pith. sign in

arxiv: 2605.29181 · v1 · pith:J72NHMRWnew · submitted 2026-05-27 · 🪐 quant-ph · cs.NA· math.NA· physics.comp-ph

A Variational Quantum Algorithm for Nonlinear Finite Element Analysis of Hyperelastic Materials

Uditnarayan Kouskiya , Caglar Oskay This is my paper

Pith reviewed 2026-06-29 11:09 UTC · model grok-4.3

classification 🪐 quant-ph cs.NAmath.NAphysics.comp-ph
keywords variational quantum algorithmhyperelastic materialsfinite element analysisnonlinear elasticityNISQ devicesNeoHookean modelpolynomial approximationpotential energy minimization
0
0 comments X

The pith

A hybrid quantum-classical method uses polynomial approximations to solve nonlinear hyperelastic finite element problems on NISQ devices.

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

The paper develops a variational quantum formulation for nonlinear elasticity in hyperelastic materials by leveraging the potential energy structure and hybrid optimization. Polynomial approximations convert the nonlinear strain energy density into a form that parameterized quantum circuits can evaluate. The approach is shown to work on a one-dimensional NeoHookean model discretized with first- and second-order finite elements under nonhomogeneous boundary conditions, with experiments tracking how approximation order trades off accuracy against computational cost.

Core claim

By introducing polynomial approximations of the nonlinear strain energy density, the energy functional of hyperelastic finite element problems becomes compatible with variational quantum algorithms; the resulting hybrid quantum-classical optimization recovers approximate solutions for the one-dimensional NeoHookean bar using both linear and quadratic shape functions and nonhomogeneous boundary data.

What carries the argument

Hybrid variational optimization of a polynomially approximated strain-energy functional evaluated by parameterized quantum circuits on finite-element discretizations of hyperelastic materials.

If this is right

  • The method accommodates nonhomogeneous boundary conditions within the variational quantum framework.
  • Both first- and second-order finite-element shape functions can be used without breaking compatibility with the quantum circuit representation.
  • Accuracy and efficiency vary systematically with the degree of the polynomial approximation to the strain energy.
  • The formulation demonstrates practical feasibility on near-term quantum hardware for at least this class of one-dimensional nonlinear problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the polynomial approximation technique generalizes without prohibitive growth in circuit depth, similar variational quantum formulations could be attempted for two- and three-dimensional hyperelastic problems.
  • The same energy-based reduction might apply to other nonlinear constitutive models whose strain-energy functions admit low-order polynomial fits.
  • Scalability tests on larger meshes would reveal whether the hybrid loop remains tractable once the number of degrees of freedom exceeds the capacity of current NISQ devices.

Load-bearing premise

Polynomial approximations of the nonlinear strain energy density remain accurate enough to preserve the essential physics of hyperelasticity.

What would settle it

Numerical comparison of the quantum-optimized nodal displacements against a classical nonlinear finite-element solver for the same 1D NeoHookean bar, with the discrepancy growing beyond a fixed tolerance as the polynomial degree is increased.

Figures

Figures reproduced from arXiv: 2605.29181 by Caglar Oskay, Uditnarayan Kouskiya.

Figure 1
Figure 1. Figure 1: VQA Framework. 2.3 Variational Quantum Algorithm The overall structure of the proposed VQA framework is depicted in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Approximations to the logarithmic nonlinearity. The Taylor series expansion is valid only [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Primitive circuits to evaluate (a) ⟨pˆ| qˆ⟩, (b) ⟨pˆ|Dqˆ|pˆ⟩, (c) ⟨pˆ|Ab|pˆ⟩, and (d) ∥ℓp∥ 2 ∥ℓq∥ ⟨ˆℓp|Dℓˆq | ˆℓp⟩; The green, red, and purple regions denote the input, QNPU, and block￾encoding phases, respectively. In (c), the two red qubit registers correspond to the ancillas required by the Adder circuit. In (d), ancillas associated with the block-encoding components are omitted for clarity. The action … view at source ↗
Figure 4
Figure 4. Figure 4: Scaling of QNPU circuit depth (including block-encoding where applicable) with the [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: VQA performance (20-run average) for Taylor orders 3–5 (T3–T5) and IHT (P3). The [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance of different models as a function of [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: VQA scaling for n = 3, 4, and 5 qubits using a third-order Taylor expansion (20-run average). The profiles of u V Q (in mm) and the iteration-cost histories are shown in (a) and (b), respectively. Performance metrics are summarized in Table (c). Vertical dotted lines in (b) indicate the average convergence iterations Navg for the minimum ansatz layer depth required by each value of n. reported metrics (see… view at source ↗
Figure 8
Figure 8. Figure 8: VQA performance comparison for different formulations (20-run average). The formula [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
read the original abstract

