pith. sign in

arxiv: 2606.27734 · v1 · pith:UHPNSLQ4new · submitted 2026-06-26 · 🪐 quant-ph · cond-mat.mtrl-sci· cond-mat.str-el

Fault tolerant computation of the static structure factor and finite size effects

Rishabh Bhardwaj , Alexander Reed Mu\~noz , Travis E. Jones , John Golden This is my paper

Pith reviewed 2026-06-29 04:59 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mtrl-scicond-mat.str-el
keywords fault-tolerant quantum computingstatic structure factorfinite-size effectsperiodic materialsBloch orbitalsdensity fluctuationsHadamard testtwist averaging
0
0 comments X

The pith

Quantum post-processing of the static structure factor corrects the leading two-body finite-size error in periodic ground-state calculations after twist averaging.

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

The paper develops a fault-tolerant strategy to estimate the small-momentum static structure factor S(q) and thereby remove the dominant residual finite-size error once one-body shell effects have been averaged out. The method encodes the relevant operator in a Bloch-orbital basis, block-encodes it through the density operator, and extracts its ground-state expectation value with an amplified Hadamard test. Adaptive global and local binary searches locate the infrared fitting window needed to reconstruct the correction. The resulting overhead scales as tilde O((N_b N_k)^3) and uses only tilde O(N_b N_k) logical qubits, remaining subdominant to the main energy-estimation routine and avoiding the plane-wave overhead of full Hamiltonian simulation.

Core claim

After twist averaging suppresses one-body shell effects, the leading two-body finite-size correction is obtained from the ground-state expectation value of the static structure factor operator S(q) at small momentum, formulated in the Bloch-orbital basis, block-encoded via the density operator, and measured by amplified Hadamard test; the procedure requires only adaptive infrared-window searches and incurs subleading cost relative to energy estimation.

What carries the argument

The small-momentum static structure factor S(q), block-encoded through the density operator and measured by amplified Hadamard test.

If this is right

  • The two-body correction can be computed without repeating the full ground-state energy estimation on successively larger supercells.
  • The qubit and gate cost of the correction remains subleading to the primary energy-estimation algorithm.
  • The method replaces down-sampling with targeted infrared density-correlation measurements.
  • Only tilde O(N_b N_k) logical qubits are needed instead of the larger plane-wave basis required for Hamiltonian simulation.

Where Pith is reading between the lines

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

  • The same block-encoding and Hadamard-test machinery could be reused to measure other two-body density correlators that appear in finite-size or embedding corrections.
  • The adaptive binary-search procedure for the fitting window may extend to other infrared extrapolations that arise in periodic quantum simulations.
  • Because the correction is obtained after the main energy estimation, it can be applied as a post-processing step to any fault-tolerant algorithm that already produces the ground state in the Bloch basis.

Load-bearing premise

Once one-body shell effects are removed by twist averaging, the remaining finite-size error is dominated by long-wavelength density fluctuations captured by S(q).

What would settle it

A direct numerical comparison, on a small periodic system with known exact finite-size error, showing that the S(q)-based correction fails to reproduce the difference between finite-cell and thermodynamic-limit energies.

Figures

Figures reproduced from arXiv: 2606.27734 by Alexander Reed Mu\~noz, John Golden, Rishabh Bhardwaj, Travis E. Jones.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic overview of finite-size mitigation as a quantum post-processing task of the quantum workflow of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration of twist averaging. Each twist shifts the underlying finite [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic illustration of why the leading two-body finite-size correction is dominated by the small- [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic illustration of the global adaptive search for the infrared cutoff. Left: [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic illustration of the local windowed search for the infrared cutoff. Left: for each candidate cutoff [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Quantum circuit for the block encoding of the density operator away from the Γ-point. The circuit implements the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Quantum circuit for the block encoding of the Γ-point density operator ˆρ [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Local logarithmic exponent of the static structure factor, [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
read the original abstract

Fault-tolerant quantum algorithms offer a promising pathway for estimating the ground-state energies of periodic materials that are beyond the practical reach of classical electronic-structure methods. A remaining challenge is finite-size mitigation: quantum algorithms evaluate a finite supercell or finite Brillouin-zone mesh, while materials properties are defined in the thermodynamic limit. In this work we develop a quantum post-processing strategy for the leading two-body finite-size correction. After one-body shell effects are reduced by twist averaging, the dominant residual error is controlled by long-wavelength density fluctuations, which are encoded in the small-momentum static structure factor $S(q)$. We formulate the corresponding operator in a Bloch-orbital basis, construct its block encoding through the density operator, and estimate its ground-state expectation value using an amplified Hadamard test. We also introduce adaptive global and local binary search procedures for identifying the infrared fitting window used to reconstruct the two-body finite size error correction. The resulting cost remains subleading relative to the main ground-state energy estimation routine: the structure-factor correction has leading $\tilde{O}(N_bN_k)^3$ dependence on the Bloch-orbital basis size, avoids the large plane-wave prefactor of full Hamiltonian simulation, and requires only $\tilde{O}(N_bN_k)$ logical qubits. This provides a fault-tolerant alternative to down-sampling, replacing repeated energy calculations on larger cells with targeted measurements of the infrared density correlations that control the finite-size effects.

