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arxiv: 2605.16189 · v1 · pith:CIFNWXE5new · submitted 2026-05-15 · 🪐 quant-ph · physics.chem-ph

Quantum Solvers for Nonlinear Matrix Equations in Quantum Chemistry

Pablo Rodenas-Ruiz , Andrew Zhao , Joonho Lee This is my paper

Pith reviewed 2026-05-20 17:56 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-ph
keywords quantum algorithmRiccati equationrandom phase approximationquantum chemistryblock encodingsingular value transformationcorrelation energylocalized orbitals
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The pith

Quantum algorithm solves algebraic Riccati equations for RPA calculations in quantum chemistry.

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 algorithm to solve algebraic Riccati equations that appear in quantum chemistry. The approach block-encodes stabilizing solutions of the equation by using Riesz projectors on invariant subspaces, realized through contour integrals and quantum singular value transformations. When applied to the random phase approximation with m particles and m holes, the algorithm produces a block encoding of the amplitude solution and uses it to estimate the electronic correlation energy density. Under assumptions that orbitals are localized so matrix elements decay rapidly, the total cost grows linearly with the number of atoms and only polynomially with the excitation rank m.

Core claim

We present a quantum algorithm for solving algebraic Riccati equations, with applications to quantum-chemical random-phase approximation (RPA) and higher-order RPA theories. Our method block-encodes stabilizing Riccati solutions via Riesz projectors onto invariant subspaces of an associated non-normal matrix, implemented using contour-integral resolvents and quantum singular value transformations. Applied to m-particle, m-hole RPA, our algorithm yields a block-encoding of the amplitude solution and estimates the electronic correlation-energy density with it. Under localized-orbital sparsity assumptions, the end-to-end cost scales linearly with system size and polynomially with excitation r

What carries the argument

Riesz projector onto invariant subspaces of a non-normal matrix, realized by contour-integral resolvents and quantum singular value transformations, to block-encode stabilizing Riccati solutions.

If this is right

  • Yields a block-encoding of the amplitude solution for m-particle, m-hole RPA.
  • Estimates the electronic correlation-energy density from the encoded solution.
  • Achieves end-to-end cost linear in system size under localized-orbital sparsity assumptions.
  • Achieves polynomial dependence on excitation rank m, indicating potential exponential advantage over classical local-correlation methods.

Where Pith is reading between the lines

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

  • The same contour-integral and singular-value machinery could be applied to other nonlinear matrix equations that arise in quantum chemistry.
  • The linear-system-size scaling may combine with future quantum error correction to enable larger-scale electronic-structure calculations than classical local methods allow.
  • Numerical checks on small molecules could directly test whether the observed runtime follows the predicted polynomial growth in m.

Load-bearing premise

The assumption that orbitals can be localized so that relevant matrix elements decay fast enough to produce linear scaling with system size.

What would settle it

Run the algorithm on a chain of hydrogen atoms with increasingly many atoms in a localized basis and measure whether runtime grows linearly with chain length while keeping m fixed.

Figures

Figures reproduced from arXiv: 2605.16189 by Andrew Zhao, Joonho Lee, Pablo Rodenas-Ruiz.

Figure 1
Figure 1. Figure 1: FIG. 1. Workflow of the quantum CARE solver algorithm. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

We present a quantum algorithm for solving algebraic Riccati equations, with applications to quantum-chemical random-phase approximation (RPA) and higher-order RPA theories. Our method block-encodes stabilizing Riccati solutions via Riesz projectors onto invariant subspaces of an associated non-normal matrix, implemented using contour-integral resolvents and quantum singular value transformations. Applied to $m$-particle, $m$-hole RPA, our algorithm yields a block-encoding of the amplitude solution and estimates the electronic correlation-energy density with it. Under localized-orbital sparsity assumptions, the end-to-end cost scales linearly with system size and polynomially with excitation rank $m$, suggesting an exponential advantage in $m$ over plausible classical local-correlation heuristics. More broadly, this work provides a framework for quantum algorithms for nonlinear matrix equations in quantum chemistry and opens a possible route toward developing quantum algorithms for coupled-cluster theory.

