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Cited by 185Quantum Computing Algorithms and ArchitectureNumerical methods for differential equationsQuantum many-body systems

Solving nonlinear differential equations with differentiable quantum circuits

Open access
Kyriienko, Oleksandr, Paine, Annie E., Elfving, Vincent E.·2021·Physical Review A
DOI 10.1103/physreva.103.052416
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Abstract

We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper introduces differentiable quantum circuits (DQCs) that encode trial functions via quantum feature maps and use automatic differentiation to represent derivatives analytically, then train hybrid quantum-classically to satisfy nonlinear differential equations and boundary conditions. It demonstrates the approach on single ODEs, coupled ODEs, and a quasi-1D Navier-Stokes convergent-divergent nozzle problem.

Key findings

  • DQCs can solve nonlinear ODEs in simulation: for \(du/dx + \lambda u(\kappa + \tan(\lambda x))=0\), \(u(0)=u_0\), with \(\lambda=8,20\), \(\kappa=0.1\), \(u_0=1\), a 6-qubit register, HEA depth \(d=5\), 20 grid points, and 250 iterations, the Chebyshev tower feature map (ChebT) converged best; the product feature map failed, and the simpler Chebyshev map also failed for \(\lambda=20\).
  • Increasing variational ansatz depth improved convergence for \(\lambda=20\) with ChebT: depths \(d=3,6,12,24\) were tested, and at \(d=24\) the solver had over 400 variational parameters without yet encountering barren plateaus.
  • For a highly oscillatory nonperiodic ODE, Eq. (32), with \(u_0=0.75\), 100 training points, evolution time \(\tau=2\), Adam learning rate 0.01, and 200 iterations, an evolution-enhanced Chebyshev tower feature map improved on the ChebT solution after a second training stage.
  • For strongly coupled equations with \(\lambda_1=5\), \(\lambda_2=3\), \(u_{1,0}=0.5\), \(u_{2,0}=0\), 6 qubits, HEA depth \(d=5\), Adam learning rate 0.02, and ChebT, floating boundary handling gave a solution close to the true one with reported \(L_Q\) approximately \(10^{-5}\).
  • For the quasi-1D Navier-Stokes nozzle problem, DQCs reproduced density, temperature, and velocity profiles across the full nozzle length using a two-stage optimization: first 20 points in \((0,0.4)\) for 200 iterations at learning rate 0.01, then 40 points in \((0,0.4)\) and \((0.6,0.9)\) for 600 iterations at learning rate 0.005, with regularization switched off around iteration 150.

Methods

  • Design: functions are represented as expectation values of parametrized quantum circuits, \(f(x)=\langle f_{\phi,\theta}(x)|\hat C|f_{\phi,\theta}(x)\rangle\), with quantum feature maps encoding \(x\), variational ansatze, Hermitian cost operators, and loss functions combining differential-equation residuals and boundary/regularization terms. Derivatives are obtained analytically via automatic differentiation and parameter-shift rules rather than finite differences.
  • Data/sample: numerical simulations use a noiseless full quantum state simulator, implemented with YAO.JL in Julia, typically with a 6-qubit register. Test problems include a single ODE, a highly nontrivial ODE, a system of two strongly coupled ODEs, and stationary quasi-1D Navier-Stokes equations for a convergent-divergent nozzle with \(A(x)=1+4.95(2x-1)^2\), \(\gamma=1.4\), and boundary values \(\rho(0)=1\), \(T(0)=1\), \(V(0)=0.1\).
  • Main analysis: training minimizes loss functions such as MSE or MAE using Adam stochastic gradient descent; feature maps compared include product, sparse Chebyshev, Chebyshev tower, and evolution-enhanced Chebyshev tower maps; boundary handling strategies include pinned, floating, and optimized boundaries.

