pith. sign in

arxiv: 2506.13534 · v3 · pith:3HULVF5Snew · submitted 2025-06-16 · 🪐 quant-ph

Quantum algorithm for solving generalized eigenvalue problems with application to the Schr\"odinger equation

Grzegorz Rajchel-Mieldzio\'c , Szymon Pli\'s , Emil Zak This is my paper

Pith reviewed 2026-05-22 13:28 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum algorithmgeneralized eigenvalue problemSchrödinger equationpseudospectral collocationsingular value spectrumamplitude amplificationphase estimationquantum chemistry
0
0 comments X

The pith

A quantum algorithm estimates eigenvalues by locating minima in the singular value spectrum of parameterized matrix families.

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

The paper develops a quantum method to solve generalized eigenvalue problems by scanning a parameter and using amplitude amplification with phase estimation to find where the smallest singular value reaches zero. This is applied to formulate a quantum version of the pseudospectral collocation method for the Schrödinger equation. The approach avoids explicit matrix inversion and its associated instability. Resource estimates show scaling up to square root of the problem size N for well-behaved potentials, compared to linear scaling classically. The method targets computation of multiple eigenvalues in dense spectra for molecular and solid-state systems.

Core claim

The authors claim that eigenvalues of a parameterized matrix family can be identified by using quantum amplitude amplification and phase estimation to minimize the singular value spectrum, with the minimum reaching zero precisely at eigenvalue locations. Applied to the Schrödinger equation, this yields a quantum collocation algorithm whose fault-tolerant resource costs scale as the square root of N rather than linearly for certain potentials, while bypassing the numerical issues of inverting the overlap matrix in the generalized eigenvalue problem.

What carries the argument

Quantum amplitude amplification combined with phase estimation applied to the singular value spectrum of a parameterized matrix family to detect exact zero minima.

If this is right

  • The quantum collocation method achieves up to square-root scaling in problem size N for well-behaved potentials.
  • The algorithm avoids numerical instability by scanning the parameter instead of inverting a matrix.
  • It supports extraction of multiple excited-state energies in systems with dense spectra.
  • The approach may apply to high-dimensional molecular simulations in photodynamics and quasi-continuum regimes.

Where Pith is reading between the lines

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

  • The same singular-value minimization strategy could extend to other parameterized linear algebra tasks in quantum chemistry beyond the Schrödinger equation.
  • Avoiding inversion may reduce error accumulation when the overlap matrix is ill-conditioned.
  • Small-scale numerical tests on toy molecular potentials could directly check whether the quantum overhead remains favorable.

Load-bearing premise

The parameterized matrix family arising from the pseudospectral collocation method can be prepared and manipulated on a quantum computer so that amplitude amplification and phase estimation locate the singular value minima accurately and with acceptable overhead.

What would settle it

Implementation on a small test instance of the Schrödinger equation where the observed runtime exceeds the claimed square-root scaling or the recovered eigenvalues deviate from known classical values by more than the expected precision.

Figures

Figures reproduced from arXiv: 2506.13534 by Emil Zak, Grzegorz Rajchel-Mieldzio\'c, Szymon Pli\'s.

Figure 1
Figure 1. Figure 1: A sketch of the generic landscape of the eigenval [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A quantum circuit realizing the search for eigenvalues of a matrix that are close to the target value [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the matrix inverse method described in Sec. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The same setup as in Fig [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the matrix inverse method described in Sec. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sketch of the quantum algorithm for searching energies for a given collocation Hamiltonian defined by the collocation [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: We extend the validity of the cutoff [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

Accurate computation of multiple eigenvalues of quantum Hamiltonians is essential in quantum chemistry, materials science, and molecular spectroscopy. Estimating excited-state energies is challenging for classical algorithms due to exponential scaling with system size, posing an even harder problem than ground-state calculations. We present a quantum algorithm for estimating eigenvalues and singular values of parameterized matrix families, including solving generalized eigenvalue problems that frequently arise in quantum simulations. Our method uses quantum amplitude amplification and phase estimation to identify matrix eigenvalues by locating minima in the singular value spectrum. We demonstrate our algorithm by proposing a quantum-computing formulation of the pseudospectral collocation method for the Schr\"odinger equation. We estimate fault-tolerant quantum resource requirements for the quantum collocation method, showing favorable scaling in the size of the problem $N$ (up to $\widetilde{\mathcal{O}}(\sqrt{N})$) compared to classical implementations with $\widetilde{\mathcal{O}}(N)$, for certain well-behaved potentials. Additionally, unlike the standard collocation method, which results in a generalized eigenvalue problem requiring matrix inversion, our algorithm circumvents the associated numerical instability by scanning a parameterized matrix family and detecting eigenvalues through singular value minimization. This approach is particularly effective when multiple eigenvalues are needed or when the generalized eigenvalue problem involves a high condition number. In the fault-tolerant era, our method may thus be useful for simulating high-dimensional molecular systems with dense spectra involving highly excited states, such as those encountered in molecular photodynamics or quasi-continuum regimes in many-body and solid-state systems.