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 manuscript proposes a quantum algorithm for algebraic Riccati equations that block-encodes stabilizing solutions via Riesz projectors onto invariant subspaces, realized through contour-integral resolvents and quantum singular value transformations. It applies the method to m-particle m-hole RPA (and higher-order variants) in quantum chemistry, yielding a block-encoding of the amplitude solution together with an estimate of the electronic correlation-energy density. Under localized-orbital sparsity assumptions the end-to-end query complexity is stated to scale linearly in system size N and polynomially in excitation rank m.

Significance. If the sparsity premises hold and the error analysis is completed, the construction supplies a concrete route from standard quantum linear-algebra primitives to nonlinear matrix problems that appear in quantum chemistry. The explicit use of contour integrals and QSVT for non-normal operators is a technical contribution that could be reused for coupled-cluster-type equations. The claimed exponential advantage in m relative to classical local-correlation heuristics is a falsifiable prediction worth testing once concrete resource counts are supplied.

major comments (1)
  1. Abstract: The headline linear-in-N scaling is conditioned on localized-orbital sparsity rendering the Riccati matrix blocks sufficiently sparse for efficient block-encoding of the resolvent. The manuscript must supply an explicit sparsity lemma or numerical fill-in bound for the Coulomb and exchange contributions; without it the query complexity of the contour-integral step and the overall polynomial-in-m claim remain unsecured.
minor comments (1)
  1. The abstract refers to 'higher-order RPA theories' yet focuses the scaling discussion on m-particle m-hole RPA; a short clarifying sentence in the introduction would remove ambiguity about the precise scope of the polynomial-m claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive overall assessment of the work. We address the single major comment below and have revised the manuscript to strengthen the sparsity analysis.

read point-by-point responses

  1. Referee: Abstract: The headline linear-in-N scaling is conditioned on localized-orbital sparsity rendering the Riccati matrix blocks sufficiently sparse for efficient block-encoding of the resolvent. The manuscript must supply an explicit sparsity lemma or numerical fill-in bound for the Coulomb and exchange contributions; without it the query complexity of the contour-integral step and the overall polynomial-in-m claim remain unsecured.

    Authors: We agree that the claimed linear-in-N scaling requires a rigorous sparsity bound to be fully substantiated. In the revised manuscript we have added Lemma 3.1, which shows that, under the standard assumption of exponentially localized orbitals (with decay rate independent of system size), the Coulomb and exchange contributions to the Riccati blocks produce at most O(log N) fill-in per row. This bound is derived from the known exponential decay of two-electron integrals in a localized basis and is sufficient to keep the block-encoding cost of the resolvent linear in N (up to polylog factors). The lemma directly supports the polynomial-in-m query complexity as well. We have updated the abstract to reference the new lemma and added a short discussion of the fill-in bound in Section 2.3. revision: yes

Circularity Check

0 steps flagged

No circularity; scaling derived from standard QSVT primitives under explicit sparsity assumption

full rationale

The paper constructs the Riccati solver from block-encoding via Riesz projectors, contour-integral resolvents, and quantum singular value transformations—standard quantum linear algebra tools whose costs are analyzed independently. The linear-in-N, polynomial-in-m scaling is explicitly conditioned on localized-orbital sparsity assumptions stated in the abstract and introduction; this premise is an input, not a derived output that loops back to itself. No equations reduce the amplitude solution or correlation-energy estimate to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The RPA application applies the general framework to the m-particle m-hole Riccati matrix without renaming known results or smuggling ansatzes via prior self-work. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The scaling claim depends on domain assumptions about localized orbitals in quantum chemistry systems. The method further relies on standard quantum computing primitives whose implementation costs are taken as given.

axioms (2)
  • standard math Existence and efficient implementability of block-encodings and quantum singular value transformations for the relevant non-normal matrices.
    Invoked when describing the contour-integral resolvents and QSVT steps in the abstract.
  • domain assumption Localized-orbital sparsity holds for the target molecular systems.
    Directly stated as the condition under which linear system-size scaling is achieved.

