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A time-marching quantum algorithm for simulation of the nonlinear Lorenz dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A quantum time-marching algorithm simulates the chaotic Lorenz dynamics using a recursive evolution operator that needs only a linear number of copies of the initial state.

desk verdict Recursive embedding is real, but doubly exponential postselection makes the claimed quantum speed-up collapse. read the letter →

arxiv 2506.21354 v1 pith:II6J2BGC submitted 2025-06-26 quant-ph nlin.CD

Efstratios Koukoutsis , George Vahala , Min Soe , Kyriakos Hizanidis , Linda Vahala , Abhay K. Ram This is my paper · ORCID
classification quant-phnlin.CD MSC 81P6837D4565L06 PACS 03.67.Ac05.45.Ac
keywords time-marchingquantumalgorithmLorenzsystemHadamardproductblockencodinglinearcombinationofunitariesrecursivestructurepostselectionchaoticattractor
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 tries to establish that nonlinear, dissipative, chaotic dynamics, which normally sit outside the linear and unitary framework of quantum mechanics, can be simulated by a quantum time-marching algorithm. Its construction writes one step of a second-order Runge-Kutta discretization of the Lorenz equations as a nonlinear state preparation followed by a linear non-unitary update, then chains steps through the recursion $U_j = U_1 U_{j-1}^{\otimes 3}$. The claimed payoff is a circuit whose width scales linearly with the number of time steps, needing only $4N_t - 1$ copies of the initial state, with a query count of $O(3^{N_t})$, improving on earlier nonlinear-solver proposals. The paper also claims that classically running this quantum scheme reproduces both the chaotic Lorenz attractor and the period-doubling limit cycles, matching high-order adaptive solvers. A sympathetic reader would care because, if true, the construction offers a general template for embedding polynomial nonlinear ODEs into postselected quantum circuits with linear growth in the number of state copies as the number of time steps grows.

What carries the argument

The load-bearing object is the recursive evolution operator $U_j = U_1 U_{j-1}^{\otimes 3}$, together with the re-use of copy registers that makes the copy count linear. The nonlinear step $U_{nl}$ is implemented through the Hadamard product, the componentwise multiplication of amplitudes given by $|\psi\rangle \odot |\phi\rangle = \sum_i \psi_i \phi_i |i\rangle$, realized by a shift-operator select unitary followed by postselection; the polynomial terms of the Lorenz nonlinear state are sums of such products combined through the linear-combination-of-unitaries (LCU) method. The non-unitary matrix $\hat{A}_2$ is applied via an SVD-LCU block encoding, a unitary dilation whose top-left block is $\hat{A}_2$ divided by its spectral norm, with success probability $\|\hat{A}_2|\psi\rangle\|^2 / a^2$. The recursion exists because the no-cloning theorem forbids re-preparing the unknown evolved state, so the circuit runs $2j-1$ branches in parallel and replenishes initial-state copies with a controlled preparation operator $C\hat{U}_{\psi_n}^{\otimes 2}$.

What would settle it

Run the actual postselected circuit on a quantum simulator for, say, $N_t = 8$ time steps with $\delta t = 10^{-3}$ and count how often every ancillary qubit reads 0; the expected count is essentially zero because $(1/\sigma_{\max}^2)^{3^8}$ is astronomically small, so the algorithm would be unusable unless the claimed amplitude amplification is explicitly implemented and its overhead counted.

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Extended reading notes

Core claim

The paper claims that the $j$-step evolution operator for a second-order Runge-Kutta discretization of the Lorenz system can be defined recursively as $U_j = U_1 U_{j-1}^{\otimes 3}$, building each $j$-step advance from three parallel copies of the $(j-1)$-step advance. Because the single-step operator $U_1 = U_{A_2} U_{nl}$ is made from a Hadamard-product nonlinear state preparation followed by a block-encoded non-unitary matrix, the full $N_t$-step simulation is a postselected circuit requiring only $N_{\text{copies}} = 4N_t - 1$ copies of the initial state, with $N_{\text{queries}} = O(3^{N_t})$ single-step calls. The paper further claims that this copy count is an exponential improvement over the earlier time-marching scheme and a quadratic improvement over repeated-measurement and related approaches, while preserving a quantum speed-up in the dimension of the underlying ODE system. A classical implementation of the same recursive scheme is shown to reproduce the chaotic Lorenz attractor and the P1, P2, P4, and P6 limit cycles of the period-doubling route, in qualitative agreement with a high-order adaptive solver.

Load-bearing premise

The whole scheme depends on measuring every auxiliary qubit in the zero state at the end, and the chance of that happening falls like $(1/\sigma_{\max}^2)^{3^{N_t}}$; the paper assumes amplitude amplification can turn this negligible probability into a usable one without showing how.

