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REVIEW 4 major objections 5 minor 61 references

Predicting fermionic densities using a Projected Quantum Kernel method

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A projected quantum kernel built from the measured observables of a four-qubit Rydberg reservoir predicts the ground-state densities of 1D fermionic systems, outperforming a classical linear kernel and matching an RBF kernel once the…

desk verdict A careful but modest numerical study of a projected quantum kernel for 1D density prediction; the main novelty is the error-vs-time characterization, and the central claim is plausible but rests on a truncated-basis target that is only spot-checked. read the letter →

arxiv 2504.14002 v2 pith:BBWPLYUU submitted 2025-04-18 quant-ph cond-mat.str-elcs.LG

Francesco Perciavalle , Francesco Plastina , Michele Pisarra , Nicola Lo Gullo This is my paper · ORCID
classification quant-phcond-mat.str-elcs.LG
keywords projectedquantumkernelsupportvectorregressionRydbergreservoirfermionicdensityKohn-Shamequationsfunctionaltheorymachinelearning1Dchemistry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a support vector regressor armed with a projected quantum kernel can predict the ground-state density of two-fermion 1D systems from the external potential alone, without solving the Kohn-Sham equations. The kernel is built from single-spin and two-spin measurements on a four-qubit Rydberg reservoir whose detunings encode the potential features. Across two model systems (an H2-like molecule and a triple-well potential), the prediction error shows the same shape in measurement time: a short plateau, a sharp drop, then stabilization. At stabilized times the quantum kernel beats a linear kernel and performs on par with an RBF kernel, and the authors read this as evidence that a minimal quantum reservoir can serve as a feature map for density prediction.

What carries the argument

The central object is the projected quantum kernel (PQK). It is $K(v_k,v_l) = m_k \cdot m_l$, where $m_k$ is the vector of measured expectation values $\langle \sigma_j^z \rangle$ and $\langle \sigma_i^z \sigma_j^z \rangle$ at time $t^*$, evolved under a four-qubit Rydberg Hamiltonian with the potential features encoded in local detunings. This mapping enlarges the feature space from $N_f$ inputs to ten observables, and the time-dependent dynamics make the kernel a tunable, nonlinear feature map; the reservoir interaction is what lifts degeneracies among the observables. The regression targets are the coefficients $\{u^{(\ell)}\}$ of the density in a truncated basis formed from single-particle eigenstates of left, center, and right well potentials, orthonormalized via Gram-Schmidt.

What would settle it

Take a triple-well configuration with barrier heights near zero so the three wells merge into one broad well, compute the exact KS density $n_k$ and its truncated expansion $n_{\mathrm{approx}}$, and evaluate the L1 distance; if the distance is large and the PQK prediction tracks $n_{\mathrm{approx}}$ rather than $n_k$, the method's accuracy is limited by the basis, not by the quantum kernel. Also, a statistical check across the full test distribution (not just selected samples in Fig. 2(c)) comparing $n_k$ and $n_{\mathrm{approx}}$ would settle whether the truncation error is negligible on average.

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Extended reading notes

Core claim

The paper's central claim is that the map from potential parameters to fermionic density can be learned by an SVR whose kernel is the inner product of observable vectors $m_k = \{\langle \sigma_j^z \rangle\} \cup \{\langle \sigma_i^z \sigma_j^z \rangle\}$ taken after a fixed evolution time $t^*$. With four interacting Rydberg qubits and the right choice of $t^*$, the PQK error on held-out density profiles is systematically lower than a linear kernel in the two test problems, and sits in the same range as an RBF kernel. The authors also identify a robust qualitative law: the error is flat at short times, drops rapidly once the reservoir dynamics have had time to spread information, and then stabilizes; the drop time is set by the interaction strength, and the non-addressed qubits become distinguishable only when interactions break the encoding symmetry. The method's best operating window, for the H2 problem, is around $\pi \leq t^*\Omega_{\max} \leq 3\pi/2$, where errors are competitive with RBF.

Load-bearing premise

The training targets are coefficients of the density in a truncated basis of single-particle well states for the three potential regions; if a test potential yields a density that this basis cannot capture, the SVR target stops being a faithful proxy for the true density.