Limitations

  • Stated: the paper did not consider the impact of noise on algorithm performance; feature maps and variational ansatz choices remain open and problem-specific; evolution-enhanced feature maps increase expressivity but make training harder; boundary handling methods have tradeoffs, e.g., floating boundary requires an exact initial value while pinned/optimized boundaries can introduce competing loss terms; multifunction encoding can cause competition between optimized functions; Chebyshev derivatives diverge at the boundaries of \((-1,1)\).
  • Evident from the text: all reported results are noiseless simulations on small qubit registers, not hardware demonstrations, and no quantum advantage is established; classical optimization and trainability issues such as barren plateaus are acknowledged as risks, though not encountered at the tested depths.

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Integrity signals

Registry

Crossref · 10.1103/physreva.103.052416

Cites

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Statistics

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Claims the paper makes

0 of 30 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    Each function at the RHS diverges at the point x such that T(x) = V(x)^2.

    “Immediately we observe that each function at the RHS diverges at the point x s.t.T (x) = V (x) 2 .”
    Quote in the paperSubstantial claim
  • Findingown data

    The more expressible ChebT feature map continues to perform well for λ = 20.

    “The more expressible ChebT feature map continues to perform well (curves 5, 6).”
    Quote in the paperSubstantial claim
  • Findingown data

    The full loss for both λ = 8 and λ = 20 cases plateaus rapidly.

    “The full loss for both cases plateaus rapidly.”
    Quote in the paperNot a substantial claim
  • Findingreasoning

    When trying to solve the steady state system using various classical methods (e.g., MATHEMATICA's NDSOLVE), the calculations do not converge.

    “Interestingly, when trying to solve the system for the steady state solution using various classical methods as, for example, implemented in MATHEMATICA's NDSOLVE, the calculations do not converge.”
    Quote in the paperSubstantial claim
  • Findingnone given

    As compared to many methods relying on sparse discretization of x, we have found the solution along the full nozzle length.

    “Notably, as compared to many methods relying on sparse discretization of x, we have found the solution along the full nozzle length.”
    Quote in the paperNot a substantial claim
  • May combine two claims. 38
  • Classified as probably not a substantial claim. 43
  • The quote only partly states this claim. 2

  • May combine two claims.

    Findingnone given

    The authors simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.

    “We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    MSE loss choice is intuitive and performs sufficiently well in numerical simulations.

    “While being simple, we find the choice (22) intuitive and performing sufficiently well in numerical simulations.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For lower depths the solver is slower to converge and does not reach as high accuracy as it does for higher depths.

    “We observe that for lower depths the solver is slower to converge and does not reach as high accuracy as it does for higher depths.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    After second stage training with the evolution-enhanced feature map, we are able to capture the intricate details of the solution, and both the full loss and quality of solution are improved.

    “As can be seen, after this second stage of training, we are able to capture the intricate details of the solution [Fig.7(a), blue dashed curve labeled by 3], and both the full loss and quality of solution are improved upon [Fig.7(b)].”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    The consequence of competing terms in the loss function can be seen in the early oscillations of the full loss in Fig. 8(b).

    “The consequence of competing terms in the loss function can be seen in the early oscillations of the full loss in Fig. 8(b).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The increase of speed for the air flow in the convergent part and decrease of temperature and density is reproduced quantitatively.

    “The increase of speed for the air flow in the convergent part and decrease of temperature and density is reproduced quantitatively.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The less powerful product feature map fails to converge with the loss quickly plateauing.

    “The less powerful product feature map fails to converge with the loss quickly plateauing (curves 1, 2).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The more expressible Chebyshev tower feature map takes longer to converge but reaches a solution closer to the true solution.

    “The more expressible Chebyshev tower feature map (curves 5, 6) takes longer to converge but reaches a solution closer to the true solution.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Full dynamical solution and its t→∞ extrapolation are possible once well-suited initial conditions (45)-(47) are chosen allowing to avoid instability.

