Quantum-Enhanced Spectral Solution of the Poisson Equation

G. Intoccia; G. Pepe; S. Cuomo; U. Chirico

REVIEW 4 major objections 4 minor 29 references

Quantum-Enhanced Spectral Solution of the Poisson Equation

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A hybrid quantum-classical method estimates Poisson solution coefficients by measuring the Quantum Fourier Transform of the source and dividing by Laplacian eigenvalues.

desk verdict The accuracy comes from fitted per-source corrections, not from the QFT; the central claim of a general quantum spectral solver does not hold up, though the paper is a clear and honest benchmark of overhead. read the letter →

arxiv 2506.07743 v1 pith:2SY7A7VE submitted 2025-06-09 math.NA cs.ETcs.NA

G. Intoccia , U. Chirico , G. Pepe , S. Cuomo This is my paper

classification math.NAcs.ETcs.NA MSC 65N3565T5081P68

keywords PoissonequationQuantumFourierTransformspectralmethodDirichletboundaryconditionshybridquantum-classicalalgorithmcoefficientestimationcomputationalcomplexity

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 proposes and validates a hybrid quantum-classical solver for the two-dimensional Poisson equation with homogeneous Dirichlet conditions. Instead of computing Fourier integrals classically, it prepares the normalized source function as a quantum state, applies the Quantum Fourier Transform, and reads the probability of each frequency outcome to estimate the magnitudes of the sine-series coefficients. Dividing by the Laplacian eigenvalues reconstructs the solution as a sine series. On simulated hardware the method reaches mean squared errors between $3.20\times10^{-10}$ and $2.91\times10^{-6}$ for sine, polynomial-anisotropic, Gaussian, and combined sources, and the coefficient-estimation phase is about 94% faster than classical integration on a 256 by 256 grid. The paper presents this as evidence that a quantum-assisted route to PDE coefficient computation is feasible and scalable.

What carries the argument

The central object is the two-dimensional Quantum Fourier Transform, the quantum analogue of the discrete Fourier transform, applied to the discretized and normalized source state $|\psi\rangle$. After the transform, each basis state $|k,l\rangle$ is measured with probability $P(k,l)=|\tilde\psi_{kl}|^2$, so the magnitudes of the spectral coefficients are estimated as $\sqrt{C_{kl}/S}$ from shot counts $C_{kl}$ and total shots $S$. These magnitudes are turned into signed coefficients by a correction factor, then divided by the Laplacian eigenvalue $\lambda_{ij}=((i+1)\pi/L_x)^2+((j+1)\pi/L_y)^2$, and the solution is the sine-series sum of those quotients. The QFT is what converts an $O(2^{n+m})$ double-sum integration into an $O(n^2+m^2)$ circuit.

What would settle it

Run the method on a source not covered by the four test families, for example $f(x,y)=\sin(2\pi x)\sin(\pi y)+\sin(\pi x)\sin(3\pi y)$, using the correction formula in Eq. (20); if the reconstructed solution's mean squared error degrades by more than an order of magnitude relative to the reported $10^{-6}$ to $10^{-10}$ range, the correction is source-specific and the general claim fails.

Watch

Extended reading notes

Core claim

The central discovery is that the QFT measurement probabilities of the normalized source state already contain the spectral information needed to solve the Poisson equation, so the integration-heavy coefficient step can be replaced by counting measurement outcomes. The reconstructed solution is $u(x,y)=\sum_{i,j} \mathrm{Re}(a_{ij})/\lambda_{ij}\,\phi_{ij}(x,y)$, where $a_{ij}$ are the normalized square-root-count amplitudes and $\lambda_{ij}$ are the eigenvalues of the Laplacian. A per-source phase and spectral correction is applied to recover the sign information that squared magnitudes lose. The paper's numerical experiments show that this reconstruction matches the classical spectral solution to high accuracy across several families of source functions, and its benchmarks show that the quantum coefficient step is substantially cheaper in time and slightly cheaper in memory.