Editorial extensions

If this is right

  • For the Lorenz discretization, simulating $N_t$ steps requires $4N_t - 1$ copies of the initial state and $O(3^{N_t})$ single-step queries, an exponential improvement in copy count over the earlier time-marching scheme and a quadratic improvement over repeated-measurement approaches.
  • For a general $d_s$-dimensional polynomial ODE system of degree $p$ under a $K$-th order discretization, the claimed complexity is $O(d^2 \log^2 d \cdot p \cdot N_t, (T^K/\epsilon)^{1/(K-1)})$ with $N_{\text{copies}} = O(N_t)$; when the original dimension is much smaller than the augmented dimension $d$, this is an exponential speed-up over classical ODE solvers.
  • A successful all-zero measurement at the output yields exactly the second-order RK2 update, so a successful run tracks the discretized Lorenz trajectory faithfully rather than an approximation of it.
  • The classical execution of the recursive scheme reproduces the chaotic attractor at $\beta = 0.58$ and the P1, P2, P4, and P6 limit cycles of the period-doubling route, indicating that the discretization is faithful and that the recursion itself does not corrupt the dynamics.
  • The recursive structure is offered as a promising primitive for other quantum algorithms, and quantum compression gadgets are anticipated to further improve the resource scalings.

Reading between the lines

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

  • The numerical demonstrations run the classical RK2 scheme, not the postselected quantum circuit, so they establish the discretization's fidelity but do not probe the exponentially small success probability a real quantum execution would face.
  • The $O(3^{N_t})$ gate complexity means that, for the fixed three-dimensional Lorenz system, the quantum circuit depth grows exponentially with integration time; any practical advantage over classical integration would require a subexponential or amplified construction that this paper does not provide.
  • The Hadamard-product-plus-LCU nonlinear state preparation is a modular component that could be reused in other quantum integrators; if a later algorithm solves the postselection bottleneck, this part can be imported directly.
  • The claimed $O(\operatorname{polylog}(d_s))$ dimension scaling, if realized, would matter most for high-dimensional PDE discretizations, but only after the success-probability and amplitude-amplification questions are settled.
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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

3 major / 5 minor

Summary. This paper proposes a quantum time-marching algorithm for a second-order Runge-Kutta discretization of the Lorenz system. The single time step is written as a linear operator A_2 acting on an augmented nonlinear state, with the nonlinear state prepared via Hadamard products and LCU and the non-unitary evolution implemented by SVD-LCU block encoding. The multi-step evolution is organized recursively as U_j = U_1 U_{j-1}^{⊗3}, and the paper claims a linear number of copies of the initial state, O(3^{N_t}) total queries, and an exponential speed-up in dimensionality. Numerical sections compare a classical evaluation of the RK2 scheme with Mathematica solutions for chaotic and periodic attractors.

Significance. The recursive construction is a genuine contribution: the explicit circuit blocks for the Hadamard-product nonlinear preparation and the SVD-LCU block encoding are self-contained, and the copy-count result N_copies = 4N_t - 1 is a concrete structural improvement over earlier time-marching proposals. The classical validation against Mathematica demonstrates that the RK2 discretization itself reproduces the expected Lorenz attractors. However, the paper does not demonstrate that the postselected quantum circuit can be turned into a usable algorithm: the postselection success probability is exponentially small in 3^{N_t}, no amplitude-amplification protocol is supplied, and the numerical demonstrations exercise only the classical finite-difference scheme, not the quantum circuit. The central 'quantum algorithm' claim therefore needs substantial additional support or a major reframing.