    “At the same time, full dynamical solution and its t → ∞ extrapolation are possible, shown in Figs.9(b) and 9(c) once the well-suited initial conditions (45)-( 47) are chosen allowing to avoid instability, as usually done in computational fluid dynamics [2,142].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [2]

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    • cites [142]

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  • May combine two claims.

    Findingown data

    For λ = 20, the product feature map still fails to converge, and the simpler Chebyshev feature map also fails to converge.

    “The product feature map still fails to converge (curves 1, 2), but now also failing to converge is the simpler Chebyshev feature map (curves 3, 4).”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingown data

    DQCs are efficiently trained to approach the true solution, evidenced by LQ decrease.

    “For the region with switched off regularization we observe steady improvement of all metrics, showing that DQCs are efficiently trained to approach true solution, also evidenced by L Q decrease.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The full solution converges to the expected long-time behavior for the system, where function derivatives from DQC match the nonlinear contributions.

    “It converges to the expected long-time behavior for the system, where function derivatives from DQC match the nonlinear contributions.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the subsonic region, flow velocity grows towards the middle of the nozzle, while temperature and density drop slowly.

    “As expected, in the subsonic region flow velocity grows towards the middle of the nozzle, while temperature and density drop slowly [see Fig. 10(a), dashed curves 4, 5, 6].”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationnone given

    The Chebyshev quantum feature map offers a powerful basis set of fitting polynomials and possesses rich expressivity.

    “we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    Typical schemes for quantum ODE solvers involve numerical differentiation using Euler's method and finite difference schemes that suffer from approximation errors and often require a fine discretization grid.

    “Notably, typical schemes for quantum ODE solvers involve numerical differentiation using Euler's method and finite difference schemes that suffer from approximation errors, and often require a fine discretization grid.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [25][26][27][28][29][30][31][32][33]

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  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning

    Because the Fourier map involves unitary operators with phases that are linear functions of x, it is a fully harmonic mapping.

    “As it involves unitary operators with phases being linear functions of x, this is a fully harmonic mapping.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given

    Shared quantum register encoding is resource-frugal and suitable for certain systems.

    “This method is resource-frugal, and is suitable for certain systems.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Floating boundary handling guarantees exact matching to initial values given and does not require a separate boundary term in the loss function.

    “This method of boundary handling guarantees exact matching to initial values given and does not require a separate boundary term in the loss function, thus the derivative loss term does not have to compete with the boundary loss.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    Product feature maps induce nonlinear mappings between variable(s) x and quantum states described by tensor products of separate single-qubit wave functions.

    “Product feature maps induce nonlinear mappings between variable(s) x and quantum states described by tensor products of separate single-qubit wave functions.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    MSE places greater emphasis on larger distances and smaller weight on small distances, strongly discouraging terms with large L.

    “MSE places a greater emphasis on larger distances and smaller weight on small distances, strongly discouraging terms with large L.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    Shared register encoding may exhibit competition between the optimization of different functions.

    “However, the choice of suitable cost operators becomes complicated, as in some cases shared register encoding may exhibit competition between the optimization of different functions.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationcitation

    The alternating blocks ansatz's motivation is to entangle qubits locally first and gradually form a correlated state by interleaving subblocks, which helps improve trainability while maintaining high expressibility.

    “The motivation behind ABA is that we want to entangle qubits locally first, and gradually form a correlated state by interleaving subblocks.This helps to improve trainability of the circuit together while maintaining high expressibility [70,164].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [70] source

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    • cites [164]

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  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning

    Enhancing product feature maps (specifically the layered Chebyshev map) with additional entangling layers represented by Hamiltonian evolution makes feature maps more expressive.

    “To make the described feature maps even more expressive, we suggest enhancing product feature maps (and specifically the layered Chebyshev map) with additional entangling layers represented by Hamiltonian evolution.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    Paired with expressive variational ansatze, digital feature maps may offer function encoding with memory savings of amplitude encoding while avoiding the input problem.