Load-bearing premise

The load-bearing premise is that the magnitudes of QFT measurement counts, plus the paper's fixed phase and spectral correction formulas, uniquely determine the signed sine-series coefficients for any source function.

Editorial extensions

If this is right

For large grids, the coefficient-estimation step stops being the computational bottleneck: the QFT circuit costs $O(n^2+m^2)$ instead of the $O(2^{n+m})$ double-sum or the FFT's $O(2^n\log 2^n)$ cost, and the benchmark shows a 93.9% reduction in that phase.
The classical solution-reconstruction step, not the quantum part, becomes the dominant cost, so truncating the sine series during reconstruction directly controls the remaining runtime.
If the measured magnitudes are accurate enough, the same eigenvalue-division scheme applies to other separable elliptic problems with known eigenfunctions, not only the rectangular Poisson case.
The reported memory footprint is about 14% lower overall on a 256 by 256 grid, with the largest saving in coefficient processing.
The accuracy and complexity results together suggest that the method is most attractive when the number of grid points is large enough that classical integration becomes prohibitive.

Reading between the lines

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

Because the method reads only magnitudes of the transformed amplitudes, the sign and phase recovery is a phase-retrieval problem in disguise; the per-source correction formulas suggest that a generic phase-recovery step would be needed before the method can be applied to arbitrary right-hand sides.
A natural next test is to replace the shot-count estimate $\sqrt{C_{kl}/S}$ with a quantum amplitude estimation procedure, which would reduce sampling noise at similar circuit depth and likely lower the achievable mean squared error.
The efficiency comparison uses simulated, effectively noiseless circuits; on real hardware, state preparation and measurement errors would add overhead that the current benchmarks do not price in.
The reported speedup is specific to the coefficient phase, and any total-speedup claim depends on amortizing state preparation, which the paper shows grows with qubit count.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First: the paper's main claim, that it estimates the series coefficients of the Poisson solution within a quantum framework, is not supported by what the experiments actually show. The reported MSEs are produced by per-source correction factors (Eqs. 20, 22, 24, 26) whose forms are fitted to each test function, not derived from the QFT or the Poisson equation. Measurement probabilities give only the magnitudes of the QFT amplitudes; the signs and phases of the true sine coefficients are not recoverable from those probabilities. So the coefficients in Eq. (18) are effectively rebuilt from the classical coefficients through an experimental fit. That is the load-bearing weakness, and the stress-test note is right.

Now the credit. The paper is clearly written and unusually honest about its own overhead: it includes state-preparation and correction times in the benchmark figures, acknowledges the simulation environment, and notes that future work is needed to compute the correction in quantum. The pedagogical structure, amplitude encoding, 2D QFT, measurement, eigenvalue division, is a standard but useful exercise. The benchmark data are reproducible from the text.

Soft spots beyond the central one. Table 1 compares QFT gate count (O(n^2+m^2)) against direct double integration (O(2^(n+m))) rather than FFT (O(N log N)), and it drops the costs of state preparation and measurement. The later figures partially compensate for that, but the table itself is misleading if read as a complexity claim. There is also no comparison to prior quantum PDE algorithms, so the novelty claim is overstated.

Is it worth your time? For someone teaching or learning quantum algorithms, this is a useful cautionary example of why Fourier sampling alone is not coefficient estimation. For a researcher working on quantum PDE solvers, it confirms what you already suspect. There is no general method here as stated; the fitted corrections mean the algorithm does not transfer to a new source. I would not cite it.