major comments (3)
  1. [Section 3.4, Eq. (67)] The viability of the algorithm rests on Eq. (67), but that equation is not a complete success-probability expression. It counts only the (1/σ_max^2)^{3^{N_t}} factor from the repeated block-encoded A_2 gates and omits the state-dependent postselection amplitudes of the U_nl preparation in Eqs. (45)-(48), which multiply over the entire recursive tree. Even ignoring that omission, the stated bound is already fatal: for σ_max < 1.01 and N_t = 10, (1/σ_max^2)^{3^{10}} is below 10^{-500}, and a quadratic speedup from amplitude amplification would still require an exponential number of repetitions. The manuscript only states that amplitude amplification is 'required' at the end of Section 3.4 and provides no reflection operator, no oracle for the marked subspace, and no end-to-end success-probability estimate. Please supply a complete amplitude-amplification construction and a rigorous total runtime analysis, or explicitly reframe the result as a postselected circuit and remove the claim that it constitutes an efficient quantum simulation.
  2. [Section 4, Figures 11-17] The numerical demonstrations in Section 4 are classical evaluations of the predictor-corrector scheme in Eqs. (10)-(11), not executions of the quantum circuits in Figures 7 and 8. They therefore cannot validate the postselection step, the Eq. (67) estimate, or any amplitude-amplification strategy. The agreement with the Mathematica solver is useful evidence that the RK2 discretization is accurate, but it is not evidence about the behavior of the proposed quantum algorithm. Please either provide quantum-circuit-level simulations, even for small N_t, or clearly label Section 4 as validating only the classical finite-difference scheme.
  3. [Equations (64)-(66) and following paragraph] The total query count O(3^{N_t}) in Eq. (64), together with the postselection overhead in Eq. (67), means the expected runtime of the algorithm grows exponentially in the number of time steps. For the fixed three-dimensional Lorenz system, a classical RK2 solver runs in O(N_t) time, so the claimed 'exponential speed-up' is at most a statement about a hypothetical high-dimensional system and is not a speed-up in integration time for Lorenz itself. The paper should state this distinction explicitly and avoid implying that the Lorenz simulation is efficient in the number of time steps.
minor comments (5)
  1. [Section 3.1.1] The text says 'we adopt the following convection in terms of notation'; 'convection' should be 'convention'.
  2. [Section 3.4 and Conclusions] The word 'discetization' should be 'discretization' in both places where it appears.
  3. [Equation (65)] The notation O(60d2pNt, T2/ϵ) is ambiguous; please write the full dependence on d, p, N_t, T, and ε with explicit exponents.
  4. [Figure 2 caption] The caption contains garbled text ('G #denotes' and 'LEFſCIRCLE /Cþrcle'); these rendering issues should be fixed before publication.
  5. [Equation (20)] The encoding in Eq. (20) is written for a 3-qubit state while the second-order scheme later uses 4 qubits; please add a sentence explaining the analogous 4-qubit amplitude encoding, since the notation is otherwise easy to misread.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central recursive construction is self-contained and the numerical targets are benchmarked against an external ODE solver.

full rationale

The paper's main derivation chain is not circular. The recursive operator U_j = U_1 U_{j-1}^{otimes 3} in Eq. (50) is a direct construction from the single-step evolution operator, and the copy-count result in Eq. (59) follows from the explicit re-usage scheme in Eqs. (52)-(58). The nonlinear-state preparation via Hadamard products and the SVD-LCU block encoding are taken from independent prior work (Refs. 27, 33, 34, 37), not from the authors' own results. The numerical demonstrations in Section 4 compare the second-order difference scheme against a higher-order adaptive Mathematica ODE solver, which is an external benchmark rather than a fitted input. The self-citations present (Refs. 39, 40, 51) are peripheral and are not load-bearing for the central claim. The main caveat is a missing demonstration rather than circularity: Section 3.4, around Eq. (67), states that amplitude amplification is required for a successful measurement to occur with non-vanishing probability, but no reflection oracle, marking oracle, or success-amplitude estimate is provided. This is an incomplete support for the algorithm's practicality, and the classical runs do not exercise the postselected circuit. However, none of this makes the derivation equivalent to its inputs by construction, so the circularity burden remains low.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The algorithm introduces no new physical entities, forces, or dimensions. It relies on standard quantum information machinery, a hand-chosen time step, and two unproven practical assumptions: that deferred postselection can be handled by amplitude amplification, and that the fixed-step RK2 scheme is an accurate proxy for the continuous Lorenz dynamics. The counted free parameters and axioms cover everything the central claim depends on beyond the cited building blocks.

free parameters (1)
  • Discretization time step δt = 10^-3, 5×10^-4, 2.5×10^-4
    Chosen by hand for each attractor class in Section 4: 10^-3 for P1/P2/P4 limit cycles, 5×10^-4 for the P6 cycle, and 2.5×10^-4 for the chaotic case. No convergence criterion or error bound justifies these choices, yet all numerical demonstrations depend on them.
assumptions (4)
  • standard math The no-cloning theorem prevents copying unknown quantum states, so multiple copies of the evolving state must be produced from known prepared copies of the initial state.
    Invoked in Section 3.3 to justify the recursive parallel evolution of initial-state copies and the absence of an operator U_c that clones the unknown intermediate state.
  • standard math A non-unitary matrix A can be block-encoded when a ≥ ∥A∥ using the SVD-LCU decomposition of Eqs. (41)-(44).
    Used in Section 3.2 as the core mechanism for implementing the non-unitary time-advancement matrix A_2.
  • domain assumption Postselection on zero ancilla outcomes can be deferred and ultimately corrected by amplitude amplification, with an available reflection oracle and known or estimable success amplitude.
    Section 3.4 states that amplitude amplification is required at the output, but does not construct the amplification operator or analyze its cost with the recursive structure. This is load-bearing for the algorithmic claim.
  • domain assumption The second-order explicit predictor-corrector scheme (Eqs. (10)-(11)) with the hand-chosen δt faithfully reproduces the continuous Lorenz dynamics in the displayed parameter windows.
    All numerical comparisons in Section 4 rest on this assumption, and only qualitative visual agreement is provided rather than quantitative convergence or error metrics.