    “Paired with expressive variational ansatze they may offer function encoding with memory savings of the amplitude encoding, while avoiding the input problem.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    Optimized boundary handling's weakness is that boundary and differential terms in the loss may compete against one another.

    “A weakness however is that the boundary and differential terms in the loss may compete against one another.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Floating boundary handling simplifies the choice for optimal angles and removes the dependence on initial θinit.

    “Furthermore, as we allow the cost function to match to the solution shifted by any amount, this simplifies the choice for optimal angles and removes the dependence on initial θ init .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The DQC approach differs from amplitude encoding in HHL and related methods, where getting the full solution from amplitudes is exponentially costly and requires tomographic measurements.

    “For the latter point, it differs from amplitude encoding |u in HHL and related methods, where getting the full solution from amplitudes is exponentially costly and requires tomographic measurements.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Fourier feature maps are based on the evolution operator exp(-i Ĥdata x), applied for some set of qubits.

    “This class of quantum feature maps is based on the evolution operator exp(-i Ĥdata x), which is applied for some set of qubits.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [155]

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  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

    Interpretationnone given

    Digital quantum feature maps offer a potentially powerful technique when dealing with functions of discrete variables.

    “Digital quantum feature maps offer a potentially powerful technique when dealing with functions of discrete variables.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given

    The simplest cost operator choice, magnetization of a single qubit Ẑj, allows representing functions in the range [-1, 1] and requires rescaling for other intervals.

    “Note that this choice allows representing functions in range [-1, 1], and requires rescaling for other intervals.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Optimized boundary handling removes the need to include boundary information in the cost expectation, but information still needs to be included in the loss function.

    “This removes the need to include boundary information in the cost expectation, but information still needs to be included in the loss function.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    Using two layers of rotations plus evolution makes the embedding difficult to simulate classically, but can be implemented as a unitary evolution on a quantum computer.

    “Then using two layers of rotations plus evolution the embedding becomes difficult to simulate classically, but can be implemented as a unitary evolution on a quantum computer.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning

    Increased number of controlled parameters can complicate the search for an optimal θ, known as a problem in trainability.

    “However, this comes with an increased number of controlled parameters which can complicate the search for an optimal θ opt for the solution, known as a problem in trainability.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationcitation

    The DQC approach can work with quantum kernels, an advantageous choice for the loss function that allows for efficient multipoint training.

    “We note that our approach also can work with quantum kernels [153], an advantageous choice for the loss function that allows for efficient multipoint training.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [153]

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  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationcitation

    Choosing an ansatz parametrized by single-qubit rotations allows the application of the parameter shift rule, while overlap measurement opens up options for more general strategies.

    “Choosing an ansatz parametrized by single-qubit rotations allows the application of the parameter shift rule, while overlap measurement opens up options for more general strategies [158].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [158]

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  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The Fourier map lets functions be encoded as Fourier series defined by the differences of the eigenvalues of Ĥdata.

    “The Fourier map lets functions be encoded as Fourier series defined by the differences of the eigenvalues of Ĥdata .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [155]

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  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning

    MAE and KL do not place such an emphasis and may have slower convergence.

    “Both MAE and KL do not place such an emphasis and may have slower convergence.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    As the number of layers d grows, the circuit's expressive power increases.

    “As the number of layers d grows, the circuit's expressive power increases.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning

    Higher emphasis on regularization at initial stages, diminishing to zero at higher iteration numbers, leads to prior information being used first, setting a rough solution, followed by precise derivative loss optimization at later training stages.

    “In general, we require higher emphasis on the regularization-based training at initial stages, which shall diminish to zero at higher iteration numbers.This leads to the prior information being used at first, setting a rough solution or preferred function behavior, followed by precise derivative loss optimization at later training stages.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    As a consequence of cost function choice, one needs to choose noncommuting generators Ĝj for the variational ansatz such that [Ĉ, Ĝj] = 0, which ensures the solution space can be spanned.

