re{H(t)}2 − Im{H(t)}2 . (E4)

  75. [83]

    • For H(t) = 1 1 − 2ti 1 + 2ti −1 , (E5) K(t) = 2 − 4t2 2 −2 2 − 4t2 , (E6) which is real but not symmetric

    Example • Any real symmetric H leads to a real symmetric K. • For H(t) = 1 1 − 2ti 1 + 2ti −1 , (E5) K(t) = 2 − 4t2 2 −2 2 − 4t2 , (E6) which is real but not symmetric. • For H =   1 −i 2i i −1 0 −2i 0 −1   , (E7) K =   6 0 0 0 2 −2 0 −2 5   , (E8) which is real and sy...

Pith tools

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