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

1 major / 1 minor

Summary. The paper introduces a quantum algorithm for solving generalized eigenvalue problems by locating minima in the singular value spectrum of parameterized matrix families using amplitude amplification and phase estimation. It applies this framework to a quantum version of the pseudospectral collocation method for the Schrödinger equation, estimating fault-tolerant resource requirements and claiming up to ~O(sqrt(N)) scaling for well-behaved potentials compared to classical ~O(N). The method avoids matrix inversion to improve numerical stability for multiple eigenvalues.

Significance. If the central claims regarding efficient quantum implementation hold, this work could provide a useful tool for computing excited states in high-dimensional quantum systems where classical methods struggle with dense spectra. The avoidance of generalized eigenvalue problem instabilities is a notable strength. Credit is due for framing the approach around standard quantum primitives and providing resource estimates, though the practical advantage hinges on operator implementation costs.

major comments (1)
  1. [Abstract and quantum collocation formulation] The favorable scaling claim of ~O(sqrt(N)) relies on the assumption that the unitary or block-encoding for the parameterized matrix family (incorporating differential operators and potential multiplication) can be implemented with polylog(N) gate cost. For the pseudospectral collocation on an N-point grid, the multiplication by a general potential typically incurs costs that may not be polylogarithmic without additional structure, which could undermine the reported advantage over classical O(N). A detailed complexity analysis or explicit circuit construction for the operator is required to substantiate this.
minor comments (1)
  1. [Abstract] The abstract mentions 'estimating fault-tolerant quantum resource requirements' but does not specify the model (e.g., gate count, depth, or number of qubits) or provide concrete numbers; including a brief summary of these estimates would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the complexity assumptions underlying our scaling claims. We address this point directly below and have revised the manuscript to strengthen the discussion of operator implementations and resource estimates.

read point-by-point responses

  1. Referee: [Abstract and quantum collocation formulation] The favorable scaling claim of ~O(sqrt(N)) relies on the assumption that the unitary or block-encoding for the parameterized matrix family (incorporating differential operators and potential multiplication) can be implemented with polylog(N) gate cost. For the pseudospectral collocation on an N-point grid, the multiplication by a general potential typically incurs costs that may not be polylogarithmic without additional structure, which could undermine the reported advantage over classical O(N). A detailed complexity analysis or explicit circuit construction for the operator is required to substantiate this.

    Authors: We agree that the reported scaling advantage is conditional on efficient implementation of the block-encoding or unitary for the parameterized family. The manuscript explicitly qualifies the O(sqrt(N)) claim as holding 'for certain well-behaved potentials,' where the potential multiplication operator admits a polylog(N) implementation (e.g., via low-degree polynomial approximations, smooth functions amenable to quantum arithmetic, or structured potentials such as harmonic oscillators). The differential operators in the pseudospectral collocation are handled via quantum Fourier transforms or equivalent techniques that are known to be polylogarithmic. We have revised the manuscript to include an expanded complexity analysis section that spells out these assumptions, cites standard results on efficient quantum implementations of multiplication and differentiation operators on grids, and clarifies the conditions under which the advantage holds versus cases with unstructured potentials. While a fully explicit gate-by-gate circuit for an arbitrary potential is outside the scope of the present work, the revised text outlines the high-level construction and resource counting used in our fault-tolerant estimates. revision: yes