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Pith. "Pith review of A time-marching quantum algorithm for simulation of the nonlinear Lorenz dynamics." pith.science (2026). https://pith.science/paper/II6J2BGC

@misc{pith2026250621354,
  author       = {Pith},
  title        = {Pith review of: A time-marching quantum algorithm for simulation of the nonlinear Lorenz dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/II6J2BGC}},
  note         = {Machine review of arXiv:2506.21354}
}
read the original abstract

Simulating nonlinear classical dynamics on a quantum computer is an inherently challenging task due to the linear operator formulation of quantum mechanics. In this work, we provide a systematic approach to alleviate this difficulty by developing a quantum algorithm that implements the time evolution of a second order time-discretized version of the Lorenz model. The Lorenz model is a celebrated system of nonlinear ordinary differential equations that has been extensively studied in the contexts of climate science, fluid dynamics, and chaos theory. Our algorithm possesses a recursive structure and requires only a linear number of copies of the initial state with respect to the number of integration time-steps. This provides a significant improvement over previous approaches, while preserving the characteristic quantum speed-up in terms of the dimensionality of the underlying differential equations system, that similar time-marching quantum algorithms have previously demonstrated. Notably, by classically implementing the proposed algorithm, we showcase that it accurately captures the structural characteristics of the Lorenz system, reproducing both regular attractors--limit cycles--and the chaotic attractor within the chosen parameter regime.

Figures

Figures reproduced from arXiv: 2506.21354 by the authors.

Figure 1
Figure 1. Quantum circuit implementation of Hadamard product |ψ⟩ ⊙ |ϕ⟩ between single-qubit states. A projection measurement operator Pˆ = |0⟩⟨0| ⊗ 1ˆ 2×2 is applied to the first register for a successful implementation. As depicted in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Quantum circuits for the Hadamard product between two n-qubit states. (a) Schematic implementation of the Uˆ h select operator. The symbol G# denotes a uniform quantum multiplex gate [28,29]. (b) Quantum circuit im￾plementation of the Sˆ− operation acting on the the |k⟩ basis expressed in its binary form, |k⟩ = |kn−1kn−2 . . . k1k0⟩b . The Xˆ gate is the Pauli-x matrix. Decomposition of the Sˆ− operator ( [PITH_FUL… view at source ↗
Figure 3
Figure 3. Quantum circuit implementation of the Uˆ nl operator of Equation (30) mediating the nonlinear polynomial transformation in Equation (21). (a) Schematic implementation of Uˆ nl. (b) Explicit implementation of the uniform multiplexed operation Uˆ LCU select in Equation (32). The decomposition for each of the nj-qubit Uˆ j h operations can be derived from the respective one in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Explicit quantum circuit preparation of the nonlinear state |ψnl⟩ in Equation (33). Implementation of Uˆ 2 h operator has been presented at [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Quantum circuit implementation of the non-unitary operator Aˆ through the general block encoding Uˆ A in Equations (39), (40). Out of the various available dilation techniques [36–40] for Uˆ A, we employ an m = 1 block encoding constructed via classical Singular Value …
Figure 6
Figure 6. Figure 6: |0⟩ Hˆ Rˆ z Hˆ |ψ⟩ Vˆ † Wˆ A | ˆ ψ⟩ SVD-LCU UˆA [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: illustrates the quantum circuit implementation corresponding to the aforementioned tasks for the second-order, single-step, time advancement of the Lorenz dynamics. G# |0⟩ e Hˆ Rˆ z Hˆ [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Schematic quantum circuit implementation of the time-marching quantum algorithm for the steps described in 1-7 in [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: The dependence of the largest singular value of Aˆ 2 matrix in Equation (14), σmax, on the discretization time step δt for the typical time steps used in Section 4. 4. Numerical Demonstrations While the proposed quantum solver exhibits a quantum speed-up in terms of th…
Figure 10
Figure 10. Figure 10: The bifurcation diagram for z(β) for 0.54 < β < 0.58, with σ = 10, ρ = 28. The bifurcation diagrams for x(β) and y(β) are qualitatively similar and with exactly the same bifurcation points in β. In the following, we shall compare the Lorenz regular (limit cycles) and …
Figure 11
Figure 11. Figure 11: The chaotic attractor for β = 0.58, σ = 10 and ρ = 28, showing both (a) its y-z projection and (b) the 3D plot. Initial conditions: x(0) = 0.1, y(0) = −1.1, z(0) = 10.1. These plots were generated by a higher order adaptive time Mathematica ODE solver. The lobes of th…
Figure 12
Figure 12. Figure 12: The chaotic attractor for β = 0.58, and with the same initial conditions as in [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: P1-limit cycle with β = 0.52 and time step δt = 10−3 : (a) y − z projection, (b) the 3D attractor. 4.2.2. Period doubling limit cycles On increasing β, to β = 0.55 we encounter a period-2 (P2) limit cycle, depicted in [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: P2-limit cycle, with β = 0.55 and time step δt = 10−3 : (a) y − z projection, (b) the 3D attractor. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: P2 2 -limit cycle, with β = 0.56 and time step δt = 10−3 : (a) y-z projection, (b) the 3D attractor. The period-doubling limit cycles found in Figures 13- 15 with the second-order method can be considered rudimentary in terms of their period-doubling behavior as the p…
Figure 16
Figure 16. Figure 16: P6-limit cycle for β = 0.5648 and a reduced time step δt = 5 × 10−4 : (a) y-z projection, (b) the 3D attractor. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: P6-limit cycle generated by a higher order adaptive time Mathematica ODE solver. (a) y-z projection, (b) the 3D attractor. Compared to the second-order scheme result in [PITH_FULL_IMAGE:figures/full_fig_p021_17.png]