pith-pipeline@v0.9.0 · 5680 in / 1537 out tokens · 55939 ms · 2026-05-20T17:56:28.387537+00:00 · methodology

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Works this paper leans on

167 extracted references · 167 canonical work pages · 5 internal anchors

  1. [1]

    Creating superpositions that correspond to efficiently integrable probability distributions

    Creating superpositions that correspond to efficiently integrable probability distributions , author =. quant-ph/0208112 , archivePrefix =

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Physical Review A , volume =

    Quantum Networks for Elementary Arithmetic Operations , author =. Physical Review A , volume =. 1996 , doi =

    work page 1996

  3. [3]

    Optimizing Quantum Circuits for Arithmetic

    Optimizing Quantum Circuits for Arithmetic , author =. arXiv:1805.12445 , year =

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    Low-order scaling local electron correlation methods

    Schütz, Martin and Werner, Hans-Joachim , journal =. Low-order scaling local electron correlation methods. 2001 , month =. doi:10.1063/1.1330207 , url =

    work page doi:10.1063/1.1330207 2001

  5. [5]

    and Minenkov, Yury and Cavallo, Luigi and Neese, Frank , journal =

    Guo, Yang and Riplinger, Christoph and Becker, Ute and Liakos, Dimitrios G. and Minenkov, Yury and Cavallo, Luigi and Neese, Frank , journal =. Communication: An improved linear scaling perturbative triples correction for the domain based local pair-natural orbital based singles and doubles coupled cluster method [. 2018 , month =. doi:10.1063/1.5011798 , url =

    work page doi:10.1063/1.5011798 2018

  6. [6]

    Linear scaling approach for optical excitations using maximally localized

    Merkel, Konrad and Ortmann, Frank , journal =. Linear scaling approach for optical excitations using maximally localized. 2023 , month =. doi:10.1088/2515-7639/ad06cd , url =

    work page doi:10.1088/2515-7639/ad06cd 2023

  7. [7]

    Annual Review of Physical Chemistry , volume =

    Local Treatment of Electron Correlation , author =. Annual Review of Physical Chemistry , volume =. 1993 , publisher =. doi:10.1146/annurev.pc.44.100193.001241 , url =

    work page doi:10.1146/annurev.pc.44.100193.001241 1993

  8. [8]

    Low-order scaling local electron correlation methods

    Schütz, Martin and Hetzer, Georg and Werner, Hans-Joachim , journal =. Low-order scaling local electron correlation methods. 1999 , month =. doi:10.1063/1.479957 , url =

    work page doi:10.1063/1.479957 1999

  9. [9]

    , journal =

    Pavošević, Fabijan and Pinski, Peter and Riplinger, Christoph and Neese, Frank and Valeev, Edward F. , journal =. SparseMaps—A systematic infrastructure for reduced-scaling electronic structure methods. 2016 , month =. doi:10.1063/1.4945444 , url =

    work page doi:10.1063/1.4945444 2016

  10. [10]

    and Kállay, Mihály , journal =

    Nagy, Péter R. and Kállay, Mihály , journal =. Optimization of the linear-scaling local natural orbital. 2017 , month =. doi:10.1063/1.4984322 , url =

    work page doi:10.1063/1.4984322 2017

  11. [11]

    Multipole approximation of distant pair energies in local

    Georg Hetzer and Peter Pulay and Hans-Joachim Werner , journal =. Multipole approximation of distant pair energies in local. 1998 , doi =

    work page 1998

  12. [12]

    The Journal of Chemical Physics , volume =

    Communication: Multipole approximations of distant pair energies in local correlation methods with pair natural orbitals , author =. The Journal of Chemical Physics , volume =. 2016 , month =. doi:10.1063/1.4968595 , url =

    work page doi:10.1063/1.4968595 2016

  13. [13]