This manuscript explores a variational quantum formulation for nonlinear elasticity problems arising from hyperelastic material models, targeting near term noisy intermediate scale quantum (NISQ) devices. The approach leverages the potential energy structure of hyperelasticity and employs a hybrid quantum classical framework in which the energy functional is evaluated using parameterized quantum circuits and optimized through classical routines. To enable implementation on current quantum hardware, polynomial approximations of the nonlinear strain energy density are introduced, yielding a representation compatible with variational quantum algorithms. The methodology is demonstrated on a one dimensional NeoHookean material model using finite element discretizations with first and second order shape functions and nonhomogeneous boundary conditions. Numerical experiments investigate the influence of the polynomial approximation order on the accuracy and efficiency of the proposed approach, illustrating its feasibility for near term quantum devices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a hybrid quantum-classical variational algorithm for nonlinear finite element analysis of hyperelastic materials. It approximates the nonlinear strain-energy density by polynomials to enable evaluation on NISQ hardware, then minimizes the resulting energy functional via parameterized quantum circuits and classical optimization. The method is illustrated on a one-dimensional Neo-Hookean bar discretized with linear and quadratic elements subject to non-homogeneous boundary conditions; numerical experiments examine the effect of polynomial order on solution accuracy and computational cost.

Significance. A working demonstration on even a simplified 1D hyperelastic problem would constitute a first step toward quantum-assisted nonlinear mechanics. The polynomial approximation strategy is a concrete technical contribution that preserves the variational structure while remaining compatible with existing VQA primitives. However, the absence of reported error norms, convergence rates, or classical benchmarks in the supplied description limits the immediate significance; if such metrics were added and showed acceptable accuracy, the work would usefully document feasibility for near-term devices.

major comments (2)
  1. [Abstract, §4] Abstract and §4 (numerical experiments): the feasibility claim rests on numerical experiments that investigate polynomial order, yet no quantitative error metrics (e.g., L2 displacement error, energy error), convergence tables, or direct comparisons against classical FEM solvers are supplied. Without these data the demonstration remains qualitative and does not yet substantiate the accuracy/efficiency statements.
  2. [§3.2] §3.2 (polynomial approximation): the claim that the polynomial representation “preserves the essential physics” is asserted but not accompanied by an a-priori error bound or a numerical study of how the approximation error propagates into the finite-element solution for the chosen Neo-Hookean model. A concrete bound or at least tabulated residual norms versus polynomial degree would be required to make the approximation step load-bearing rather than heuristic.
minor comments (2)
  1. [§2] Notation for the strain-energy density W and its polynomial surrogate W_p should be introduced once and used consistently; the current text alternates between symbols without explicit definition.
  2. [§4] Figure captions for the 1D bar results should state the element type, polynomial degree, and boundary conditions used in each panel to allow immediate comparison with the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments point by point below and will revise the manuscript to incorporate additional quantitative analysis.

read point-by-point responses

  1. Referee: [Abstract, §4] Abstract and §4 (numerical experiments): the feasibility claim rests on numerical experiments that investigate polynomial order, yet no quantitative error metrics (e.g., L2 displacement error, energy error), convergence tables, or direct comparisons against classical FEM solvers are supplied. Without these data the demonstration remains qualitative and does not yet substantiate the accuracy/efficiency statements.

    Authors: We agree that the current presentation of the numerical results in §4 is primarily qualitative. In the revised manuscript we will add explicit L2 displacement and total-energy error norms, tabulated convergence data versus polynomial degree, and side-by-side comparisons with classical FEM solutions obtained with the same discretizations. These additions will be placed in a new subsection of §4 and referenced from the abstract. revision: yes

  2. Referee: [§3.2] §3.2 (polynomial approximation): the claim that the polynomial representation “preserves the essential physics” is asserted but not accompanied by an a-priori error bound or a numerical study of how the approximation error propagates into the finite-element solution for the chosen Neo-Hookean model. A concrete bound or at least tabulated residual norms versus polynomial degree would be required to make the approximation step load-bearing rather than heuristic.