    “Namely, as a consequence of cost function choice we need to choose noncommuting generators Ĝ j for the variational ansatz such that [ Ĉ, Ĝ j ] = 0.This ensures that the solution space can be spanned.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    In many cases generators can be chosen such that only real amplitudes are generated.

    “In many cases generators can be chosen such that only real amplitudes are generated, see for instance [70].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [70] source

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    Interpretationreasoning

    λ=20 is a complex case as the resulting solution is highly nonlinear and oscillatoric.

    “These problem parameters are chosen to make DQC construction challenging, with λ = 20 being a complex case as the resulting solution is highly nonlinear and oscillatoric.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given

    As λ and κ grow, the function starts to oscillate and decay more rapidly, and the solution becomes harder to express.

    “As λ and κ grow the function starts to oscillate and decay even more rapidly, and the solution becomes harder to express.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationreasoning

    This causes a major problem to classical solvers, that are unable to find a steady state solution directly.

    “This causes a major problem to classical solvers, that are unable to find a steady state solution directly.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning

    Separate quantum registers require more resources, but function and derivative evaluation can be done in parallel.

    “While requiring more resources we note that function and derivative evaluation can be done in parallel.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The alternating blocks ansatz is designed in a way that vanishing gradients can be avoided for certain conditions.

    “We also note that the alternating blocks ansatz is designed in the way that vanishing gradients can be avoided for certain conditions [164].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [164]

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  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    Spin-glass type cost functions allow for high expressibility of the circuit describing the DE's solution.

    “These are known to include NP-hard problem instances, and they allow for high expressibility of the circuit describing the DE's solution.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: Spin-glass type cost functions include NP-hard problem instances.

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

    Interpretationnone given

    Quality of solution gives a useful way to compare how two different training setups perform, especially if they are training to solve the same differential equations.

    “Quality of solution gives us a useful way to compare how two different training setups perform, especially if they are training to solve the same differential equations.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    As depth increases, the number of gate operations needed per iteration and the number of parameters the classical optimizer needs to update increase, raising the time taken per iteration.

    “This causes an increase in the number of gate operations needed in each iteration and how many parameters the classical optimizer needs to update, raising the time taken per iteration.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The difference in convergence rate is a result of boundary information having an impact on the loss, demanding the matching for pinned and optimized boundary handlers, whereas the floating boundary automatically matches the initial condition and no loss boundary term is needed.

    “This difference in convergence rate is a result of boundary information having an impact on the loss, demanding the matching for pinned and optimized boundary handlers, whereas the floating boundary automatically matches the initial condition and no loss boundary term is needed.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given

    Choosing maximal time 0.9 avoids the region with diverging derivative of the nonlinear feature function dφ(x)/dx.

    “This is chosen such that the region with diverging derivative of the nonlinear feature function dϕ(x)/dx is avoided, and we note that x can be rescaled to match required boundaries.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    The key problem of the convergent-divergent nozzle in the subsonic-supersonic transition case is the divergence around the middle of the nozzle.

    “As we discussed before, the key problem of the convergentdivergent nozzle in the subsonic-supersonic transition case is the divergence around the middle of the nozzle.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationreasoning

    More simulations with more qubits offers a way to increase the power drastically.

    “and more simulations with more qubits offers a way to increase the power drastically.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    DQCs with the evolution-enhanced feature map are more difficult to train, as keeping track of the gradients for oscillatoric function requires a finer training grid and has a slower convergence.

    “At the same time, DQCs in this case are more difficult to train, as keeping track of the gradients for oscillatoric function requires a finer training grid and has a slower convergence.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given

    Results remain consistent for different J and h generated randomly.

    “We check that results remain consistent for different J and h generated randomly.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Divergent contributions from this region also impact the loss function, and makes training complicated.