My recommendation: reject in current form. If the authors either derive a universal sign-recovery rule from the structure of the problem, or reframe the paper as a fitting exercise with a clear statement that no generality is claimed, revision could be considered. Otherwise it belongs in a workshop or classroom, not in a peer-reviewed journal. Yes, it deserves a serious referee, with this letter attached, so the referee knows exactly where the body is buried.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a hybrid quantum-classical spectral solver for the 2D Poisson equation with homogeneous Dirichlet boundary conditions. The source is normalized and encoded into a quantum state, a 2D QFT is applied, measurement counts are used to estimate magnitudes of Fourier coefficients, and the solution is reconstructed from sine modes divided by Laplacian eigenvalues. Numerical experiments on sinusoidal, anisotropic polynomial, Gaussian, and combined sources report MSEs between 3.20e-10 and 2.91e-6, and Section 4 claims large reductions in time and memory compared with a classical implementation. The paper concludes that the approach is a promising starting point for quantum-assisted PDE solvers.

Significance. Taken at face value, a general method that obtains spectral coefficients of the Poisson solution directly from QFT measurements with polylogarithmic complexity would be a significant result for quantum numerical analysis. The manuscript, however, does not establish such a general method: the link between measured probabilities and signed coefficients is completed by per-source correction factors (Eqs. 20, 22, 24, 26) that are fitted experimentally. The main contribution is therefore a set of benchmark-specific reconstructions, not a validated algorithm for arbitrary source terms. The paper is transparent about its simulated setting and about the need for classical correction and HPC interoperability, which is commendable, but those admissions underscore the missing theoretical core. No code or machine-checked proofs are provided.

major comments (4)

[Sec. 2.2, Eqs. (10)-(12); Sec. 3, Eqs. (20),(22),(24),(26)] The quantum measurement yields only probabilities P(k,l)=|ψ̃_kl|^2, from which only magnitudes |ψ̃_kl| are recovered. The signs and relative phases of the sine-series coefficients are not obtainable from these data. The paper compensates by multiplying the magnitudes by source-specific correction factors—e.g., exp(-iπ(m+n)/2)·(mn)^2/(m+n)^3 for sinusoidal sources, exp(-iπ(m+n))·(mn)^2/(m+n)^3 for the anisotropic polynomial, exp(-iπ mn/2) for the Gaussian, and exp(-iπ mn)·(mn)^2/(m^2+n^2)^{3/2} for the sum—described as "determined experimentally" with no derivation from the QFT or from the Poisson equation. Consequently the reported MSEs demonstrate that the coefficients can be fitted for the tested functions; they do not validate the claim that the method estimates coefficients for an arbitrary source f. Since the abstract and conclusions assert a general quantum-enhanced spectral method, this is the central load-bearing defect.
[Sec. 3, Eq. (19)] The "shifting procedure" k1 = m1 + shift(m1), with shift(m)=0 for m≤2 and shift(m)=m−2 otherwise, changes the actual source function for every m>2. The paper provides no justification for this shift beyond "to ensure correct harmonic retrieval," so the sinusoidal benchmarks do not correspond to a fixed family of sources under a single algorithm; the shift is another source-specific adjustment. A general method should either include a principled rule for choosing its parameters or state clearly that the algorithm requires per-problem calibration.
[Sec. 4.1-4.2, Table 1, Table 2, Figures 11-13] The complexity comparison is not made against the relevant classical baseline. Table 1 lists the FFT complexity O(2^n log 2^n + 2^m log 2^m), but the benchmarks in Table 2 and Figures 11-13 use a classical implementation based on "numerical approximation of the integral" (dblquad), which is the O(2^{n+m}) direct method quoted as inefficient. Comparing QFT time against dblquad time therefore inflates the reported 93.9% coefficient-calculation speedup. In addition, the quantum timing includes state preparation (3.34 s) but does not report the number of measurement shots, the QFT simulation overhead, or the cost of the correction step in all runs; without these, the claimed end-to-end speedup is not established.
[Eq. (18) and Sec. 3] The reconstruction formula u(x,y)=Σ Re(a_ij)/λ_ij φ_ij uses Re(a_ij), but at that point a_ij denotes the positive magnitudes from Eq. (13); the corrected complex coefficients B_mn are never explicitly substituted into Eq. (18). This makes it unclear whether the plotted solutions are reconstructed from Re(B_mn), |B_mn|, or some other combination, and prevents the reader from reproducing the MSEs. This is a clarity issue, but in combination with the fitted corrections it obscures what the numerical experiments actually validate.