Circularity Check

0 steps flagged

No circularity in the quantum algorithm derivation

full rationale

The paper constructs a quantum algorithm for generalized eigenvalue problems by directly applying standard primitives (amplitude amplification and phase estimation) to locate singular-value minima in a parameterized matrix family obtained from the pseudospectral collocation discretization of the Schrödinger equation. Resource estimates yielding O(sqrt(N)) scaling for well-behaved potentials follow from the assumed efficient block-encoding of the differential and multiplication operators, which is an external implementability assumption rather than a self-referential definition or fitted parameter. No load-bearing step reduces by construction to the inputs via self-definition, renaming of known results, or self-citation chains; the formulation is self-contained against external quantum and numerical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard quantum computing primitives and the assumption of efficient quantum implementation of the collocation operator family; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The parameterized matrix family corresponding to the pseudospectral collocation discretization can be efficiently implemented on a quantum computer.
    This is required for the claimed resource scaling to hold when using amplitude amplification and phase estimation.

pith-pipeline@v0.9.0 · 5811 in / 1217 out tokens · 33200 ms · 2026-05-22T13:28:38.623546+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages

  1. [1]

    The same setup as in Fig

    × 10-6 2.5 × 10-6 Minimal singular value ofH (E) Figure 4. The same setup as in Fig. 3 with 35 (blue) and 36 (orange) basis functions. The condition numbers respectively 1.21 × 1016 and 5.47 × 1016. Two different sets of basis functions are combined to account for the parity of the solutions to the Schrödinger equation. by the following formula VM(x) = De...

    work page

  2. [2]

    Comparison of the matrix inverse method described in Sec

    × 10-6 3.5 × 10-6 Minimal singular value ofH (E) Figure 5. Comparison of the matrix inverse method described in Sec. IIIB with the classical landscape scanning method for solving the collocation Schrödinger equation for the 1D Morse potential, given by Eq. (34), using 35 basis functions, with the condition number ofB†B equal to 1.23 × 1016. The curves hav...

    work page

  3. [3]

    For instance, for the collocation matrix ele- ments Bij we have the following representation: Bij = Ce −(βi−γj)2 , (37) for some constants β and γ and a normalization constant C

    Quantum-random oracle model (QROM) imple- menting matrix elements of B and B′′ in block- encoding can be implemented with quantum arith- metic. For instance, for the collocation matrix ele- ments Bij we have the following representation: Bij = Ce −(βi−γj)2 , (37) for some constants β and γ and a normalization constant C. Similarly, alsoB′′ has a compact f...

    work page

  4. [4]

    We also note that the matrices B′′ and VdiagB are expected to share many common non-zero elements

    The leading contribution to Mmax is expected to come from the potential energy function andB′′, as due to the normalization of the wavefunctions, the max-norm of matrix B is independent of its size N, i.e., ||B||max = O(1). We also note that the matrices B′′ and VdiagB are expected to share many common non-zero elements. This is most ev- ident when Gaussi...

    work page

  5. [5]

    For equal-width distributed Gaussian basis func- tions, since bothB and B′′ are circulant matrices, it suffices to encode a single column for each matrix with QROM to reduce the T-gate cost greatly

    work page

  6. [6]

    If the potentialˆV is given in a functional form, the cost of block-encoding it scales asO(polylog(N/ε)) via quantum arithmetic [71]

    work page

  7. [7]

    Gaussian basis functions are localized in space, leading to logarithmic sparsity of the residue ma- trix M(α), in which cased = O(log N)

    work page

  8. [8]

    What is more, the norm of the second derivative matrixB′′ will not grow as we go to higher dimensions in the spatial sense of the Schrödinger equation

    The max-norm Mmax of B is independent of the number of basis functions as we increase the di- mensionality of the matrices. What is more, the norm of the second derivative matrixB′′ will not grow as we go to higher dimensions in the spatial sense of the Schrödinger equation. With the above assumptions, the quantum landscape scanning complexity is compared...

    work page

  9. [9]

    The widths of the minima depend on the number of basis functions

    Because we aim to measure at least one grid pointα capturing the corresponding minimum, the choice of grid must be adjusted accordingly. The widths of the minima depend on the number of basis functions. To successfully find the energies using both landscape methods, it might be necessary to adaptively choose precisionε of singular values, i.e., the number...

    work page

  10. [10]

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

    work page 2010

  11. [11]

    Helms, J

    S.Lee, J.Lee, H.Zhai, Y.Tong, A.M.Dalzell, A.Kumar, P. Helms, J. Gray, Z.-H. Cui, W. Liu, M. Kastoryano, R.Babbush, J.Preskill, D.R.Reichman, E.T.Campbell, E. F. Valeev, L. Lin, and G. K.-L. Chan, Evaluating the evidence for exponential quantum advantage in ground- state quantum chemistry, Nature Communications 14, 10.1038/s41467-023-37587-6 (2023)

    work page doi:10.1038/s41467-023-37587-6 2023

  12. [12]