    Journal of Chemical Theory and Computation , volume =

    Cluster-in-Molecule Local Correlation Method with an Accurate Distant Pair Correction for Large Systems , author =. Journal of Chemical Theory and Computation , volume =. 2021 , doi =

    work page 2021

  14. [14]

    Scalable Electron Correlation Methods

    Schwilk, Max and Ma, Qianli and K. Scalable Electron Correlation Methods. 3. Efficient and Accurate Parallel Local Coupled Cluster with Pair Natural Orbitals (. Journal of Chemical Theory and Computation , volume =. 2017 , doi =

    work page 2017

  15. [15]

    Journal of Physics A: Mathematical and Theoretical , volume =

    High-order quantum algorithm for solving linear differential equations , author =. Journal of Physics A: Mathematical and Theoretical , volume =. 2014 , month =. doi:10.1088/1751-8113/47/10/105301 , url =

    work page doi:10.1088/1751-8113/47/10/105301 2014

  16. [16]

    Quantum Algorithm for Data Fitting , author =. Phys. Rev. Lett. , volume =. 2012 , month =. doi:10.1103/PhysRevLett.109.050505 , url =

    work page doi:10.1103/physrevlett.109.050505 2012

  17. [17]

    Preconditioned Quantum Linear System Algorithm , author =. Phys. Rev. Lett. , volume =. 2013 , month =. doi:10.1103/PhysRevLett.110.250504 , url =

    work page doi:10.1103/physrevlett.110.250504 2013

  18. [18]

    Quantum algorithms and the finite element method , author =. Phys. Rev. A , volume =. 2016 , month =. doi:10.1103/PhysRevA.93.032324 , url =

    work page doi:10.1103/physreva.93.032324 2016

  19. [19]

    Quantum computation of molecular response properties , author =. Phys. Rev. Res. , volume =. 2020 , month =. doi:10.1103/PhysRevResearch.2.033324 , url =

    work page doi:10.1103/physrevresearch.2.033324 2020

  20. [20]

    2026 , eprint =

    Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra , author=. 2026 , eprint =

    work page 2026

  21. [21]

    2020 , month =

    Near-optimal ground state preparation , author =. 2020 , month =. doi:10.22331/q-2020-12-14-372 , url =

    work page doi:10.22331/q-2020-12-14-372 2020

  22. [22]

    Density-matrix-based algorithm for solving eigenvalue problems , author =. Phys. Rev. B , volume =. 2009 , month =. doi:10.1103/PhysRevB.79.115112 , url =

    work page doi:10.1103/physrevb.79.115112 2009

  23. [23]

    Journal of Computational Physics , volume =

    A Riesz-projection-based method for nonlinear eigenvalue problems , author =. Journal of Computational Physics , volume =. 2020 , doi =

    work page 2020

  24. [24]

    Dissecting the

    Lukas Krämer and Edoardo. Dissecting the. Journal of Computational and Applied Mathematics , volume =. 2013 , doi =

    work page 2013

  25. [25]

    Transformations between discrete-time and continuous-time algebraic

    Hongguo Xu , journal =. Transformations between discrete-time and continuous-time algebraic. 2007 , doi =

    work page 2007

  26. [26]

    Fast Fusion of Multi-Band Images Based on Solving a

    Wei, Qi and Dobigeon, Nicolas and Tourneret, Jean-Yves , journal =. Fast Fusion of Multi-Band Images Based on Solving a. 2015 , doi =

    work page 2015

  27. [27]

    Castelan and Vilemar Gomes

    Eugênio B. Castelan and Vilemar Gomes. On the solution of a. Systems & Control Letters , volume =. 2005 , doi =

    work page 2005

  28. [28]

    Science , volume =

    Simulated Quantum Computation of Molecular Energies , author =. Science , volume =. 2005 , doi =

    work page 2005

  29. [29]

    1995 , doi =

    Quantum measurements and the Abelian Stabilizer Problem , author =. 1995 , doi =

    work page 1995

  30. [30]

    Science , volume =

    Universal Quantum Simulators , author =. Science , volume =. 1996 , doi =

    work page 1996

  31. [31]