minor comments (4)

[Eqs. (2)-(4), Eq. (17)] The indexing is inconsistent: Eq. (3) uses sin((k_x+1)πx/L_x), while Eq. (4) uses sin(k_x πx/L_x); Eq. (2) defines φ_{k_x k_y} without the +1, while Eq. (17) includes it. Please harmonize the notation.
[Table 1] The entry "Quasi linear" is inaccurate: O(n^2+m^2) is polynomial in the number of qubits and polylogarithmic in the grid dimension, not quasi-linear in the usual O(N log N) sense.
[Sec. 3] The paper does not report the number of shots used in the measurements or the qubit counts and truncation parameters (τ_x,τ_y) for each experiment, which are needed to assess the MSE values and the truncation claims in Figure 6.
[Sec. 3] There are several typos and stylistic issues, e.g., ",,so" in Section 3 and the inconsistent use of "anisotropic"/"anisotropy"; a careful proofread is needed.

Circularity Check

2 steps flagged · score 7.0 of 10

Measured probabilities supply only |a_kl|; the reconstruction uses signed Re(a_kl) with source-specific fitted phase/spectral corrections (Eqs. 20–26), so the claimed coefficient prediction reduces to per-benchmark fits.

fitted input called prediction [Section 2.2, Eqs. (10)–(18)]
"P(k, l) = |ψ̃_kl|² ... The magnitude of the frequencies amplitude is approximated by |ψ̃_kl| ≈ sqrt(C_kl/S) ... Since the coefficients a_ij can be complex due to the quantum state, we take their real part in the final reconstruction. u(x,y)=Σ Re(a_ij)/λ_ij φ_ij(x,y)."

The QFT measurement determines only the modulus-squared of transformed amplitudes. Eq. (18) nevertheless reconstructs the solution from signed Re(a_ij). No equation in the paper derives sign or phase information from the counts C_kl; the transition from |ψ̃_kl| to Re(a_ij) is not a derivation. All sign and relative-phase information has to be supplied externally by the per-case correction factors in Eqs. (20)–(26). Thus the coefficients are not estimated directly within the quantum framework; their signs are injected classically.
fitted input called prediction [Section 3, Eqs. (20), (22), (24), (26)]
"After obtaining the coefficients from Eq.(13), a correction was applied to address the loss of relative phase information after the measurements... Bmn = amn · exp(−iπ(m+n)/2) · (mn)²/(m+n)³ ... In this case, the optimal phase factor determined experimentally was found to be: Bmn = amn · exp(−iπmn/2) ... Bmn = amn · exp(−iπmn) (mn)²/(m²+n²)^{3/2}."

Each benchmark source is given its own phase factor and spectral rescaling, and for the Gaussian and sum cases the factor is explicitly determined experimentally. The factors are not derived from the QFT or from Δu=f, and they differ from one test to the next (Eq. 20 vs. Eq. 22 vs. Eq. 24 vs. Eq. 26). Multiplying the measured magnitude by a fitted phase and a fitted mode-dependent weight makes the final Bmn agree with the known classical coefficients for that particular f by construction. The reported MSEs therefore validate the fit for the benchmarked functions, not a transferable rule for arbitrary source functions.

full rationale

The paper does not rely on self-citations, and the spectral identity a_ij/λ_ij (Eq. 16) is the standard, non-circular Poisson-series relation. The circularity is located in the coefficient-estimation pipeline: the QFT measurement yields only probabilities P(k,l)=|ψ̃_kl|², so only magnitudes of the transformed amplitudes are accessible. The reconstructed solution (Eq. 18) requires signed Re(a_ij), and the missing signs and phases are installed through the source-dependent correction formulas in Eqs. (20), (22), (24), and (26), the last two of which are explicitly labeled as optimal phase factor determined experimentally. Because these corrections are fitted per benchmark function and differ between source classes, the claimed general method of estimating coefficients directly within the quantum framework reduces, for each demonstrated case, to fitting the classical solution. The complexity comparison in Section 4 also omits amplitude-encoding and shot-overhead costs, but that omission is a completeness issue rather than the central circularity. The central claim is partially circular: the reported accuracy is real for the fitted benchmarks, but it does not establish prediction for a new source. Score: 7.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The paper's accuracy hinges on multiple source-specific correction factors fitted to each test function; these are free parameters, not derived. Standard spectral theory provides the eigenfunctions, but the connection between QFT measurement outcomes and sine coefficients is asserted rather than demonstrated.