    Authors: Section 4 already contains numerical experiments that track solution accuracy as a function of polynomial degree for the Neo-Hookean model. To strengthen the theoretical grounding we will add (i) a short a-priori error estimate based on standard polynomial approximation theory for the strain-energy density and (ii) a table of residual norms (difference between exact and approximated energy density) versus polynomial degree, together with a brief discussion of how these residuals affect the finite-element solution. These items will be inserted into §3.2. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper introduces an explicit polynomial approximation to the nonlinear strain-energy density as a design choice to enable VQA compatibility on NISQ hardware, then demonstrates the resulting hybrid method numerically on a 1D Neo-Hookean bar using standard FEM shape functions and boundary conditions. The central result is a feasibility illustration whose accuracy is assessed by direct comparison to classical solutions; no step equates a fitted parameter to a claimed prediction by construction, no uniqueness theorem is imported from self-citation, and the optimization loop remains a standard VQA-classical FEM hybrid without self-referential reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from variational quantum algorithms and finite-element theory together with the modeling choice of polynomial approximation; no new entities are postulated.

free parameters (1)
  • polynomial approximation order
    Chosen to trade accuracy against quantum-circuit depth; value not fixed by first principles.
axioms (2)
  • domain assumption The hyperelastic potential energy functional can be evaluated to sufficient accuracy by a parameterized quantum circuit
    Invoked to justify the hybrid quantum-classical loop.
  • domain assumption Polynomial truncation of the strain-energy density preserves the essential nonlinear response for the chosen finite-element discretizations
    Required for the approximated problem to remain physically meaningful.

pith-pipeline@v0.9.1-grok · 5673 in / 1258 out tokens · 51789 ms · 2026-06-29T11:09:19.134000+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 26 canonical work pages · 4 internal anchors

  1. [1]

    A data compressor for FPGA-based state vector quantum simulators

    Kaijie Wei et al. “A data compressor for FPGA-based state vector quantum simulators”. In:Proceedings of the 14th International Symposium on Highly Efficient Accelerators and Reconfigurable Technologies. HEART ’24. Porto, Portugal: Association for Computing Ma- chinery, 2024, pp. 63–70.isbn: 9798400717277.doi:10.1145/3665283.3665293.url:https: //doi.org/10...

    work page doi:10.1145/3665283.3665293.url:https: 2024

  2. [2]

    A Survey of Important Issues in Quantum Com- puting and Communications

    Zebo Yang, Maede Zolanvari, and Raj Jain. “A Survey of Important Issues in Quantum Com- puting and Communications”. In:IEEE Communications Surveys & Tutorials25.2 (2023). A broad survey of architectures, algorithms, and challenges in quantum computing and com- munication., pp. 1059–1100.doi:10.1109/COMST.2023.3241234

    work page doi:10.1109/comst.2023.3241234 2023

  3. [3]

    Improved Quantum Algorithms for Linear and Nonlinear Differential Equations

    Andrew M. Childs, Jin-Peng Liu, and Aaron Ostrander. “Improved Quantum Algorithms for Linear and Nonlinear Differential Equations”. In:Quantum7 (2023). High-level development of quantum algorithms for both linear and nonlinear differential equations., p. 913.doi: 10.22331/q-2023-02-02-913

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

  4. [4]

    Review and perspectives in quantum computing for partial differential equations in structural mechanics

    Giorgio Tosti Balducci et al. “Review and perspectives in quantum computing for partial differential equations in structural mechanics”. In:Frontiers in Mechanical EngineeringVol- ume 8 - 2022 (2022).issn: 2297-3079.doi:10 . 3389 / fmech . 2022 . 914241.url:https : / / www . frontiersin . org / journals / mechanical - engineering / articles / 10 . 3389 / ...

    work page arXiv 2022

  5. [5]

    Quantum algorithm for linear differential equations with exponentially improved dependence on precision

    Dominic W. Berry.Quantum algorithm for linear differential equations. arXiv:1701.03684. 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  6. [6]