Editorial extensions

If this is right

  • If correct, KS-free density prediction for these model systems needs only four qubits, making the method testable on current Rydberg hardware.
  • The universal plateau-drop-stabilize error curve provides a practical rule for choosing measurement time: aim for the stabilized window instead of tuning the pulse naively.
  • The finding that reservoir interactions improve performance suggests interaction strength is a useful control knob for encoding quality, not just a source of noise.
  • Extending the pipeline with a second SVR that maps predicted densities to energies would give a complete bypass of the KS self-consistency loop.

Reading between the lines

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

  • Since the training target is itself a truncated basis expansion, the method's ceiling is the representational power of the three-well basis; away from the sampled potential ranges (e.g., very low barriers that merge the wells), the reported advantage over linear and RBF kernels may not transfer even if the kernel is perfect.
  • A direct test of the reservoir's contribution would be to replace the PQK with a random but fixed nonlinear feature map of the same dimension and compare errors; if the quantum dynamics matter beyond dimension, the PQK should win across measurement times.
  • The plateau-drop structure suggests the kernel's effective rank grows with time; measuring the observable-vector covariance as a function of $t^*$ could quantify when the feature space saturates, and whether the optimal window coincides with maximal rank.
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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

4 major / 5 minor

Summary. The paper proposes a support vector regression (SVR) scheme based on a projected quantum kernel (PQK) to predict the ground-state density of 1D two-fermion systems, specifically a one-dimensional H2 molecule and a triple-well potential. The kernel is constructed from the measured magnetizations and correlation functions of a four-qubit Rydberg reservoir, with potential features encoded into the detunings of the reservoir Hamiltonian. The SVR is trained on the coefficients of the density in a fixed three-region single-particle basis (Appendix C), and the prediction error is defined in Eq. (9) as the L1 difference between the predicted density and the truncated basis expansion n^(approx) of the Kohn-Sham density. The authors compare PQK-SVR with linear and RBF kernel SVRs and report that, for sufficiently large measurement times, the PQK method outperforms the linear kernel and is competitive with the RBF kernel. They also analyze the dependence of the error on reservoir parameters such as interaction strength, Rabi frequency, and detuning, and observe a common plateau–drop–stabilization behavior of the error as a function of measurement time.

Significance. If the central claim holds, the paper provides a concrete and platform-relevant demonstration that a very small (four-qubit) quantum reservoir can act as a feature map for predicting densities in model fermionic systems, with numerical protocols that are described in detail (Pulser simulations, scikit-learn training, and appendices with formulas). The strength of the paper is its careful numerical setup: 20 random data sets, explicit hyperparameter grids, and a systematic scan over reservoir parameters. The observation that the reservoir needs a minimum interaction time before the error drops, and that the error then stabilizes at a value competitive with classical kernels, is a potentially useful design principle for reservoir-based quantum kernels. However, the significance of the result as a route to 'bypassing Kohn-Sham equations' is currently limited by two methodological issues: the error is measured against a truncated-basis proxy rather than the full KS density, and the headline comparison is made at a measurement time chosen after inspecting the hidden-test error curves. These issues narrow the scope of the claim as presently stated.