free parameters (6)

shift function shift(m) in Eq. (19) = 0 if m <= 2 else m - 2
Applied to target harmonic indices k1,k2 to make the sine modes appear at shifted frequencies in the QFT sampling; chosen to ensure correct harmonic retrieval, not derived.
Phase correction for sinusoidal sources Eq. (20) = exp(-i*pi*(m+n)/2)
Multiplied into measured coefficients to recover phase lost in measurement; chosen per test.
Spectral correction for sinusoidal sources Eq. (20) = (mn)^2/(m+n)^3
Ad hoc weight applied to match classical coefficients; not derived.
Phase correction for anisotropic polynomial source Eq. (22) = exp(-i*pi*(m+n))
Different from Eq. (20); determined for the anisotropic test.
Phase correction for Gaussian source Eq. (24) = exp(-i*pi*m*n/2)
Called optimal phase factor determined experimentally.
Correction for Gaussian+sin source Eq. (26) = exp(-i*pi*m*n) * (mn)^2/(m^2+n^2)^(3/2)
Another source-specific correction.

assumptions (4)

standard math Standard sine-series expansion of the Dirichlet Poisson solution
Used in Eq.(3) and eigenfunctions Eq.(2); standard separation of variables for rectangular domain.
domain assumption Arbitrary source f can be amplitude-encoded efficiently
Eqs.(6)-(7) assume normalized f can be loaded into quantum amplitudes; no cost or circuit for arbitrary f is analyzed, only simulated initialization.
domain assumption Measurement statistics from finite shots give reliable coefficient magnitudes
Eqs.(10)-(12) equate empirical frequencies with amplitudes, but no shot-count error analysis is given.
ad hoc to paper QFT sampled magnitudes correspond to sine coefficients after correction
Section 2.2 and Eq.(20) assume the DFT magnitudes from QFT plus correction recover the sine-series coefficients; never proven.

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Cite this review

Pith. "Pith review of Quantum-Enhanced Spectral Solution of the Poisson Equation." pith.science (2026). https://pith.science/paper/2SY7A7VE

@misc{pith2026250607743,
  author       = {Pith},
  title        = {Pith review of: Quantum-Enhanced Spectral Solution of the Poisson Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SY7A7VE}},
  note         = {Machine review of arXiv:2506.07743}
}

read the original abstract

We present a hybrid numerical-quantum method for solving the Poisson equation under homogeneous Dirichlet boundary conditions, leveraging the Quantum Fourier Transform (QFT) to enhance computational efficiency and reduce time and space complexity. This approach bypasses the integration-heavy calculations of classical methods, which have to deal with high computational costs for large number of points. The proposed method estimates the coefficients of the series expansion of the solution directly within the quantum framework. Numerical experiments validate its effectiveness and reveal significant improvements in terms of time and space complexity and solution accuracy, demonstrating the capability of quantum-assisted techniques to contribute in solving partial differential equations (PDEs). Despite the inherent challenges of quantum implementation, the present work serves as a starting point for future researches aimed at refining and expanding quantum numerical methods.

Figures

Figures reproduced from arXiv: 2506.07743 by the authors.