    Y. Su, D. W. Berry, N. Wiebe, N. Rubin, and R. Bab- bush, Fault-tolerant quantum simulations of chemistry in first quantization, PRX Quantum2, 040332 (2021)

    work page 2021

  13. [13]

    Babbush, C

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. Mc- Clean, A. Paler, A. Fowler, and H. Neven, Encoding elec- tronic spectra in quantum circuits with linear t complex- ity, Phys. Rev. X8, 041015 (2018)

    work page 2018

  14. [14]

    J. Lee, D. W. Berry, C. Gidney, W. J. Huggins, J. R. Mc- Clean, N. Wiebe, and R. Babbush, Even more efficient quantum computations of chemistry through tensor hy- percontraction, PRX Quantum2, 030305 (2021)

    work page 2021

  15. [15]

    von Burg, G

    V. von Burg, G. H. Low, T. Häner, D. S. Steiger, M. Rei- her, M. Roetteler, and M. Troyer, Quantum computing enhanced computational catalysis, Phys. Rev. Res. 3, 033055 (2021)

    work page 2021

  16. [16]

    Deka and E

    K. Deka and E. Zak, Simultaneously optimizing sym- metry shifts and tensor factorizations for cost-efficient fault-tolerant quantum simulations of electronic hamil- tonians, Journal of Chemical Theory and Computation 21, 4458–4465 (2025)

    work page 2025

  17. [17]

    Richerme, M

    P. Richerme, M. C. Revelle, C. G. Yale, D. Lobser, A. D. Burch, S. M. Clark, D. Saha, M. A. Lopez-Ruiz, A. Dwivedi, J. M. Smith, S. A. Norrell, A. Sabry, and S. S. Iyengar, Quantum computation of hydrogen bond dynamics and vibrational spectra, The Journal of Phys- ical Chemistry Letters14, 7256–7263 (2023)

    work page 2023

  18. [18]

    L. A. Mariano, V. H. A. Nguyen, J. B. Petersen, M. Björnsson, J. Bendix, G. R. Eaton, S. S. Eaton, and A. Lunghi, The role of electronic excited states in the spin-lattice relaxation of spin-1/2 molecules, Science Ad- vances 11, 10.1126/sciadv.adr0168 (2025)

    work page doi:10.1126/sciadv.adr0168 2025

  19. [19]

    P. R. Bunker, P. Jensen, and C. Jungen, Molecular symmetry and spectroscopy, Physics Today52, 63–64 (1999)

    work page 1999

  20. [20]

    Kittel and P

    C. Kittel and P. B. Kahn, Quantum theory of solids, American Journal of Physics33, 517–518 (1965)

    work page 1965

  21. [21]

    B. M. Weight, X. Li, and Y. Zhang, Theory and mod- eling of light-matter interactions in chemistry: current and future, Physical Chemistry Chemical Physics 25, 31554–31577 (2023)

    work page 2023

  22. [22]

    Mukherjee, M

    S. Mukherjee, M. Pinheiro, B. Demoulin, and M. Bar- batti, Simulations of molecular photodynamics in long timescales, Philosophical Transactions of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences 380, 10.1098/rsta.2020.0382 (2022)

    work page doi:10.1098/rsta.2020.0382 2020

  23. [23]

    A. C. Peet and W. Yang, The collocation method for calculating vibrational bound states of molecular sys- tems—with application to ar–hcl, The Journal of Chem- ical Physics 90, 1746–1751 (1989)

    work page 1989

  24. [24]

    Helgaker, P

    T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure Theory (John Wiley & Sons, Ltd, 2000)

    work page 2000

  25. [25]

    Carrington, Perspective: Computing (ro-)vibrational spectra of molecules with more than four atoms, The Journal of Chemical Physics 146, 10.1063/1.4979117 (2017)

    T. Carrington, Perspective: Computing (ro-)vibrational spectra of molecules with more than four atoms, The Journal of Chemical Physics 146, 10.1063/1.4979117 (2017). 17

    work page doi:10.1063/1.4979117 2017

  26. [26]