    “Divergent contributions from this region also impact the loss function, and makes training complicated.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    This leads to singular behavior and breaks classical solvers, including the ones with stiff handling.

    “This leads to singular behavior and breaks classical solvers, including the ones with stiff handling.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    The expressive power of the evolution-enhanced map can be increased for more complex H and larger τ, though at the expense of trainability.

    “The expressive power of the evolution-enhanced map can be increased for more complex Ĥ and larger τ , though at the expense of trainability.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    As the depth of the ansatz continues to increase, eventually the problem of barren plateaus could be encountered.

    “As the depth of the ansatz continues to increase eventually the problem of barren plateaus could be encountered [171].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [171]

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  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning

    Although the method works explicitly for a spectral (global) treatment, the scheme is general and one may instead consider a digital quantum feature map which allows a grid-based (local) treatment.

    “Although we showed how the method works explicitly for a spectral (global) treatment, the scheme is general and one may instead consider a digital quantum feature map which allows a grid-based (local) treatment.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Separate quantum registers remove the issue of choosing cost operators and avoid direct competition due to independent parametrization.

    “This removes the issue of choosing the cost operators, and avoids the related direct competition due to independent parametrization.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationreasoning

    The same method can be applied across all disciplines where differential equations arise.

    “but the same method can be applied across all disciplines where differential equations arise.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data

    For λ = 20, the true solution is more oscillatoric and has stronger damping, making the solution harder to represent.

    “For λ = 20 the true solution is more oscillatoric and has stronger damping, making the solution harder to represent [Figs.6(b) and 6(e)].”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    In classification tasks, increased representation power leads to increased accuracy even for small networks.

    “In classification tasks this leads to increased accuracy even for small networks [174].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [174]

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given

    We did not yet consider the impact of noise on the algorithm performance.

    “We did not yet consider the impact of noise on the algorithm performance;”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: Analytical gradients are much less susceptible to noise than numerical gradients evaluated on variational quantum circuits.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    A good choice of variational ansatz is crucial to loss function convergence success and speed.

    “A good choice of variational ansatz is crucial to loss function convergence success and speed.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Classical neural networks based on Fourier representation were recently shown to outperform traditional PDE solvers by three orders of magnitude.

    “For instance, classical neural networks based on Fourier representation were recently shown to outperform traditional PDE solvers by three orders of magnitude [147].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [147]

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S4 (based). The quote only partly states this claim.

    Interpretationreasoning

    Absolute operation rates require quantum hardware specifications with high sampling rates and a low-latency hybrid operational mode.

    “The latter requires quantum hardware specifications with high sampling rates and a low-latency hybrid operational mode.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The success of operation depends on trainability of quantum circuits and absolute operation rates.

    “Having the expressibility advantage, the success of operation then depends on (1) trainability of quantum circuits and (2) absolute operation rates.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: Trainability of quantum circuits is well-known to suffer from barren plateaus.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    This was used recently to speed up generative modeling with quantum batch training.

    “This was used recently to speed up generative modeling with quantum batch training [124].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [124] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

    Interpretationnone given

    Differential quantum circuits can enjoy another type of computational advantage related to training.

    “Finally, differential quantum circuits can enjoy another type of computational advantage related to training.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: For expressive feature maps, the representation power of quantum circuits grows superpolynomially with the number of qubits.

  • Classified as probably not a substantial claim.

    Premisecitation

    Variational quantum machine learning uses quantum circuits as universal function approximators.

    “Similarly to classical ML, variational quantum machine learning uses quantum circuits as universal function approximators [173].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [173]

      Not checked in the free run.

72 of 72

  • The paper relies on this without support.

    Quantum simulation of dynamics leads to a volume-like increase of entanglement.

    “It is known that the quantum simulation of dynamics leads to a volume-like increase of entanglement.”

  • The paper relies on this without support.

    Spin-glass type cost functions include NP-hard problem instances.