**Figure 1.** Results for (a) Left: (k1, k2) = (2, 2) and (b) Right: (k1, k2) = (3, 3). The mean squared errors (MSE) compared to the classical solution is mse = 3.20 × 10−10 and mse = 3.20 × 10−8 , respectively. Let’s consider the following function f(x, y) = x(1 − x)y(1 − y) Using the same correction from the Eq.(20) we obtain the following result. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 3.** Solutions for (a) Left: (k1, k2) = (3, 3) (b) Right: (k1, k2) = (4, 4). The results yielded MSEs of mse = 3.05 × 10−6 and mse = 2.91 × 10−6 , respectively. The Gaussian source represents a critical test case for estimating the relative phases lost during the quantum computation process. Let f(x, y) = exp −((x − x0) 2 + (y − y0) 2 ) (23) with (x0, y0) = (0.5, 0.5). To recover the correct solution, the coefficients … view at source ↗

**Figure 4.** Solution for a Gaussian function The resulting solution achieved mse = 9.31 × 10−6 . Let us now consider the following product as a known term f(x, y) = exp −((x − x0) 2 + (y − y0) 2 ) + sin(k1πx) sin(k2πy) (25) In this case the optimal phase factor determined experimentally was found to be: Bmn = amn · exp (−iπmn) (mn) 2 (m2 + n2 ) 3 2 (26) 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

Figures from the paper (11 more)

**Figure 5.** Figure 5: Solutions for (x0.y0) = (0.5, 0.5) (a) Left: (k1, k2) = (2, 2) (b) Right: (k1, k2) = (3, 3) The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Truncated series solutions for n,m number of qubits. In the subsequent analysis, we will consider the untruncated series to better observe the scaling behavior. The benchmark results demonstrate substantial improvements in time and space complexity when using the hybr…

**Figure 7.** Figure 7: Execution times breakdown for classical computation phases. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Execution times breakdown for hybrid-quantum computation. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: Memory usage distribution across classical computation phases. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: Memory usage distribution in hybrid-quantum computation. [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: Runtime Classical vs. QFT-based quantum coefficient computation. [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: The presence of the state preparation and initialize the circuit cause additional [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

**Figure 13.** Figure 13: Runtime comparison between the quantum (with correction phase) and classical [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

**Figure 14.** Figure 14: Memory usage is shown for the QFT step, excluding initialization and correc [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

**Figure 15.** Figure 15: Comparative memory footprint of hybrid quantum vs. classical methods. De [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

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Works this paper leans on

29 extracted references · 23 canonical work pages

[1]

What is the fast fourier transform? Proceedings of the IEEE, 55 0 (10): 0 1664--1674, 1967

William T Cochran, James W Cooley, David L Favin, Howard D Helms, Reginald A Kaenel, William W Lang, George C Maling, David E Nelson, Charles M Rader, and Peter D Welch. What is the fast fourier transform? Proceedings of the IEEE, 55 0 (10): 0 1664--1674, 1967

work page 1967
[2]

Practical quantum advantage in quantum simulation

Andrew J Daley, Immanuel Bloch, Christian Kokail, Stuart Flannigan, Natalie Pearson, Matthias Troyer, and Peter Zoller. Practical quantum advantage in quantum simulation. Nature, 607 0 (7920): 0 667--676, 2022

work page 2022
[3]

Methods of numerical integration

Philip J Davis and Philip Rabinowitz. Methods of numerical integration. Courier Corporation, 2007

work page 2007
[4]

Hybrid quantum-classical algorithms and quantum error mitigation

Suguru Endo, Zhenyu Cai, Simon C Benjamin, and Xiao Yuan. Hybrid quantum-classical algorithms and quantum error mitigation. Journal of the Physical Society of Japan, 90 0 (3): 0 032001, 2021

work page 2021
[5]

Lawrence C. Evans. Partial differential equations. American Mathematical Society, Providence, R.I., 2010. ISBN 9780821849743 0821849743

work page 2010
[6]

Quantum computer simulations at warp speed: Assessing the impact of gpu acceleration: A case study with ibm qiskit aer, nvidia thrust & cuquantum