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Works this paper leans on

51 extracted references · 25 canonical work pages

  1. [1]

    Efficient quantum algorithm for dissipative nonlinear differential equations

    Liu, J.P .; Kolden, H.O.; Krovi, H.K.; Loureiro, N.F.; Trivisa, K.; Childs, A.M. Efficient quantum algorithm for dissipative nonlinear differential equations. Proc. Natl. Acad. Sci. 2021, 118, e2026805118. https: //doi.org/10.1073/pnas.2026805118

    doi:10.1073/pnas.2026805118 2021

  2. [2]

    Improved quantum algorithms for linear and nonlinear differential equations

    Krovi, H. Improved quantum algorithms for linear and nonlinear differential equations. Quantum 2023, 7, 913. https://doi.org/10.22331/q-2023-02-02-913

    doi:10.22331/q-2023-02-02-913 2023

  3. [3]

    Three Carleman routes to the quantum simulation of classical fluids

    Sanavio, C.; Scatamacchia, R.; de Falco, C.; Succi, S. Three Carleman routes to the quantum simulation of classical fluids. Phys. Fluids 2024, 36, 057143. https://doi.org/10.1063/5.0204955

    doi:10.1063/5.0204955 2024

  4. [4]

    Quantum Algorithms for Nonlinear Dynamics: Revisiting Carleman Linearization with No Dissipative Conditions

    Wu, H.C.; Wang, J.; Li, X. Quantum Algorithms for Nonlinear Dynamics: Revisiting Carleman Linearization with No Dissipative Conditions. SIAM J. Sci. Comput. 2025, 47, A943–A970. https://doi.org/10.1137/24M1 665799

    work page doi:10.1137/24m1 2025

  5. [5]

    Koopman–von Neumann approach to quantum simulation of nonlinear classical dynamics

    Joseph, I. Koopman–von Neumann approach to quantum simulation of nonlinear classical dynamics. Phys. Rev. Res. 2020, 2, 043102. https://doi.org/10.1103/PhysRevResearch.2.043102

    work page doi:10.1103/physrevresearch.2.043102 2020

  6. [6]

    Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems

    Novikau, I.; Joseph, I. Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems. Comput. Phys. Commun. 2025, 309, 109498. https: //doi.org/https://doi.org/10.1016/j.cpc.2025.109498

    work page arXiv 2025

  7. [7]

    Simulating non-native cubic interactions on noisy quantum machines

    Shi, Y.; Castelli, A.R.; Wu, X.; Joseph, I.; Geyko, V .; Graziani, F.R.; Libby, S.B.; Parker, J.B.; Rosen, Y.J.; Martinez, L.A.; et al. Simulating non-native cubic interactions on noisy quantum machines. Phys. Rev. A 2021, 103, 062608. https://doi.org/10.1103/PhysRevA.103.062608

    work page doi:10.1103/physreva.103.062608 2021

  8. [8]

    Nonlinear solution of classical three-wave interaction via finite-dimensional quantum model

    May, M.Q.; Qin, H. Nonlinear solution of classical three-wave interaction via finite-dimensional quantum model. J. Plasma Phys. 2024, 90, 805900302. https://doi.org/10.1017/S002237782400059X

    work page doi:10.1017/s002237782400059x 2024

Show all 51 references
  1. [9]

    Simulating nonlinear optical processes on a superconducting quantum device

    Shi, Y.; Evert, B.; Brown, A.F.; Tripathi, V .; Sete, E.A.; Geyko, V .; Cho, Y.; DuBois, J.L.; Lidar, D.; Joseph, I.; et al. Simulating nonlinear optical processes on a superconducting quantum device. J. Plasma Phys. 2024, 90, 805900602. https://doi.org/10.1017/S0022377824001326

    2024 doi

  2. [10]

    Quantum simulation of nonlinear dynamical systems using repeated measurement

    Andress, J.; Engel, A.; Shi, Y.; Parker, S. Quantum simulation of nonlinear dynamical systems using repeated measurement. J. Plasma Phys. 2025, 91, E55. https://doi.org/10.1017/S0022377825000091