major comments (4)
  1. [Sec. II, Eq. (9); Sec. III B; Appendix C] The error measure in Eq. (9) compares the predicted density to n^(approx)_k, the truncated basis expansion defined in Eq. (4), not to the full Kohn-Sham density n_k. The only evidence that n_k ≈ n^(approx) is a single H2 sample shown in Fig. 2(c), with the text stating 'there is good agreement between n_k and n^(approx)_k, proving that Ntrunc = 30 is sufficient.' No statistical check over the sample distribution is provided, and no analogous check is shown for the triple-well model, where Ntrunc = 18 and barrier heights can be as low as 0.8 a.u. (Appendix A). Since the basis is built once from fixed well regions (Appendix C), configurations with low barriers that merge the wells may be poorly represented, and for those configurations E is not a faithful estimate of the true density prediction error. The authors should either report the distribution of ||n_k − n^(approx)|| over all samples for both models, or explicitly restate the claim as predicting the truncated-basis density rather than the KS density. As written, the assertion that the method 'predicts the density structure' exceeds what Eq. (9) actually measures.
  2. [Sec. III A, Figs. 2(a), 3, 4; Sec. IV] The headline comparison ('overall, the best performance of the PQK method is obtained in the interval π ≲ t*Ωmax ≲ 3π/2' and the conclusion that PQK 'often outperforms the linear kernel results and can be competitive with the RBF kernel') is drawn from error curves computed on the hidden test set. The paper does not specify a protocol for selecting t* from training data alone, so the reported advantage over linear and competitiveness with RBF is subject to selection bias. Please define a deterministic rule for choosing t* (for example, based on a validation split or on the magnetization dynamics discussed in Sec. III B) and report errors at that fixed t* for all methods, or report the error averaged over a time window that is chosen without reference to the hidden labels.
  3. [Appendix C, Eq. (C1)] The definition of the piecewise function is internally inconsistent: P(z,y) = θ(z−x) − θ(y−x) with z < y equals −1 on the interval [z,y], not +1 as stated in the text ('The function P(z,y) is one in the interval [z,y] and zero elsewhere'). This sign error propagates through Eqs. (C1)–(C4), which define the single-particle potentials used to construct the basis for the density expansion. Because the basis is load-bearing for the target coefficients, this inconsistency must be corrected (either the definition or the description) and the basis construction re-verified so that the numerical protocol is reproducible and self-consistent.
  4. [Sec. II, Fig. 2 caption; Appendix D] The grid-search procedure for the SVR hyperparameters C and ϵ (and γ for the RBF kernel) is described, but the manuscript does not state whether the grid search is performed using the training set only (e.g., by cross-validation) or using the hidden test set. If the hidden set is used for hyperparameter selection, the reported errors are optimistic and the comparison is not a fair assessment of prediction performance. Please clarify the protocol and, if necessary, modify the evaluation so that all hyperparameters, including t*, are selected without access to the test labels.
minor comments (5)
  1. [Appendix C, Eq. (C3)] The scalar product in Eq. (C3) should read ⟨Ψ(ℓ′), Ψ̃(ℓ)⟩ = ∫ dx Ψ*_ℓ′(x) Ψ̃_ℓ(x); the current expression contains Ψ̃_ℓ′(x) on the right-hand side, which appears to be a typo.
  2. [Appendix D, Eq. (D9)] In the last line of Eq. (D9), the second term of the kernel should involve ρ_t*(v_l) σ^i σ^j with i < j, consistent with the definition in Eq. (6); the current expression Tr[ρ_t*(v_l) σ^i σ^i] is dimensionally consistent only for i = j and appears to be a typographical error.
  3. [Sec. III B, Fig. 4 caption] The caption states 'In any case the global detuning term is set to ∆glob = 5 rad/µs' but the description of the panels is somewhat ambiguous; please clarify which parameters are common to panels (a), (b), and (c) and which vary.
  4. [Fig. 2(c) and Eq. (11)] The bottom panels of Fig. 2(c) appear to show signed differences between the approximated KS density and the ML densities, whereas Eq. (11) defines δn^ML as an absolute value; please clarify the plotting convention.
  5. [Abstract and Sec. IV] The abstract says the method 'outperforms the classical linear kernel method and can be competitive with the radial basis function method' at large measurement times, while the conclusion says 'often outperforms the linear kernel results and can be competitive with the RBF kernel.' These statements should be aligned and, if the post hoc time selection is corrected, the wording should reflect the protocol under which the claim holds.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kernel is defined by fixed reservoir observables, the SVR targets come from independent DFT calculations, and self-citations are background only.