    Quantum Algorithms for Linear and Nonlinear Differential Equations

    Jin-Peng Liu. “Quantum Algorithms for Linear and Nonlinear Differential Equations”. PhD thesis. University of Maryland, 2022

    2022

  7. [7]

    Quantum Algorithm for Linear Systems of Equations

    Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. “Quantum Algorithm for Linear Systems of Equations”. In:Physical Review Letters103.15 (2009), p. 150502.doi:10.1103/ PhysRevLett.103.150502

    2009

  8. [8]

    Universal Quantum Simulators

    Seth Lloyd. “Universal Quantum Simulators”. In:Science273.5278 (1996), pp. 1073–1078. doi:10.1126/science.273.5278.1073

    work page doi:10.1126/science.273.5278.1073 1996

  9. [9]

    Hamiltonian simulation using linear combinations of unitary operations

    Dominic W. Berry and Andrew M. Childs. “Hamiltonian simulation using linear combinations of unitary operations”. In:Quantum Information and Computation12 (2012), pp. 29–62

    2012

  10. [10]

    A Grand Unification of Quantum Algorithms

    John M. Martyn et al. “A Grand Unification of Quantum Algorithms”. In:PRX Quantum2 (2021), p. 040203.doi:10.1103/PRXQuantum.2.040203

    work page doi:10.1103/prxquantum.2.040203 2021

  11. [11]

    Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Computing

    Yigit Suba¸ sı, Rolando D. Somma, and Davide Orsucci. “Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Computing”. In:Physical Review Letters 122.6 (2019), p. 060504.doi:10.1103/PhysRevLett.122.060504

    work page doi:10.1103/physrevlett.122.060504 2019

  12. [12]

    Quantum algorithm and circuit design solving the Poisson equation

    Yudong Cao et al. “Quantum algorithm and circuit design solving the Poisson equation”. In: New Journal of Physics15.1 (2013), p. 013021.doi:10.1088/1367-2630/15/1/013021

    work page doi:10.1088/1367-2630/15/1/013021 2013

  13. [13]

    Quantum algorithms and the finite element method

    Ashley Montanaro and Sam Pallister. “Quantum algorithms and the finite element method”. In:Physical Review A93.3 (2016), p. 032324.doi:10.1103/PhysRevA.93.032324

    work page doi:10.1103/physreva.93.032324 2016

  14. [14]

    Quantum computing in the nisq era and beyond.Quantum2, 79 (2018)

    John Preskill. “Quantum Computing in the NISQ era and beyond”. In:Quantum2 (2018), p. 79.doi:10.22331/q-2018-08-06-79. 32

    work page internal anchor Pith review doi:10.22331/q-2018-08-06-79 2018

  15. [15]

    Noisy intermediate-scale quantum algorithms

    Kishor Bharti et al. “Noisy intermediate-scale quantum algorithms”. In:Rev. Mod. Phys.94 (1 Feb. 2022), p. 015004.doi:10.1103/RevModPhys.94.015004.url:https://link.aps. org/doi/10.1103/RevModPhys.94.015004

    work page doi:10.1103/revmodphys.94.015004.url:https://link.aps 2022

  16. [16]

    A variational eigenvalue solver on a photonic quantum processor

    Alberto Peruzzo et al. “A variational eigenvalue solver on a photonic quantum processor”. In:Nature Communications5.1 (2014), p. 4213.doi:10.1038/ncomms5213.url:https: //doi.org/10.1038/ncomms5213

    work page doi:10.1038/ncomms5213.url:https: 2014

  17. [17]

    Variational Quantum Linear Solver

    Carlos Bravo-Prieto et al. “Variational Quantum Linear Solver”. In:Quantum7 (Nov. 2023), p. 1188.doi:10.22331/q-2023-11-22-1188.url:https://doi.org/10.22331/q-2023- 11-22-1188

    work page doi:10.22331/q-2023-11-22-1188.url:https://doi.org/10.22331/q-2023- 2023

  18. [18]

    An implementation of the finite element method in hybrid classical/quantum computers

    A. Arora, B. M. Ward, and C. Oskay. “An implementation of the finite element method in hybrid classical/quantum computers”. In:Finite Elements in Analysis and Design248 (2025), p. 104354

    2025

  19. [19]

    Performance Study of Variational Quantum Linear Solver for Linear Elastic Problems