    Hu, T.-S

    X.-G. Hu, T.-S. Ho, and H. Rabitz, The collocation method based on a generalized inverse multiquadric basis for bound-state problems, Computer Physics Communi- cations 113, 168–179 (1998)

    work page 1998

  27. [27]

    Brown and T

    J. Brown and T. Carrington, Using an iterative eigen- solver to compute vibrational energies with phase- spaced localized basis functions, The Journal of Chemical Physics 143, 10.1063/1.4926805 (2015)

    work page doi:10.1063/1.4926805 2015

  28. [28]

    Hashemi, Y

    B. Hashemi, Y. Nakatsukasa, and L. N. Trefethen, Rect- angulareigenvalueproblems,AdvancesinComputational Mathematics 48, 10.1007/s10444-022-09994-8 (2022)

    work page doi:10.1007/s10444-022-09994-8 2022

  29. [29]

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,Numerical Recipes 3rd Edition: The Art of Sci- entific Computing, 3rd ed. (Cambridge University Press, USA, 2007)

    work page 2007

  30. [30]

    Ford and G

    B. Ford and G. Hall, The generalized eigenvalue problem in quantum chemistry, Computer Physics Communica- tions 8, 337–348 (1974)

    work page 1974

  31. [31]

    Dusson, M.-S

    G. Dusson, M.-S. Dupuy, and I.-M. Lygatsika, A poste- riori error estimates for schrödinger operators discretized with linear combinations of atomic orbitals (2024)

    work page 2024

  32. [32]

    Szalay, T

    V. Szalay, T. Szidarovszky, G. Czakó, and A. G. Császár, A paradox of grid-based representation techniques: accu- rate eigenvalues from inaccurate matrix elements, Jour- nal of Mathematical Chemistry50, 636 (2011)

    work page 2011

  33. [33]

    Avila and T

    G. Avila and T. Carrington, Nonproduct quadrature grids for solving the vibrational schrödinger equation, J. Chem. Phys. 131, 174103 (2009)

    work page 2009

  34. [34]

    Poirier and J

    B. Poirier and J. C. Light, Efficient distributed gaussian basis for rovibrational spectroscopy calculations, The Journal of Chemical Physics113, 211–217 (2000)

    work page 2000

  35. [35]

    Richings, I

    G. Richings, I. Polyak, K. Spinlove, G. Worth, I. Burghardt, and B. Lasorne, Quantum dynamics sim- ulations using gaussian wavepackets: the vmcg method, International Reviews in Physical Chemistry34, 269–308 (2015)

    work page 2015

  36. [36]

    Burghardt, K

    I. Burghardt, K. Giri, and G. A. Worth, Multi- mode quantum dynamics using gaussian wavepackets: The gaussian-based multiconfiguration time-dependent hartree (g-mctdh) method applied to the absorption spectrum of pyrazine, The Journal of Chemical Physics 129, 10.1063/1.2996349 (2008)

    work page doi:10.1063/1.2996349 2008

  37. [37]

    Manzhos, M

    S. Manzhos, M. Ihara, and T. Carrington, Using colloca- tion to solve the schrödinger equation, Journal of Chem- ical Theory and Computation19, 1641–1656 (2023)

    work page 2023

  38. [38]

    E. J. Zak and T. Carrington, Using collocation and a hierarchical basis to solve the vibrational schrödinger equation, The Journal of Chemical Physics 150, 10.1063/1.5096169 (2019)

    work page doi:10.1063/1.5096169 2019

  39. [39]

    Carrington, Using collocation to study the vibrational dynamics of molecules, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 248, 119158 (2021)

    T. Carrington, Using collocation to study the vibrational dynamics of molecules, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 248, 119158 (2021)

    work page 2021

  40. [40]

    Simmons and T

    J. Simmons and T. Carrington, Computing vibrational spectra using a new collocation method with a pruned basis and more points than basis functions: Avoid- ing quadrature, The Journal of Chemical Physics158, 10.1063/5.0146703 (2023)

    work page doi:10.1063/5.0146703 2023

  41. [41]

    S. Jin, S. Wu, G. Zhou, Y. Li, L. Li, B. Li, and X. Wang, A query-based quantum eigensolver, Quantum Engineer- ing 2, 10.1002/que2.49 (2020)

    work page doi:10.1002/que2.49 2020

  42. [42]

    Y. Chen, A. Gilyén, and R. de Wolf, A quantum speed- up for approximating the top eigenvectors of a matrix (2024)

    work page 2024

  43. [43]