full rationale

The derivation chain is self-contained rather than circular. The PQK kernel is defined in Eqs. (5)-(9) from reservoir observables m_k = phi(v_k), with the input features encoded in the detunings; the kernel has no fitted dependence on the target densities. The SVR targets are the coefficients {u_k^(l)} obtained by projecting the DFT/KS density n_k(x) onto the truncated basis of Eq. (4), and the same coefficients are used for all three kernels, so the comparison between PQK, linear, and RBF is fair. The reported error E in Eq. (9) compares n_ML to n_approx; although this measures error against the same truncated representation used to define the targets, it is a standard regression metric in the coefficient-induced L1 norm and not an equation that substitutes the target for the output. The only apparent concerns are soundness, not circularity: the optimal measurement time t* is selected post hoc from the error curves, and the adequacy of Ntrunc is demonstrated only for selected H2 samples in Fig. 2(c) rather than statistically for the triple-well model. Self-citations [16,20] appear only in a background sentence on quantum reservoir computing and are not load-bearing; the methodological references for the kernel and local encoding are external ([23,28,29]). No equation reduces a predicted quantity to a fitted or cited input, so no circular step is identified.

Assumptions & free parameters 6 free parameters · 3 assumptions · 0 invented entities

The method introduces no new physical entities, and the kernel is a fixed function of reservoir observables. The main unpaid assumptions are the LDA training-label fidelity, the noiseless reservoir model, and the single-particle-well basis truncation. The hand-chosen reservoir control parameters, including the post hoc selected measurement time, are listed as free parameters. SVR hyperparameters C, epsilon, and gamma are tuned by grid search identically for all kernels, so they do not bias the comparison.

free parameters (6)
  • measurement time t* = optimal interval t*Omega_max in [pi, 3pi/2], identified post hoc on hidden data
    The central error-versus-time claim depends on choosing t*; the paper selects the best interval from the test error curves, making it a hand-chosen operating point.
  • global Rabi frequency Omega_glob = 5 rad/microsecond (H2); 5 or 7.5 rad/microsecond (triple-well)
    Reservoir control parameter, scanned to test robustness; it affects the slope of the error drop.
  • local detuning amplitude Delta_loc = -3.5 rad/microsecond (H2); -1 rad/microsecond (triple-well)
    Controls the encoding sensitivity; strong values speed up the error drop but can worsen the stabilized error.
  • global detuning Delta_glob = 0 rad/microsecond (H2); 5 rad/microsecond (triple-well)
    Baseline detuning common to all qubits; it contributes to Omega_max and is scanned.
  • nearest-neighbor interaction VNN = 0.5, 2, 4, 8 rad/microsecond
    Rydberg interaction strength; the paper links it to the feature-space inhomogeneity that causes the error drop.
  • homogeneous detuning on non-addressed sites v_homo = 0.5
    Fixed value for qubits j=1,3 in the H2 encoding; a hand-chosen control parameter.
assumptions (3)
  • domain assumption The LDA exchange-correlation functional of Ref. [46], V_xc = (-1.19 + 1.77n - 1.37n^2)n^0.604, produces ground-state densities accurate enough to serve as training targets.
    All targets n_k(x) are generated by the KS equation with this LDA; the paper treats these densities as ground truth and does not benchmark against exact solutions.
  • domain assumption The reservoir evolves unitarily and without noise from the initial state |gggg>, with observables evaluated from the exact state vector at t*.
    The simulations use Pulser's ideal statevector dynamics; real Rydberg implementations include decay and dephasing, which are not modeled.
  • domain assumption The basis functions defined in Appendix C with Ntrunc=30 (H2) or 18 (triple-well) accurately reproduce the true KS densities.
    The SVR predicts expansion coefficients in this basis, so the method inherits any truncation error; the agreement between n and n_approx is shown for a specific potential configuration in Fig. 2(c), not quantified over all samples.

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Cite this review

Pith. "Pith review of Predicting fermionic densities using a Projected Quantum Kernel method." pith.science (2026). https://pith.science/paper/BBWPLYUU

@misc{pith2026250414002,
  author       = {Pith},
  title        = {Pith review of: Predicting fermionic densities using a Projected Quantum Kernel method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BBWPLYUU}},
  note         = {Machine review of arXiv:2504.14002}
}
read the original abstract

We use a support vector regressor based on a projected quantum kernel method to predict the density structure of 1D fermionic systems of interest in quantum chemistry and quantum matter. The kernel is built on with the observables of a quantum reservoir implementable with interacting Rydberg atoms. Training and test data of the fermionic system are generated using a Density Functional Theory approach. We test the performance of the method for several Hamiltonian parameters, finding a general common behavior of the error as a function of measurement time. At sufficiently large measurement times, we find that the method outperforms the classical linear kernel method and can be competitive with the radial basis function method.