    Xiang Rao and Kou Du. “Performance Study of Variational Quantum Linear Solver for Linear Elastic Problems”. In:Computational and Experimental Simulations in Engineering. Ed. by Kun Zhou. Vol. 168. Mechanisms and Machine Science. Cham: Springer, 2024, pp. 80–94. isbn: 978-3-031-68775-4.doi:10.1007/978-3-031-68775-4_6

    work page doi:10.1007/978-3-031-68775-4_6 2024

  20. [20]

    Divergence-free algorithms for solving nonlinear differential equations on quantum computers

    Katsuhiro Endo and Kazuaki Z. Takahashi. “Divergence-free algorithms for solving nonlinear differential equations on quantum computers”. In:Phys. Rev. Res.8 (1 Jan. 2026), p. 013057. doi:10.1103/2hfp-qyz1.url:https://link.aps.org/doi/10.1103/2hfp-qyz1

    work page doi:10.1103/2hfp-qyz1.url:https://link.aps.org/doi/10.1103/2hfp-qyz1 2026

  21. [21]

    A quantum nonlinear solver based on the asymptotic numerical method

    Yongchun Xu et al. “A robust quantum nonlinear solver based on the asymptotic numerical method”. In:arXiv preprint arXiv:2412.03939(2024).doi:10.48550/arXiv.2412.03939. url:https://arxiv.org/abs/2412.03939

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2412.03939 2024

  22. [22]

    Variational quantum algorithms for nonlinear problems

    Michael Lubasch et al. “Variational quantum algorithms for nonlinear problems”. In:Phys. Rev. A101 (1 Jan. 2020), p. 010301.doi:10.1103/PhysRevA.101.010301.url:https: //link.aps.org/doi/10.1103/PhysRevA.101.010301

    work page doi:10.1103/physreva.101.010301.url:https: 2020

  23. [23]

    Quantum variational solving of nonlinear and multidimensional par- tial differential equations

    Abhijat Sarma et al. “Quantum variational solving of nonlinear and multidimensional par- tial differential equations”. In:Phys. Rev. A109 (6 June 2024), p. 062616.doi:10.1103/ PhysRevA.109.062616.url:https://link.aps.org/doi/10.1103/PhysRevA.109.062616

    work page doi:10.1103/physreva.109.062616 2024

  24. [24]

    Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation

    Yuki Sato et al. “Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation”. In:Phys. Rev. A104 (5 Nov. 2021), p. 052409.doi:10.1103/ PhysRevA.104.052409.url:https://link.aps.org/doi/10.1103/PhysRevA.104.052409

    work page doi:10.1103/physreva.104.052409 2021

  25. [25]

    Boundary treatment for variational quantum simulations of partial differ- ential equations on quantum computers

    Paul Over et al. “Boundary treatment for variational quantum simulations of partial differ- ential equations on quantum computers”. In:Computers and Fluids288 (2025), p. 106508. issn: 0045-7930.doi:https : / / doi . org / 10 . 1016 / j . compfluid . 2024 . 106508.url: https://www.sciencedirect.com/science/article/pii/S0045793024003396

    2025

  26. [26]

    Quantum time-marching algorithms for solving linear transport problems including boundary conditions

    Sergio Bengoechea, Paul Over, and Thomas Rung. “Quantum time-marching algorithms for solving linear transport problems including boundary conditions”. In:arXiv preprint arXiv:2511.04271(2025).doi:10.48550/arXiv.2511.04271. arXiv:2511.04271 [quant-ph]. url:https://arxiv.org/abs/2511.04271

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2511.04271 2025

  27. [27]

    Dover Publications, 1994

    Jerrold E Marsden and Thomas JR Hughes.Mathematical Foundations of Elasticity. Dover Publications, 1994. 33

    1994

  28. [28]

    Expressibility and Entangling Ca- pability of Parameterized Quantum Circuits for Hybrid Quantum-Classical Algorithms

    Sukin Sim, Peter D. Johnson, and Al´ an Aspuru-Guzik. “Expressibility and Entangling Ca- pability of Parameterized Quantum Circuits for Hybrid Quantum-Classical Algorithms”. In: Advanced Quantum Technologies2.12 (2019), p. 1900070.doi:https://doi.org/10.1002/ qute.201900070. eprint:https://advanced.onlinelibrary.wiley.com/doi/pdf/10. 1002/qute.201900070.u...