    2025 doi

  3. [11]

    Variational quantum algorithms for nonlinear problems

    Lubasch, M.; Joo, J.; Moinier, P .; Kiffner, M.; Jaksch, D. Variational quantum algorithms for nonlinear problems. Phys. Rev. A 2020, 101, 010301. https://doi.org/10.1103/PhysRevA.101.010301

    2020 doi

  4. [12]

    Nonlinear dynamics as a ground-state solution on quantum computers

    Pool, A.J.; Somoza, A.D.; Mc Keever, C.; Lubasch, M.; Horstmann, B. Nonlinear dynamics as a ground-state solution on quantum computers. Phys. Rev. Res. 2024, 6, 033257. https://doi.org/10.1103/PhysRevResearch. 6.033257

    2024 doi

  5. [13]

    A Hybrid Quantum Solver for the Lorenz System

    Fathi Hafshejani, S.; Gaur, D.; Dasgupta, A.; Benkoczi, R.; Gosala, N.R.; Iorio, A. A Hybrid Quantum Solver for the Lorenz System. Entropy 2024, 26. https://doi.org/10.3390/e26121009

    2024 doi

  6. [14]

    A quantum algorithm to solve nonlinear differential equations, 2008, [arXiv:quant- ph/0812.4423]

    Leyton, S.K.; Osborne, T.J. A quantum algorithm to solve nonlinear differential equations, 2008, [arXiv:quant- ph/0812.4423]

    2008 arXiv

  7. [15]

    Quantum algorithm for nonlinear differential equations, 2020, [arXiv:quant-ph/2011.06571]

    Lloyd, S.; Palma, G.D.; Gokler, C.; Kiani, B.; Liu, Z.W.; Marvian, M.; Tennie, F.; Palmer, T. Quantum algorithm for nonlinear differential equations, 2020, [arXiv:quant-ph/2011.06571]

    2020 arXiv

  8. [16]

    Finding flows of a Navier-Stokes fluid through quantum computing

    Gaitan, F. Finding flows of a Navier-Stokes fluid through quantum computing. npj Quantum Inf. 2020, 06. https://doi.org/10.1038/s41534-020-00291-0

    2020 doi

  9. [17]

    Quantum Computers for Weather and Climate Prediction: The Good, the Bad, and the Noisy

    Tennie, F.; Palmer, T.N. Quantum Computers for Weather and Climate Prediction: The Good, the Bad, and the Noisy. Bull. Am. Meteorol. Soc. 2023, 104, E488 – E500. https://doi.org/10.1175/BAMS-D-22-0031.1

    2023 doi

  10. [18]

    Quantum algorithm for nonlinear Burgers’ equation for high-speed compressible flows

    Esmaeilifar, E.; Ahn, D.; Myong, R.S. Quantum algorithm for nonlinear Burgers’ equation for high-speed compressible flows. Phys. Fluids 2024, 36, 106110. https://doi.org/10.1063/5.0231994

    2024 doi

  11. [19]

    The Lorenz model and the method of Carleman embedding

    Andrade, R.; Rauh, A. The Lorenz model and the method of Carleman embedding. Physics Letters A 1981, 82, 276–278. https://doi.org/https://doi.org/10.1016/0375-9601(81)90644-7. 24 of 25

    1981 doi

  12. [20]

    Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems

    Lewis, D.; Eidenbenz, S.; Nadiga, B.; Suba¸ sı, Y. Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems. Quantum 2024, 8, 1509. https://doi.org/10.22331/q-2024-10-24-1509

    2024 doi

  13. [21]

    Challenges for quantum computation of nonlinear dynamical systems using linear representations, 2024, [arXiv:quant-ph/2202.02188]

    Lin, Y.T.; Lowrie, R.B.; Aslangil, D.; Suba¸ sı, Y.; Sornborger, A.T. Challenges for quantum computation of nonlinear dynamical systems using linear representations, 2024, [arXiv:quant-ph/2202.02188]

    2024 arXiv

  14. [22]

    Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

    Engel, A.; Smith, G.; Parker, S.E. Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms. Phys. Plasmas 2021, 28, 062305. https://doi.org/10.1063/5.0040313

    2021 doi

  15. [23]

    The Painleve property for partial differential equations

    Weiss, J.; Tabor, M.; Carnevale, G. The Painleve property for partial differential equations. J. Math. Phys. 1983, 24, 522–526. https://doi.org/10.1063/1.525721

    1983 doi

  16. [24]

    Quantum dynamics in Krylov space: Methods and applications

    Nandy, P .; Matsoukas-Roubeas, A.S.; Martinez-Azcona, P .; Dymarsky, A.; del Campo, A. Quantum dynamics in Krylov space: Methods and applications. Phys. Reports 2025, 1125-1128, 1–82. https://doi.org/https: //doi.org/10.1016/j.physrep.2025.05.001