Figures

Figures reproduced from arXiv: 2504.14002 by the authors.

Figure 1
Figure 1. Sketch of the method: the two-particle fermionic ground state density is found using the KS equations. For any [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 5
Figure 5. Sketch of the triple-well potential defined in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Scaling of the error with the number of training [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figures from the paper (1 more)
Figure 7
Figure 7. Figure 7: Error in function of the measurement time [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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Reference graph

Works this paper leans on

61 extracted references · 35 canonical work pages

  1. [1]

    Hearst, S

    M. Hearst, S. Dumais, E. Osuna, J. Platt, and B. Scholkopf, Support vector machines, IEEE Intelligent Systems and their Applications 13, 18–28 (1998)

    work page 1998

  2. [2]

    Kecman, Support vector machines – an introduction, in Support Vector Machines: Theory and Applications (Springer Berlin Heidelberg, 2005) p

    V. Kecman, Support vector machines – an introduction, in Support Vector Machines: Theory and Applications (Springer Berlin Heidelberg, 2005) p. 1–47

    work page 2005

  3. [3]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cour- napeau, M. Brucher, M. Perrot, and E. Duchesnay, Scikit-learn: Machine learning in Python, Journal of Ma- chine Learning Research 12, 2825 (2011)

    2011

  4. [4]

    K. P. Murphy, Probabilistic Machine Learning: An intro- duction (MIT Press, 2022)

    work page 2022

  5. [5]

    Muller, S

    K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda, and B. Scholkopf, An introduction to kernel-based learning algorithms, IEEE Transactions on Neural Networks 12, 181–201 (2001)

    work page 2001

  6. [6]

    Carleo, I

    G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborov´ a, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019)

    2019

  7. [7]

    Brockherde, L

    F. Brockherde, L. Vogt, L. Li, M. E. Tuckerman, K. Burke, and K.-R. M¨ uller, Bypassing the kohn-sham equations with machine learning, Nature Communica- tions 8, 10.1038/s41467-017-00839-3 (2017)

    doi:10.1038/s41467-017-00839-3 2017

  8. [8]

    Kohn and L

    W. Kohn and L. J. Sham, Self-consistent equations in- cluding exchange and correlation effects, Phys. Rev. 140, A1133 (1965)

    1965

Show all 61 references
  1. [9]

    Kohn, Nobel lecture: Electronic structure of matter— wave functions and density functionals, Rev

    W. Kohn, Nobel lecture: Electronic structure of matter— wave functions and density functionals, Rev. Mod. Phys. 71, 1253 (1999)

    1999

  2. [10]

    Hohenberg and W

    P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964)

    1964

  3. [11]

    Giuliani and G

    G. Giuliani and G. Vignale, Quantum Theory of the Elec- tron Liquid (Cambridge University Press, 2005)

    2005

  4. [12]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys.94, 015004 (2022)

    2022

  5. [13]

    Biamonte, P

    J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Na- ture 549, 195–202 (2017)

    2017

  6. [14]

    Fujii and K

    K. Fujii and K. Nakajima, Harnessing disordered- ensemble quantum dynamics for machine learning, Phys. Rev. Appl. 8, 024030 (2017)

    2017

  7. [15]

    Nakajima, K

    K. Nakajima, K. Fujii, M. Negoro, K. Mitarai, and M. Kitagawa, Boosting computational power through spatial multiplexing in quantum reservoir computing, Phys. Rev. Appl. 11, 034021 (2019)

    2019

  8. [16]

    Settino, L

    J. Settino, L. Salatino, L. Mariani, M. Channab, L. Boz- zolo, S. Vallisa, P. Barill` a, A. Policicchio, N. L. Gullo, A. Giordano, C. Mastroianni, and F. Plastina, Memory- augmented hybrid quantum reservoir computing (2024), arXiv:2409.09886 [quant-ph]