    2019

  29. [29]

    Rattew and Patrick Rebentrost.Non-Linear Transformations of Quantum Ampli- tudes: Exponential Improvement, Generalization, and Applications

    Arthur G. Rattew and Patrick Rebentrost.Non-Linear Transformations of Quantum Ampli- tudes: Exponential Improvement, Generalization, and Applications. 2023. arXiv:2309.09839 [quant-ph].url:https://arxiv.org/abs/2309.09839

    work page arXiv 2023

  30. [30]

    Stegun.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

    Milton Abramowitz and Irene A. Stegun.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Vol. 55. Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964

    1964

  31. [31]

    Karel Rektorys.Variational Methods in Mathematics, Science and Engineering. 2nd ed. Dor- drecht: D. Reidel Publishing Company, 1977

    1977

  32. [32]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang.Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge, United Kingdom: Cambridge University Press, 2010. isbn: 978-1-107-00217-3

    2010

  33. [33]

    Quantum lower bounds by polynomials

    Robert Beals et al. “Quantum lower bounds by polynomials”. In:J. ACM48.4 (July 2001), pp. 778–797.issn: 0004-5411.doi:10.1145/502090.502097.url:https://doi.org/10. 1145/502090.502097

    work page doi:10.1145/502090.502097.url:https://doi.org/10 2001

  34. [34]

    Explicit Quantum Circuits for Block Encodings of Certain Sparse Ma- trices

    Daan Camps et al. “Explicit Quantum Circuits for Block Encodings of Certain Sparse Ma- trices”. In:SIAM Journal on Matrix Analysis and Applications45.1 (Mar. 2024).doi:10. 1137/22m1484298

    2024

  35. [35]

    Direct measurement of the quantum wavefunction

    Jeff S. Lundeen et al. “Direct measurement of the quantum wavefunction”. In:Nature474.7350 (2011), pp. 188–191.doi:10 . 1038 / nature10120.url:https : / / doi . org / 10 . 1038 / nature10120

    2011

  36. [36]

    Direct measurement of large-scale quantum states via expectation values of non-Hermitian matrices

    Eliot Bolduc, Genevieve Gariepy, and Jonathan Leach. “Direct measurement of large-scale quantum states via expectation values of non-Hermitian matrices”. In:Nature Communica- tions7.1 (2016), p. 10439.doi:10.1038/ncomms10439.url:https://doi.org/10.1038/ ncomms10439

    work page doi:10.1038/ncomms10439.url:https://doi.org/10.1038/ 2016

  37. [37]

    Simple quantum algorithm to efficiently prepare sparse states

    Debora Ramacciotti, Andreea I. Lefterovici, and Antonio F. Rotundo. “Simple quantum algorithm to efficiently prepare sparse states”. In:Phys. Rev. A110 (3 Sept. 2024), p. 032609. doi:10 . 1103 / PhysRevA . 110 . 032609.url:https : / / link . aps . org / doi / 10 . 1103 / PhysRevA.110.032609

    2024

  38. [38]

    Quantum state preparation via piecewise QSVT

    Oliver O’Brien and Christoph S¨ underhauf. “Quantum state preparation via piecewise QSVT”. In:Quantum9 (July 2025), p. 1786.issn: 2521-327X.doi:10.22331/q-2025-07-03-1786. url:https://doi.org/10.22331/q-2025-07-03-1786

    work page doi:10.22331/q-2025-07-03-1786 2025

  39. [39]

    Evaluating analytic gradients on quantum hardware

    Maria Schuld et al. “Evaluating analytic gradients on quantum hardware”. In:Phys. Rev. A 99 (3 Mar. 2019), p. 032331.doi:10.1103/PhysRevA.99.032331.url:https://link.aps. org/doi/10.1103/PhysRevA.99.032331

    work page doi:10.1103/physreva.99.032331.url:https://link.aps 2019

  40. [40]

    A new approach to variable metric algorithms

    R. Fletcher. “A new approach to variable metric algorithms”. In:The Computer Journal13.3 (1970), pp. 317–322

    1970

  41. [41]

    Nocedal and S

    J. Nocedal and S. J. Wright.Numerical Optimization. 2nd ed. Springer, 2006. 34

    2006