    2025 doi

  17. [25]

    Deterministic Nonperiodic Flow

    Lorenz, E.N. Deterministic Nonperiodic Flow. J. Atmos. Sci. 1963, 20, 130 – 141. https://doi.org/10.1175/15 20-0469(1963)020<0130:DNF>2.0.CO;2

    1963 doi

  18. [26]

    An Introduction to Numerical Analysis ; Cambridge University Press, 2003

    Süli, E.; Mayers, D.F. An Introduction to Numerical Analysis ; Cambridge University Press, 2003

    2003

  19. [27]

    Non-linear quantum state transformation of spin-12

    Bechmann-Pasquinucci, H.; Huttner, B.; Gisin, N. Non-linear quantum state transformation of spin-12. Phys. Lett. A 1998, 242, 198–204. https://doi.org/https://doi.org/10.1016/S0375-9601(98)00189-3

    1998 doi

  20. [28]

    Quantum Circuits for General Multiqubit Gates

    Möttönen, M.; Vartiainen, J.J.; Bergholm, V .; Salomaa, M.M. Quantum Circuits for General Multiqubit Gates. Phys. Rev. Lett. 2004, 93, 130502. https://doi.org/10.1103/PhysRevLett.93.130502

    2004 doi

  21. [29]

    Quantum circuits with uniformly controlled one-qubit gates

    Bergholm, V .; Vartiainen, J.J.; Möttönen, M.; Salomaa, M.M. Quantum circuits with uniformly controlled one-qubit gates. Phys. Rev. A 2005, 71, 052330. https://doi.org/10.1103/PhysRevA.71.052330

    2005 doi

  22. [30]

    Elementary gates for quantum computation

    Barenco, A.; Bennett, C.H.; Cleve, R.; DiVincenzo, D.P .; Margolus, N.; Shor, P .; Sleator, T.; Smolin, J.A.; Weinfurter, H. Elementary gates for quantum computation. Phys. Rev. A 1995, 52, 3457–3467. https: //doi.org/10.1103/PhysRevA.52.3457

    1995 doi

  23. [31]

    Quantum-state preparation with universal gate decompositions

    Plesch, M.; Brukner, i.c.v. Quantum-state preparation with universal gate decompositions. Phys. Rev. A 2011, 83, 032302. https://doi.org/10.1103/PhysRevA.83.032302

    2011 doi

  24. [32]

    General Quantum Interference Principle and Duality Computer

    Long, G.L. General Quantum Interference Principle and Duality Computer. Commun. Theor. Phys. 2006, 45, 825. https://doi.org/10.1088/0253-6102/45/5/013

    2006 doi

  25. [33]

    Nonlinear transformations in quantum computation

    Holmes, Z.; Coble, N.J.; Sornborger, A.T.; Suba¸ s ı, Y.b.u. Nonlinear transformations in quantum computation. Phys. Rev. Res. 2023, 5, 013105. https://doi.org/10.1103/PhysRevResearch.5.013105

    2023 doi

  26. [34]

    Hamiltonian simulation using linear combinations of unitary operations

    Childs, A.M.; Wiebe, N. Hamiltonian simulation using linear combinations of unitary operations. Quantum Info. Comput. 2012, 12, 901–924. https://doi.org/10.26421/QIC12.11-12-1

    2012 doi

  27. [35]

    Quantum Computation and Quantum Information: 10th Anniversary Edition ; Cambridge University Press, 2010

    Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information: 10th Anniversary Edition ; Cambridge University Press, 2010

    2010

  28. [36]

    Quantum Simulation of Open Quantum Systems Using a Unitary Decomposition of Operators

    Schlimgen, A.W.; Head-Marsden, K.; Sager, L.M.; Narang, P .; Mazziotti, D.A. Quantum Simulation of Open Quantum Systems Using a Unitary Decomposition of Operators. Phys. Rev. Lett. 2021, 127, 270503. https://doi.org/10.1103/PhysRevLett.127.270503

    2021 doi

  29. [37]

    Quantum state preparation and nonunitary evolution with diagonal operators

    Schlimgen, A.W.; Head-Marsden, K.; Sager-Smith, L.M.; Narang, P .; Mazziotti, D.A. Quantum state preparation and nonunitary evolution with diagonal operators. Phys. Rev. A 2022, 106, 022414. https: //doi.org/10.1103/PhysRevA.106.022414

    2022 doi

  30. [38]

    Quantum simulation of partial differential equations: Applications and detailed analysis

    Jin, S.; Liu, N.; Yu, Y. Quantum simulation of partial differential equations: Applications and detailed analysis. Phys. Rev. A 2023, 108, 032603. https://doi.org/10.1103/PhysRevA.108.032603

    2023 doi

  31. [39]