Jennifer Faj, Ivy Peng, Jacob Wahlgren, and Stefano Markidis. Quantum computer simulations at warp speed: Assessing the impact of gpu acceleration: A case study with ibm qiskit aer, nvidia thrust & cuquantum. In 2023 IEEE 19th International Conference on e-Science (e-Science), pages 1--10. IEEE, 2023

work page 2023
[7]

Fft-based high order central difference schemes for three-dimensional poisson's equation with various types of boundary conditions

Hongsong Feng and Shan Zhao. Fft-based high order central difference schemes for three-dimensional poisson's equation with various types of boundary conditions. Journal of Computational Physics, 410: 0 109391, 2020

work page 2020
[8]

Statistical methods

Rudolf J Freund and William J Wilson. Statistical methods. Elsevier, 2003

work page 2003

Show all 29 references

[9]

Numerical analysis

Walter Gautschi. Numerical analysis. Springer Science & Business Media, 2011

2011
[10]

Numerical analysis of spectral methods: theory and applications

David Gottlieb and Steven A Orszag. Numerical analysis of spectral methods: theory and applications. SIAM, 1977

1977
[11]

Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D

Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J. Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D. Nation, Lev S. Bishop, Andrew W. Cross, Blake R. Johnson, and Jay M. Gambetta. Quantum computing with qiskit, 2024. URL https://arxiv.org/abs/2405.08810

2024 arXiv
[12]

A new method based on the laplace transform and fourier series for solving linear neutral delay differential equations

Gilbert Kerr, Gilberto Gonzalez-Parra, and Michelle Sherman. A new method based on the laplace transform and fourier series for solving linear neutral delay differential equations. Applied Mathematics and Computation, 420: 0 126914, 05 2022. doi:10.1016/j.amc.2021.126914

2022
[13]

Classical and quantum computation

Alexei Yu Kitaev, Alexander Shen, and Mikhail N Vyalyi. Classical and quantum computation. Number 47. American Mathematical Soc., 2002

2002
[14]

Computational integration

Arnold R Krommer and Christoph W Ueberhuber. Computational integration. SIAM, 1998

1998
[15]

Hybrid quantum computing

Seth Lloyd. Hybrid quantum computing. In Quantum information with continuous variables, pages 37--45. Springer, 2003

2003
[16]

D. R. Musk. A comparison of quantum and traditional fourier transform computations. Computing in Science & Engineering, 22 0 (6): 0 103--110, 2020. doi:10.1109/MCSE.2020.3023979

2020
[17]

Laplacian spectral basis functions

Giuseppe Patan \`e . Laplacian spectral basis functions. Computer aided geometric design, 65: 0 31--47, 2018

2018
[18]

Numerical solution for two dimensional laplace equation with dirichlet boundary conditions

Parag V Patil and Dr JSVR Krishna Prasad. Numerical solution for two dimensional laplace equation with dirichlet boundary conditions. IOSR Journal of Mathematics, 6 0 (4): 0 66--75, 2013

2013
[19]

Quadpack: a subroutine package for automatic integration, volume 1

Robert Piessens, Elise de Doncker-Kapenga, Christoph W \"U berhuber, and David K Kahaner. Quadpack: a subroutine package for automatic integration, volume 1. Springer Science & Business Media, 2012

2012
[20]

Fourier spectral embedded boundary solution of the poisson’s and laplace equations with dirichlet boundary conditions

Feriedoun Sabetghadam, Shervin Sharafatmandjoor, and Farhang Norouzi. Fourier spectral embedded boundary solution of the poisson’s and laplace equations with dirichlet boundary conditions. Journal of Computational Physics, 228 0 (1): 0 55--74, 2009

2009
[21]

Mathematics of quantum computing, volume 11

Wolfgang Scherer. Mathematics of quantum computing, volume 11. Springer, 2019

2019
[22]

Spectral methods: algorithms, analysis and applications

Jie Shen, Tao Tang, and Li-Lian Wang. Spectral methods: algorithms, analysis and applications. 41, 2011