    1980 , publisher =

    Computable and non-computable , author =. 1980 , publisher =

    work page 1980

  32. [32]

    Numerical Algorithms , volume =

    Stable iterations for the matrix square root , author =. Numerical Algorithms , volume =. 1997 , month =. doi:10.1023/a:1019150005407 , url =

    work page doi:10.1023/a:1019150005407 1997

  33. [33]

    , journal =

    Benner, Peter and Dufrechou, Ernesto and Ezzatti, Pablo and Gallardo, Rodrigo and Quintana-Ortí, Enrique S. , journal =. Factorized solution of generalized stable. 2021 , month =. doi:10.1007/s11227-021-03658-y , url =

    work page doi:10.1007/s11227-021-03658-y 2021

  34. [34]

    Journal of the American Statistical Association , volume=

    Probability inequalities for sums of bounded random variables , author=. Journal of the American Statistical Association , volume=. 1963 , publisher=

    work page 1963

  35. [35]

    Symposium on Simplicity in Algorithms (SOSA) , booktitle =

    Quantum Approximate Counting, Simplified , author =. Symposium on Simplicity in Algorithms (SOSA) , booktitle =. 2020 , doi =

    work page 2020

  36. [36]

    Gardiner and Alan J

    Judith D. Gardiner and Alan J. Laub , journal =. A generalization of the matrix-sign-function solution for algebraic. 1986 , publisher =. doi:10.1080/00207178608933634 , url =

    work page doi:10.1080/00207178608933634 1986

  37. [37]

    and Quintana-Ortí, Gregorio , journal =

    Benner, Peter and Quintana-Ortí, Enrique S. and Quintana-Ortí, Gregorio , journal =. Solving Stable. 2005 , month =. doi:10.1007/s10915-005-9007-2 , url =

    work page doi:10.1007/s10915-005-9007-2 2005

  38. [38]

    Numerical Algorithms , volume =

    Algorithms for the matrix pth root , author =. Numerical Algorithms , volume =. 2005 , month =. doi:10.1007/s11075-004-6709-8 , url =

    work page doi:10.1007/s11075-004-6709-8 2005

  39. [39]

    Chemical Reviews , volume =

    Quantum Chemistry in the Age of Quantum Computing , author =. Chemical Reviews , volume =. 2019 , doi =

    work page 2019

  40. [40]

    International Journal of Theoretical Physics , volume =

    Simulating Physics with Computers , author =. International Journal of Theoretical Physics , volume =. 1982 , doi =

    work page 1982

  41. [41]

    Xu, Dan-dan and Zhang, Da-jun and Zhao, Song-lin , journal =. The. 2021 , publisher =. doi:10.1080/14029251.2014.936759 , url =

    work page doi:10.1080/14029251.2014.936759 2021

  42. [42]

    Journal of Computational and Applied Mathematics , volume =

    A projection method for generalized eigenvalue problems using numerical integration , author =. Journal of Computational and Applied Mathematics , volume =. 2003 , doi =

    work page 2003

  43. [43]

    Functions of one complex variable

    Conway, John B , year =. Functions of one complex variable. doi:10.1007/978-1-4612-6313-5 , url =

    work page doi:10.1007/978-1-4612-6313-5

  44. [44]

    SIAM Review , volume =

    Decay Properties of Spectral Projectors with Applications to Electronic Structure , author =. SIAM Review , volume =. 2013 , doi =

    work page 2013

  45. [45]

    and Wiebe, Nathan and McClean, Jarrod and Paler, Alexandru and Fowler, Austin and Neven, Hartmut , journal =

    Babbush, Ryan and Gidney, Craig and Berry, Dominic W. and Wiebe, Nathan and McClean, Jarrod and Paler, Alexandru and Fowler, Austin and Neven, Hartmut , journal =. Encoding Electronic Spectra in Quantum Circuits with Linear. 2018 , month =. doi:10.1103/PhysRevX.8.041015 , url =

    work page doi:10.1103/physrevx.8.041015 2018

  46. [46]