    Quantum simulation of dissipation for Maxwell equations in dispersive media

    Koukoutsis, E.; Hizanidis, K.; Ram, A.K.; Vahala, G. Quantum simulation of dissipation for Maxwell equations in dispersive media. Future Gener. Comput. Syst. 2024, 159, 221–229. https://doi.org/https: //doi.org/10.1016/j.future.2024.05.028

    2024 doi

  32. [40]

    Quantum Implementation of Non-unitary Operations with Biorthogonal Representations

    Koukoutsis, E.; Papagiannis, P .; Hizanidis, K.; Ram, A.K.; Vahala, G.; Amaro, Ó.; Gamiz, L.I.I.; Vallis, D. Quantum Implementation of Non-unitary Operations with Biorthogonal Representations. Quantum Inf. Comput. 2025, 25, 141–155. https://doi.org/10.2478/qic-2025-0007

    2025 doi

  33. [41]

    Hamiltonian Simulation in the Interaction Picture, 2019, [arXiv:quant-ph/1805.00675]

    Low, G.H.; Wiebe, N. Hamiltonian Simulation in the Interaction Picture, 2019, [arXiv:quant-ph/1805.00675]

    2019 arXiv

  34. [42]

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

    Fang, D.; Lin, L.; Tong, Y. Time-marching based quantum solvers for time-dependent linear differential equations. Quantum 2023, 7, 955. https://doi.org/10.22331/q-2023-03-20-955

    2023 doi

  35. [43]

    Quantum amplitude amplification and estimation

    Brassard, G.; Høyer, P .; Mosca, M.; Tapp, A. Quantum amplitude amplification and estimation. InQuantum Computation and Information ; American Mathematical Society, 2002; p. 53–74. https://doi.org/10.1090/ conm/305/05215

    2002

  36. [44]

    A review of chaotification techniques for discrete-time systems

    Moysis, L.; Lawnik, M.; Baptista, M.S.; Volos, C.; Fragulis, G.F. A review of chaotification techniques for discrete-time systems. Nonlinear Dyn. 2025, 113, 14103–14119. https://doi.org/10.1007/s11071-025-10934-6. 25 of 25

    2025 doi

  37. [45]

    Exploiting recursive structures for the design of novel quantum primitives, 2024, [arXiv:quant-ph/2410.13927]

    Bao, N.; Suer, G. Exploiting recursive structures for the design of novel quantum primitives, 2024, [arXiv:quant-ph/2410.13927]

    2024 arXiv

  38. [46]

    Quantum lattice gas representation of some classical solitons

    Vahala, G.; Yepez, J.; Vahala, L. Quantum lattice gas representation of some classical solitons. Phys. Lett. A 2003, 310, 187–196. https://doi.org/https://doi.org/10.1016/S0375-9601(03)00334-7

    2003 doi

  39. [47]

    Quantum lattice representations for vector solitons in external potentials

    Vahala, G.; Vahala, L.; Yepez, J. Quantum lattice representations for vector solitons in external potentials. Phys. A: Stat. Mech. Appl. 2006, 362, 215–221. https://doi.org/https://doi.org/10.1016/j.physa.2005.09.029

    2006 doi

  40. [48]

    Twisting of filamentary vortex solitons demarcated by fast Poincaré recursion

    Yepez, J.; Vahala, G.; Vahala, L. Twisting of filamentary vortex solitons demarcated by fast Poincaré recursion. In Proceedings of the Quantum Information and Computation VII; Donkor, E.J.; Pirich, A.R.; Brandt, H.E., Eds. International Society for Optics and Photonics, SPIE, ...

    2009 doi

  41. [49]

    Unitary-quantum-lattice algorithm for two-dimensional quantum turbulence

    Zhang, B.; Vahala, G.; Vahala, L.; Soe, M. Unitary-quantum-lattice algorithm for two-dimensional quantum turbulence. Phys. Rev. E 2011, 84, 046701. https://doi.org/10.1103/PhysRevE.84.046701

    2011 doi

  42. [50]

    Qubit unitary lattice algorithm for spin-2 Bose-Einstein condensates: II - vortex reconnection simulations and non-Abelain vortices

    Vahala, G.; Soe, M.; Vahala, L. Qubit unitary lattice algorithm for spin-2 Bose-Einstein condensates: II - vortex reconnection simulations and non-Abelain vortices. Rad. Eff. Def. Solids 2020, 175, 113–119. https://doi.org/10.1080/10420150.2020.1718136

    2020

  43. [51]

    Koukoutsis, E.; Hizanidis, K.; Vahala, G.; Tsironis, C.; Ram, A.K.; Soe, M.; Vahala, L. A Quantum Walk Inspired Qubit Lattice Algorithm for Simulating Electromagnetic Wave Propagation and Scattering in Conservative and Dissipative Magnetized Plasmas, 2025, [arXiv:quant-ph/2503...

    2025 arXiv

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