    2024 arXiv

  9. [17]

    Mujal, R

    P. Mujal, R. Mart´ ınez-Pe˜ na, J. Nokkala, J. Garc´ ıa-Beni, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Oppor- tunities in quantum reservoir computing and extreme learning machines, Advanced Quantum Technologies 4, 10.1002/qute.202100027 (2021)

    2021 doi

  10. [18]

    QuCADD” and “ThAnQ

    In this case, the potential is codified by 4 features which are encoded in the detunings of all the qubits of the reservoir Hamiltonian. In Fig. 4 we report the com- parison between the error of the PQK method and those of classical linear and RBF kernel methods. In this case,...

  11. [19]

    Mart´ ınez-Pe˜ na, G

    R. Mart´ ınez-Pe˜ na, G. L. Giorgi, J. Nokkala, M. C. So- riano, and R. Zambrini, Dynamical phase transitions in quantum reservoir computing, Phys. Rev. Lett. 127, 100502 (2021)

    2021

  12. [20]

    D. Zia, L. Innocenti, G. Minati, S. Lorenzo, A. Suprano, R. D. Bartolo, N. Spagnolo, T. Giordani, V. Cimini, G. M. Palma, A. Ferraro, F. Sciarrino, and M. Paternos- tro, Quantum extreme learning machines for photonic en- tanglement witnessing (2025), arXiv:2502.18361 [quant- ph]

    2025

  13. [21]

    De Lorenzis, M

    A. De Lorenzis, M. Casado, M. Estarellas, N. Lo Gullo, T. Lux, F. Plastina, A. Riera, and J. Settino, Harnessing quantum extreme learning machines for image classifica- tion, Phys. Rev. Appl. 23, 044024 (2025)

    2025

  14. [22]

    Beaulieu, M

    D. Beaulieu, M. Kornjaca, Z. Krunic, M. Stivaktakis, T. Ehmer, S.-T. Wang, and A. Pham, Robust quantum reservoir computing for molecular property prediction (2024), arXiv:2412.06758 [quant-ph]

    2024 arXiv

  15. [23]

    Lo Monaco, M

    G. Lo Monaco, M. Bertini, S. Lorenzo, and G. Mas- simo Palma, Quantum extreme learning of molecular po- tential energy surfaces and force fields, Machine Learn- ing: Science and Technology 5, 035014 (2024)

    2024

  16. [24]

    Huang, M

    H.-Y. Huang, M. Broughton, M. Mohseni, R. Babbush, S. Boixo, H. Neven, and J. R. McClean, Power of data in quantum machine learning, Nature Communications 12, 10.1038/s41467-021-22539-9 (2021)

    2021 doi

  17. [25]

    Schnabel and M

    J. Schnabel and M. Roth, Quantum kernel meth- ods under scrutiny: A benchmarking study (2024), arXiv:2409.04406 [quant-ph]

    2024 arXiv

  18. [26]

    T. Haug, C. N. Self, and M. S. Kim, Quantum ma- chine learning of large datasets using randomized mea- surements, Machine Learning: Science and Technology 4, 015005 (2023)

    2023

  19. [27]

    Blank, D

    C. Blank, D. K. Park, J.-K. K. Rhee, and F. Petruc- cione, Quantum classifier with tailored quantum kernel, npj Quantum Information 6, 10.1038/s41534-020-0272-6 (2020)

    2020 doi

  20. [28]

    Mengoni and A

    R. Mengoni and A. Di Pierro, Kernel methods in quan- tum machine learning, Quantum Machine Intelligence 1, 65–71 (2019)

    2019

  21. [29]

    D’Amore, L

    F. D’Amore, L. Mariani, C. Mastroianni, J. Settino, and A. Vinci, Projected quantum kernel for iot data analysis, in 2024 IEEE Conference on Pervasive and Intelligent Computing (PICom) (IEEE, 2024) p. 173–177

    2024

  22. [30]

    Kornjaˇ ca, H.-Y

    M. Kornjaˇ ca, H.-Y. Hu, C. Zhao, J. Wurtz, P. Wein- berg, M. Hamdan, A. Zhdanov, S. H. Cantu, H. Zhou, R. A. Bravo, K. Bagnall, J. I. Basham, J. Campo, A. Choukri, R. DeAngelo, P. Frederick, D. Haines, J. Hammett, N. Hsu, M.-G. Hu, F. Huber, P. N. Jepsen, N. Jia, T. Karolyshy...