    Kerzner, V

    A. Kerzner, V. Gheorghiu, M. Mosca, T. Guilbaud, F. Carminati, F. Fracas, and L. Dellantonio, A square- root speedup for finding the smallest eigenvalue, Quan- tum Science and Technology9, 045025 (2024)

    work page 2024

  44. [44]

    Shao, Computing eigenvalues of diagonalizable ma- trices on a quantum computer, ACM Transactions on Quantum Computing 3, 1–20 (2022)

    C. Shao, Computing eigenvalues of diagonalizable ma- trices on a quantum computer, ACM Transactions on Quantum Computing 3, 1–20 (2022)

    work page 2022

  45. [45]

    G. H. Low and Y. Su, Quantum eigenvalue processing, in 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)(IEEE, 2024) p. 1051–1062

    work page 2024

  46. [46]

    Z. Ding, H. Li, L. Lin, H. Ni, L. Ying, and R. Zhang, Quantum multiple eigenvalue gaussian filtered search: an efficient and versatile quantum phase estimation method, Quantum 8, 1487 (2024)

    work page 2024

  47. [47]

    Zimborás, B

    Z. Zimborás, B. Koczor, Z. Holmes, E.-M. Borrelli, A. Gi- lyén, H.-Y. Huang, Z. Cai, A. Acín, L. Aolita, L. Banchi, F.G.S.L.Brandão, D.Cavalcanti, T.Cubitt, S.N.Filip- pov, G. García-Pérez, J. Goold, O. Kálmán, E. Kyoseva, M. A. C. Rossi, B. Sokolov, I. Tavernelli, and S. Man- iscalco, Myths around quantum computation before full fault tolerance: What no-...

    work page 2025

  48. [48]

    Tilly, H

    J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson, The variational quantum eigensolver: A re- viewof methodsand best practices, Physics Reports986, 1–128 (2022)

    work page 2022

  49. [49]

    Hogben,Handbook of Linear Algebra(Chapman and Hall/CRC, 2013)

    L. Hogben,Handbook of Linear Algebra(Chapman and Hall/CRC, 2013)

    work page 2013

  50. [50]

    In this case,N is the larger of the two dimensions

    work page

  51. [51]

    Later on, we generalize this problem also to singular val- ues

    work page

  52. [52]

    For normal matrices, ε-pseudospectrum are the regions ±ε around their spectrum; therefore, this notion is aligned with our goal of finding numbers close to the exact eigenvalues

    work page

  53. [53]

    More specifically, for each register, with the highest prob- ability, the outcome will be the true eigenvalueλ. How- ever, there will also be a small irrelevant part|g⟩, that will eventually not contribute much to the final estima- tion of the eigenvalue|η⟩ = √1 − γ |λ⟩ + γ P i |gi⟩, as the coefficient γ ≈ 0

    work page

  54. [54]

    G. H. Low and I. L. Chuang, Optimal hamiltonian sim- ulation by quantum signal processing, Physical Review Letters 118, 10.1103/physrevlett.118.010501 (2017)

    work page doi:10.1103/physrevlett.118.010501 2017

  55. [55]

    Camps, L

    D. Camps, L. Lin, R. Van Beeumen, and C. Yang, Ex- plicit quantum circuits for block encodings of certain sparse matrices (2022)

    work page 2022

  56. [56]

    In the above, we disregard the registers related to the QPE, as these do not change the value of the character- istic function qubit

    work page

  57. [57]

    In principle, due to the not-exact accuracy of QPE, the median qubit will also contain contributions from other states, but we drop this for clarity

    work page

  58. [58]

    Cardona and M

    M. Cardona and M. L. W. Thewalt, Isotope effects on theopticalspectraofsemiconductors,ReviewsofModern Physics 77, 1173 (2005)

    work page 2005

  59. [59]

    Kittel, Introduction to Solid State Physics, 8th ed

    C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2004)

    work page 2004

  60. [60]

    R.M.Nieminen,Electronicstructurecalculationsforma- terials design, Journal of Physics: Condensed Matter14, 2859 (2002)

    work page 2002

  61. [61]

    Boys, Proceedings of the Royal Society of London

    S. Boys, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 309, 195–208 18 (1969)

    work page 1969

  62. [62]