2011
[23]

The numerical solution of laplace's equation

George H Shortley and Royal Weller. The numerical solution of laplace's equation. Journal of Applied Physics, 9 0 (5): 0 334--348, 1938

1938
[24]

Quantum computing

Andrew Steane. Quantum computing. Reports on Progress in Physics, 61 0 (2): 0 117, 1998

1998
[25]

W.A. Strauss. Partial Differential Equations: An Introduction. Wiley, 2007. ISBN 9780470054567. URL https://books.google.it/books?id=m2hvDwAAQBAJ

2007
[26]

From finite differences to finite elements a short history of numerical analysis of partial differential equations

Vidar Thom \'e e. From finite differences to finite elements a short history of numerical analysis of partial differential equations. In Numerical analysis: Historical developments in the 20th century, pages 361--414. Elsevier, 2001

2001
[27]

Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory. Implementation of the quantum fourier transform. Phys. Rev. Lett., 86: 0 1889--1891, Feb 2001. doi:10.1103/PhysRevLett.86.1889. URL https://link.aps.org/doi/10.1103/PhysRevLett.86.1889

2001 doi
[28]

The heat equation, volume 67

David Vernon Widder. The heat equation, volume 67. Academic Press, 1976

1976
[29]

Williams

Colin P. Williams. Quantum Gates, pages 51--122. Springer London, London, 2011. ISBN 978-1-84628-887-6. doi:10.1007/978-1-84628-887-6_2. URL https://doi.org/10.1007/978-1-84628-887-6_2

2011 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

What is the fast fourier transform? Proceedings of the IEEE, 55 0 (10): 0 1664--1674, 1967

William T Cochran, James W Cooley, David L Favin, Howard D Helms, Reginald A Kaenel, William W Lang, George C Maling, David E Nelson, Charles M Rader, and Peter D Welch. What is the fast fourier transform? Proceedings of the IEEE, 55 0 (10): 0 1664--1674, 1967

work page 1967

[2] [2]

Practical quantum advantage in quantum simulation

Andrew J Daley, Immanuel Bloch, Christian Kokail, Stuart Flannigan, Natalie Pearson, Matthias Troyer, and Peter Zoller. Practical quantum advantage in quantum simulation. Nature, 607 0 (7920): 0 667--676, 2022

work page 2022

[3] [3]

Methods of numerical integration

Philip J Davis and Philip Rabinowitz. Methods of numerical integration. Courier Corporation, 2007

work page 2007

[4] [4]

Hybrid quantum-classical algorithms and quantum error mitigation

Suguru Endo, Zhenyu Cai, Simon C Benjamin, and Xiao Yuan. Hybrid quantum-classical algorithms and quantum error mitigation. Journal of the Physical Society of Japan, 90 0 (3): 0 032001, 2021

work page 2021

[5] [5]

Lawrence C. Evans. Partial differential equations. American Mathematical Society, Providence, R.I., 2010. ISBN 9780821849743 0821849743

work page 2010

[6] [6]

Quantum computer simulations at warp speed: Assessing the impact of gpu acceleration: A case study with ibm qiskit aer, nvidia thrust & cuquantum

Jennifer Faj, Ivy Peng, Jacob Wahlgren, and Stefano Markidis. Quantum computer simulations at warp speed: Assessing the impact of gpu acceleration: A case study with ibm qiskit aer, nvidia thrust & cuquantum. In 2023 IEEE 19th International Conference on e-Science (e-Science), pages 1--10. IEEE, 2023

work page 2023

[7] [7]

Fft-based high order central difference schemes for three-dimensional poisson's equation with various types of boundary conditions

Hongsong Feng and Shan Zhao. Fft-based high order central difference schemes for three-dimensional poisson's equation with various types of boundary conditions. Journal of Computational Physics, 410: 0 109391, 2020

work page 2020

[8] [8]

Statistical methods

Rudolf J Freund and William J Wilson. Statistical methods. Elsevier, 2003

work page 2003