    2024 arXiv

  23. [31]

    L. O. Wagner, E. M. Stoudenmire, K. Burke, and S. R. White, Reference electronic structure calculations in one dimension, Physical Chemistry Chemical Physics 14, 8581 (2012)

    2012

  24. [32]

    E. M. Stoudenmire, L. O. Wagner, S. R. White, and K. Burke, One-dimensional continuum electronic struc- ture with the density-matrix renormalization group and its implications for density-functional theory, Phys. Rev. Lett. 109, 056402 (2012)

    2012

  25. [33]

    P.-F. Loos, C. J. Ball, and P. M. W. Gill, Chemistry in one dimension, Physical Chemistry Chemical Physics 17, 3196–3206 (2015)

    2015

  26. [34]

    C. J. Ball and P. M. Gill, Chem1d: a software package for electronic structure calculations on one-dimensional systems, Molecular Physics 113, 1843–1857 (2015)

    2015

  27. [35]

    T. E. Baker, E. M. Stoudenmire, L. O. Wagner, K. Burke, and S. R. White, One-dimensional mimicking of elec- tronic structure: The case for exponentials, Phys. Rev. B 91, 235141 (2015)

    2015

  28. [36]

    Li, Exact analytical ground state solution of 1d H + 2 with soft coulomb potential, Journal of Mathematical Chemistry 60, 184–194 (2021)

    C. Li, Exact analytical ground state solution of 1d H + 2 with soft coulomb potential, Journal of Mathematical Chemistry 60, 184–194 (2021)

    2021

  29. [37]

    L. Li, T. E. Baker, S. R. White, and K. Burke, Pure density functional for strong correlation and the thermo- dynamic limit from machine learning, Phys. Rev. B 94, 245129 (2016)

    2016

  30. [38]

    L. Li, J. C. Snyder, I. M. Pelaschier, J. Huang, U. Ni- ranjan, P. Duncan, M. Rupp, K. M¨ uller, and K. Burke, Understanding machine-learned density functionals, In- ternational Journal of Quantum Chemistry 116, 819–833 (2015)

    2015

  31. [39]

    J. C. Snyder, M. Rupp, K. Hansen, L. Blooston, K.-R. M¨ uller, and K. Burke, Orbital-free bond breaking via ma- chine learning, The Journal of Chemical Physics 139, 10.1063/1.4834075 (2013)

    2013 doi

  32. [40]

    Gedeon, J

    J. Gedeon, J. Schmidt, M. J. P. Hodgson, J. Wetherell, C. L. Benavides-Riveros, and M. A. L. Marques, Machine learning the derivative discontinuity of density-functional theory, Machine Learning: Science and Technology 3, 015011 (2021)

    2021

  33. [41]

    Sarkar, Physics-informed neural networks for one-dimensional quantum well problems (2025), arXiv:2504.05367 [quant-ph]

    S. Sarkar, Physics-informed neural networks for one-dimensional quantum well problems (2025), arXiv:2504.05367 [quant-ph]

    2025 arXiv

  34. [42]

    Esaki and R

    L. Esaki and R. Tsu, Superlattice and negative differ- ential conductivity in semiconductors, IBM Journal of Research and Development 14, 61–65 (1970)

    1970

  35. [43]

    Harrison, Quantum Wells, Wires and Dots: Theoreti- cal and Computational Physics of Semiconductor Nanos- tructures (Wiley, 2005)

    P. Harrison, Quantum Wells, Wires and Dots: Theoreti- cal and Computational Physics of Semiconductor Nanos- tructures (Wiley, 2005)

    2005

  36. [44]

    Kasap and P

    S. Kasap and P. Capper, Springer Handbook of Electronic and Photonic Materials (Springer International Publish- ing, 2017)