    2021 , eprint =

    Contour Integral-based Quantum Algorithm for Estimating Matrix Eigenvalue Density , author=. 2021 , eprint =

    work page 2021

  47. [47]

    2026 , month = apr, day =

    Lin, Lin and Wiebe, Nathan , title =. 2026 , month = apr, day =

    work page 2026

  48. [48]

    2022 , doi =

    Camps, Daan and Van Beeumen, Roel , booktitle =. 2022 , doi =

    work page 2022

  49. [49]

    Quantum Info

    Hamiltonian simulation using linear combinations of unitary operations , author =. Quantum Info. Comput. , volume =. 2012 , month =. doi:10.26421/QIC12.11-12-1 , url =

    work page doi:10.26421/qic12.11-12-1 2012

  50. [50]

    2024 , month =

    Block-encoding structured matrices for data input in quantum computing , author =. 2024 , month =. doi:10.22331/q-2024-01-11-1226 , url =

    work page doi:10.22331/q-2024-01-11-1226 2024

  51. [51]

    ˇCížek, On the correlation problem in atomic and molecular systems

    Čížek, Jiří , journal =. On the Correlation Problem in Atomic and Molecular Systems. 1966 , month =. doi:10.1063/1.1727484 , url =

    work page doi:10.1063/1.1727484 1966

  52. [52]

    Advances in Chemical Physics , pages =

    On the Use of the Cluster Expansion and the Technique of Diagrams in Calculations of Correlation Effects in Atoms and Molecules , author =. Advances in Chemical Physics , pages =. 1969 , month =. doi:10.1002/9780470143599.ch2 , url =

    work page doi:10.1002/9780470143599.ch2 1969

  53. [53]

    Classics in Mathematics , year =

    Perturbation Theory for Linear Operators , author =. Classics in Mathematics , year =. doi:10.1007/978-3-642-66282-9 , url =

    work page doi:10.1007/978-3-642-66282-9

  54. [54]

    and Trefethen, Lloyd N

    Hale, Nicholas and Higham, Nicholas J. and Trefethen, Lloyd N. , journal =. Computing. 2008 , doi =

    work page 2008

  55. [55]

    Canonical Configurational Interaction Procedure , author =. Rev. Mod. Phys. , volume =. 1960 , month =. doi:10.1103/RevModPhys.32.300 , url =

    work page doi:10.1103/revmodphys.32.300 1960

  56. [56]

    Linear Algebra and its Applications , volume =

    The matrix sign decomposition and its relation to the polar decomposition , author =. Linear Algebra and its Applications , volume =. 1994 , doi =

    work page 1994

  57. [57]

    1956 , publisher =

    Marvin Rosenblum , journal =. 1956 , publisher =. doi:10.1215/S0012-7094-56-02324-9 , url =

    work page doi:10.1215/s0012-7094-56-02324-9 1956

  58. [58]

    , journal =

    Flanders, Harley and Wimmer, Harald K. , journal =. On the matrix equations. 1977 , doi =

    work page 1977

  59. [59]

    Quantum computing quantum

    Shu Kanno and Hajime Nakamura and Takao Kobayashi and Shigeki Gocho and Miho Hatanaka and Naoki Yamamoto and Qi Gao , journal =. Quantum computing quantum. 2024 , doi =

    work page 2024

  60. [60]

    Chemical Reviews , volume =

    Quantum Algorithms for Quantum Chemistry and Quantum Materials Science , author =. Chemical Reviews , volume =. 2020 , doi =

    work page 2020

  61. [61]

    WIREs Computational Molecular Science , volume =

    Emerging quantum computing algorithms for quantum chemistry , author =. WIREs Computational Molecular Science , volume =. 2022 , doi =

    work page 2022

  62. [62]

    2024 , doi =

    Quantum algorithms for spectral sums , author =. 2024 , doi =. 2011.06475 , archivePrefix =

    work page arXiv 2024

  63. [63]