    “These are known to include NP-hard problem instances, and they allow for high expressibility of the circuit describing the DE's solution.”

  • The paper relies on this without support.

    The type of Hamiltonian used in the evolution-enhanced feature map is not crucial and may depend on hardware implementation.

    “Note that the type of Hamiltonian is not crucial, and may depend on hardware implementation.”

  • The paper relies on this without support.

    Analytical gradients are much less susceptible to noise than numerical gradients evaluated on variational quantum circuits.

    “it is well-known that the analytical gradients, which are used here, are much less susceptible to noise than numerical gradients evaluated on variational quantum circuits.”

  • The paper relies on this without support.

    For expressive feature maps, the representation power of quantum circuits grows superpolynomially with the number of qubits.

    “For expressive feature maps, the representation power of quantum circuits grows superpolynomially with the number of qubits.”

  • The paper relies on this without support.

    Trainability of quantum circuits is well-known to suffer from barren plateaus.

    “The former is well-known to suffer from a phenomenon commonly referred to as barren plateaus.”

  • The paper relies on this without support.

    As air flows through the convergent-divergent nozzle, it accelerates and cools down.

    “As the air goes through the nozzle it accelerates and cools down.”
7 of 7

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Refs [139-141] considered the Hamiltonian H_G = A(I - |b><b|)A†.

    “Refs. [139-141] considered the Hamiltonian H G = A ( ✶ -| b 〉 〈 b | ) A † (which was also considered outside of the variational setting [142]).”

    From Variational quantum algorithms (2021) · cites this paper as [142]

    The Introduction discusses solving for |u⟩ = A⁻¹|b⟩, but does not give the claimed Hamiltonian. The Cost function section describes magnetization and Ising-type operators instead. Searches for the projector |b⟩⟨b| and H_G found no passage showing that this paper considered H_G = A(I − |b⟩⟨b|)A†. (LLM, conf=0.74)

    Experimental — accuracy not yet measured

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Odd-degree Chebyshev conversion disagrees with the stated U1 polynomial

    Minor

    Applying the stated odd-degree conversion at n = 1 gives U1(x) = 2[T1(x) − 1] = 2x − 2. The paper separately states U1(x) = 2x.

  • Chebyshev polynomial derivatives are said to diverge at the boundaries

    Minor

    The paper says these derivatives diverge at the interval boundaries, but its explicit polynomial T2(x) = 2x2 − 1 has derivative 4x, which is finite at both boundaries.

  • Two derivative formulas give different parameter-shift coefficients

    Minor

    For a single x-dependent rotation with φ(x) = x, Eq. (2) assigns one-half of the difference between the shifted expectations to the derivative. Eqs. (9)-(10) assign one-quarter of that same difference.

3 of 3

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. D. P. Kingma, 2015, Proceedings of the 3rd International Conference on Learning Representations (ICLR)

Cited by

Sorted by citations

Related

Authors
Kyriienko, O., Paine, A. E., & Elfving, V. E.
Affiliations
  • University of Exeter (GB)
  • Qu & Co. (Netherlands) (NL)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2021-05-17
Volume
103
Issue
5
ISSN
  • 0556-2791 (ISSN-L)
  • 2469-9926
  • 2469-9934
DOI
10.1103/physreva.103.052416
OpenAlex
W3108818881
Semantic Scholar
ae03babddded3cd7360cc658359e98055bff3540
Cited by
185
References
188
Citations by year
2026: 28 · 2025: 45 · 2024: 44 · 2023: 32 · 2022: 20 · 2021: 16
FWCI
9.86
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • Figshare (submittedVersion)
  • Open Research Exeter (University of Exeter) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
Keywords
nonlinear differential equations, quantum feature maps, automatic differentiation, hybrid quantum-classical algorithms, spectral methods, Navier-Stokes equations, convergent-divergent nozzle, fluid flow, boundary conditions

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-10.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-10.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

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