    G.AvilaandT.Carrington,Amulti-dimensionalsmolyak collocationmethodincurvilinearcoordinatesforcomput- ing vibrational spectra, The Journal of Chemical Physics 143, 10.1063/1.4936294 (2015)

    work page doi:10.1063/1.4936294 2015

  63. [63]

    R. B. Lehoucq, D. C. Sorensen, and C. Yang,ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Prob- lems with Implicitly Restarted Arnoldi Methods(Society for Industrial and Applied Mathematics, 1998)

    work page 1998

  64. [64]

    Myllykoski and C

    M. Myllykoski and C. C. Kjelgaard Mikkelsen, Task- based, gpu-accelerated and robust library for solv- ing dense nonsymmetric eigenvalue problems, Concur- rency and Computation: Practice and Experience 33, 10.1002/cpe.5915 (2020)

    work page doi:10.1002/cpe.5915 2020

  65. [65]

    A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum al- gorithm for linear systems of equations, Physical Review Letters 103, 10.1103/physrevlett.103.150502 (2009)

    work page doi:10.1103/physrevlett.103.150502 2009

  66. [66]

    Trenev, P

    D. Trenev, P. J. Ollitrault, S. M. Harwood, T. P. Gu- jarati, S. Raman, A. Mezzacapo, and S. Mostame, Refin- ing resource estimation for the quantum computation of vibrational molecular spectra through Trotter error anal- ysis, Quantum 9, 1630 (2025)

    work page 2025

  67. [67]

    To avoid obfuscation by too many parameters, we also assume the number of grid points in the spatial coordi- nate M is proportional to N, i.e., M = O(N ), making the ratio of the dimensions of the initial matricesB and B′′ constant

    work page

  68. [68]

    This assumption is valid for localized basis sets, such as the Gaussian basis

    work page

  69. [69]

    V. V. Williams, Y. Xu, Z. Xu, and R. Zhou, New bounds for matrix multiplication: from alpha to omega (2023)

    work page 2023

  70. [70]

    Alman and V

    J. Alman and V. V. Williams, A refined laser method and faster matrix multiplication, TheoretiCSV olume 3, 10.46298/theoretics.24.21 (2024)

    work page doi:10.46298/theoretics.24.21 2024

  71. [71]

    Saad and M

    Y. Saad and M. H. Schultz, Gmres: A generalized min- imal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Com- puting 7, 856–869 (1986)

    work page 1986

  72. [72]

    C.Musco, C.Musco,andA.Sidford,Stabilityofthelanc- zos method for matrix function approximation (2017)

    work page 2017

  73. [73]

    Note that we neglect the dependence of the Krylov space size on the number of eigenvalues requested, which is often logarithmic [62]

    work page

  74. [74]

    Lord, Matrix computations, 3rd edition, by g

    N. Lord, Matrix computations, 3rd edition, by g. h. golub and c. f. van loan. pp. 694. 1996. isbn 0 8018 5414 8, 0 8018 5413 x. (johns hopkins university press)., The Math- ematical Gazette 83, 556–557 (1999)

    work page 1996

  75. [75]

    Thus, quantum advantage can be searched already for low J’s

    Forhigher-ordertruncations, thecomplexitiesoftheclas- sical and quantum algorithms both grow linearly inJ. Thus, quantum advantage can be searched already for low J’s

    work page

  76. [76]

    In the case of vanishing derivative of the singular value dependenceontheparameter α, K mightbeproportional to roots of1/ε, making this dependence better from the quantum computing perspective

    work page

  77. [77]

    Sünderhauf, E

    C. Sünderhauf, E. Campbell, and J. Camps, Block- encoding structured matrices for data input in quantum computing, Quantum 8, 1226 (2024)

    work page 2024

  78. [78]

    Mukhopadhyay, T

    P. Mukhopadhyay, T. F. Stetina, and N. Wiebe, Quan- tum simulation of the first-quantized pauli-fierz hamilto- nian, PRX Quantum5, 010345 (2024)

    work page 2024

  79. [79]

    Quantum arithmetic with the quantum Fourier transform

    L. Ruiz-Perez and J. C. Garcia-Escartin, Quantum arith- metic with the quantum fourier transform, Quantum Information Processing 16, 10.1007/s11128-017-1603-1 (2017)

    work page doi:10.1007/s11128-017-1603-1 2017

  80. [80]

    Gosset, R

    D. Gosset, R. Kothari, and K. Wu, Quantum state prepa- ration with optimal t-count (2024)

    work page 2024

Showing first 80 references.