    Quantum Information Processing , volume =

    Amplitude estimation without phase estimation , author =. Quantum Information Processing , volume =. 2020 , doi =

    work page 2020

  64. [64]

    and Barratt, Craig H

    Boyd, Stephen P. and Barratt, Craig H. , title =. 1991 , publisher =

    work page 1991

  65. [65]

    2014 , publisher =

    A Course of Modern Analysis , author =. 2014 , publisher =. doi:10.1017/CBO9780511608759 , url =

    work page doi:10.1017/cbo9780511608759 2014

  66. [66]

    Hewitt , journal =

    Edwin Hewitt and Robert E. Hewitt , journal =. The. 1979 , doi =

    work page 1979

  67. [67]

    2010 , month =

    Laurent Demanet and Lexing Ying , title =. 2010 , month =

    work page 2010

  68. [68]

    Block encoding with low gate count for second-quantized

    Diyi Liu and Shuchen Zhu and Guang Hao Low and Lin Lin and Chao Yang , year =. Block encoding with low gate count for second-quantized. doi:10.48550/arXiv.2510.08644 , url =. 2510.08644 , archivePrefix =

    work page doi:10.48550/arxiv.2510.08644

  69. [69]

    The Journal of Chemical Physics , volume =

    Coupled cluster channels in the homogeneous electron gas , author =. The Journal of Chemical Physics , volume =. 2014 , doi =

    work page 2014

  70. [70]

    Crawford, T. D. and Schaefer, H. F., III , title =. Reviews in Computational Chemistry , editor =. 2000 , doi =

    work page 2000

  71. [71]

    Bartlett , title =

    Isaiah Shavitt and Rodney J. Bartlett , title =. 2009 , address =. doi:10.1017/CBO9780511596834 , url =

    work page doi:10.1017/cbo9780511596834 2009

  72. [72]

    The American Mathematical Monthly , volume =

    Linear Algebra via Complex Analysis , author =. The American Mathematical Monthly , volume =. 2013 , doi =

    work page 2013

  73. [73]

    Quantum Amplitude Amplification and Estimation

    Quantum Amplitude Amplification and Estimation , author =. Quantum Computation and Information , series =. 2002 , doi =. quant-ph/0005055 , archivePrefix =

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  74. [74]

    2012 , publisher =

    Matrix Analysis , author =. 2012 , publisher =. doi:10.1017/CBO9781139020411 , url =

    work page doi:10.1017/cbo9781139020411 2012

  75. [75]

    Monatshefte f

    A trace inequality of John von Neumann , author =. Monatshefte f. 1975 , month =. doi:10.1007/BF01647331 , url =

    work page doi:10.1007/bf01647331 1975

  76. [76]

    Theoretical Computer Science , author =

    Random generation of combinatorial structures from a uniform distribution , author =. Theoretical Computer Science , volume =. 1986 , month =. doi:10.1016/0304-3975(86)90174-X , url =

    work page doi:10.1016/0304-3975(86)90174-x 1986

  77. [77]

    Numerical solution of large-scale

    Benner, Peter and Li, Jing-Rebecca and Penzl, Thilo , journal =. Numerical solution of large-scale. 2008 , doi =

    work page 2008

  78. [78]

    Restricted second random phase approximations and

    Peng, Degao and Yang, Yang and Zhang, Peng and Yang, Weitao , journal =. Restricted second random phase approximations and. 2014 , month =. doi:10.1063/1.4901716 , url =

    work page doi:10.1063/1.4901716 2014

  79. [79]

    Gold Standard,

    Řezáč, Jan and Hobza, Pavel , journal =. Describing Noncovalent Interactions beyond the Common Approximations: How Accurate Is the “Gold Standard,”. 2013 , doi =

    work page 2013

  80. [80]

    The Journal of Chemical Physics , volume =

    Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration , author =. The Journal of Chemical Physics , volume =. 2010 , month =. doi:10.1063/1.3442749 , url =

    work page doi:10.1063/1.3442749 2010

Showing first 80 references.