    2017

  37. [45]

    Faist, F

    J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Quantum cascade laser, Sci- ence 264, 553–556 (1994)

    1994

  38. [46]

    Jirauschek and T

    C. Jirauschek and T. Kubis, Modeling techniques for quantum cascade lasers, Applied Physics Reviews 1, 011307 (2014)

    2014

  39. [47]

    M. T. Entwistle, M. J. P. Hodgson, J. Wetherell, B. Longstaff, J. D. Ramsden, and R. W. Godby, local density approximations from finite systems, Phys. Rev. B 94, 205134 (2016)

    2016

  40. [48]

    M. J. P. Hodgson, J. D. Ramsden, J. B. J. Chapman, P. Lillystone, and R. W. Godby, Exact time-dependent density-functional potentials for strongly correlated tun- neling electrons, Phys. Rev. B 88, 241102 (2013)

    2013

  41. [49]

    Burke, Perspective on density functional theory, The Journal of Chemical Physics 136, 10.1063/1.4704546 (2012)

    K. Burke, Perspective on density functional theory, The Journal of Chemical Physics 136, 10.1063/1.4704546 (2012)

    2012 doi

  42. [50]

    R. A. Bravo, K. Najafi, X. Gao, and S. F. Yelin, Quantum reservoir computing using arrays of rydberg atoms, PRX Quantum 3, 030325 (2022)

    2022

  43. [51]

    Browaeys and T

    A. Browaeys and T. Lahaye, Many-body physics with in- dividually controlled rydberg atoms, Nature Physics 16, 132–142 (2020)

    2020

  44. [52]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Probing many- body dynamics on a 51-atom quantum simulator, Nature 551, 579–584 (2017)

    2017

  45. [53]

    Silv´ erio, S

    H. Silv´ erio, S. Grijalva, C. Dalyac, L. Leclerc, P. J. Kar- alekas, N. Shammah, M. Beji, L.-P. Henry, and L. Hen- riet, Pulser: An open-source package for the design of pulse sequences in programmable neutral-atom arrays, Quantum 6, 629 (2022)

    2022

  46. [54]

    Blatt and C

    R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nature Physics 8, 277–284 (2012)

    2012

  47. [55]

    Bloch, J

    I. Bloch, J. Dalibard, and S. Nascimb` ene, Quantum sim- ulations with ultracold quantum gases, Nature Physics 8, 267–276 (2012)

    2012

  48. [56]

    Bentsen, T

    G. Bentsen, T. Hashizume, A. S. Buyskikh, E. J. Davis, A. J. Daley, S. S. Gubser, and M. Schleier-Smith, Treelike interactions and fast scrambling with cold atoms, Phys. Rev. Lett. 123, 130601 (2019)

    2019

  49. [57]

    Bogojeski, L

    M. Bogojeski, L. Vogt-Maranto, M. E. Tuckerman, K.-R. M¨ uller, and K. Burke, Quantum chemical accuracy from density functional approximations via machine learning, Nature Communications 11, 10.1038/s41467-020-19093- 1 (2020)

    2020 doi

  50. [58]

    Chuiko and P

    V. Chuiko and P. W. Ayers, Predicting energy of the quantum system from one- and two- electron integrals using deep learning (2025), arXiv:2504.03849 [quant-ph]

    2025

  51. [59]

    Patil, D

    S. Patil, D. Mondal, and R. Maitra, Machine learning approach towards quantum error mitigation for accurate molecular energetics (2025), arXiv:2504.07077 [quant- ph]

    2025 arXiv

  52. [60]

    Kr¨ amer, D

    S. Kr¨ amer, D. Plankensteiner, L. Ostermann, and H. Ritsch, Quantumoptics.jl: A julia framework for sim- ulating open quantum systems, Computer Physics Com- munications 227, 109–116 (2018)

    2018

  53. [61]

    Pulay, Convergence acceleration of iterative sequences

    P. Pulay, Convergence acceleration of iterative sequences. the case of scf iteration, Chemical Physics Letters73, 393 (1980)

    1980

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