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 quantum post-processing method to compute the leading two-body finite-size correction for ground-state energies of periodic materials. After twist averaging mitigates one-body shell effects, the residual error is asserted to be dominated by long-wavelength density fluctuations captured by the small-q static structure factor S(q). The authors formulate the corresponding operator in a Bloch-orbital basis, construct its block encoding from the density operator, estimate its ground-state expectation value via an amplified Hadamard test, and introduce adaptive global/local binary searches to identify the infrared fitting window. The claimed cost is sub-leading: ilde{O}((N_b N_k)^3) gate complexity with only ilde{O}(N_b N_k) logical qubits, providing a fault-tolerant alternative to repeated larger-cell simulations.

Significance. If the central premise and cost analysis hold, the approach supplies a targeted, low-overhead correction that avoids the plane-wave prefactors of full Hamiltonian simulation and reduces the need for down-sampling. This would be a practical addition to the fault-tolerant quantum electronic-structure toolkit, particularly for systems where finite-size errors are dominated by infrared density correlations.

major comments (2)
  1. [Abstract and finite-size mitigation section] Abstract and § on finite-size mitigation: the claim that twist averaging leaves two-body S(q) fluctuations as the dominant residual error is presented without bounds, counter-example analysis, or conditions of validity (e.g., gapped vs. metallic systems, incomplete Bloch basis). If this premise fails, the sub-leading cost no longer guarantees the correct thermodynamic-limit correction.
  2. [Abstract] Abstract: no derivations of the block encoding, no error analysis for the amplified Hadamard test, no numerical validation on model systems, and no explicit operator constructions are supplied, so the stated ilde{O}((N_b N_k)^3) scaling and qubit count cannot be verified from the text.
minor comments (2)
  1. [Adaptive binary search procedures] Define the infrared fitting window and the precise functional form assumed for S(q) at small q (constant vs. 1/q^2) with an equation reference.
  2. [Operator construction] Clarify whether the Bloch-orbital basis size N_b N_k already incorporates the plane-wave cutoff or is strictly the number of occupied bands times k-points.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to clarify our manuscript. Below we respond point-by-point to the major comments, indicating revisions where appropriate to strengthen the presentation and scope of the claims.

read point-by-point responses

  1. Referee: [Abstract and finite-size mitigation section] Abstract and § on finite-size mitigation: the claim that twist averaging leaves two-body S(q) fluctuations as the dominant residual error is presented without bounds, counter-example analysis, or conditions of validity (e.g., gapped vs. metallic systems, incomplete Bloch basis). If this premise fails, the sub-leading cost no longer guarantees the correct thermodynamic-limit correction.

    Authors: We agree that the conditions of validity for the dominance of two-body S(q) fluctuations after twist averaging merit explicit discussion. The claim draws from established results in the classical periodic electronic-structure literature, but the manuscript does not provide bounds or counter-examples. In revision we will add a dedicated paragraph in the finite-size mitigation section stating the assumptions (sufficiently dense k-mesh, gapped systems with complete Bloch basis) and noting that metallic or strongly correlated cases may require additional corrections. This clarifies the regime in which the sub-leading cost applies without altering the algorithm. revision: yes

  2. Referee: [Abstract] Abstract: no derivations of the block encoding, no error analysis for the amplified Hadamard test, no numerical validation on model systems, and no explicit operator constructions are supplied, so the stated ilde{O}((N_b N_k)^3) scaling and qubit count cannot be verified from the text.

    Authors: The abstract is intentionally concise. The full manuscript contains the block-encoding construction (via the density operator in the Bloch basis), the error analysis for the amplified Hadamard test (including precision and amplification overhead), and the explicit operator definitions. However, we acknowledge that these elements are not cross-referenced from the abstract and that no numerical validation on model systems is currently included. In revision we will (i) add explicit cross-references in the abstract, (ii) expand the methods section with a concise derivation summary and qubit-count derivation, and (iii) include a short numerical demonstration on the uniform electron gas to illustrate the scaling. These additions will make the cost claims directly verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation measures external physical quantity S(q) via standard quantum primitives

full rationale

The paper's central claim is a fault-tolerant algorithm to estimate the static structure factor S(q) in a Bloch basis and use its small-q behavior to correct finite-size errors after twist averaging. This is framed as an independent measurement of an external observable (density fluctuations) using block encoding of the density operator and amplified Hadamard test; no equation reduces the claimed correction to a fitted parameter, self-referential definition, or prior self-citation. The cost scaling and qubit count are derived from standard quantum simulation primitives without importing uniqueness theorems or ansatzes from the authors' prior work. The premise that twist averaging leaves two-body S(q) as dominant is an assumption about physics, not a definitional loop. No load-bearing step collapses by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that long-wavelength fluctuations dominate residual finite-size error after twist averaging and that the density operator admits an efficient block encoding whose cost is subleading to Hamiltonian simulation. No free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Dominant residual finite-size error after twist averaging arises from long-wavelength density fluctuations encoded in small-momentum S(q)
    Stated directly in the abstract as the justification for targeting S(q) for the two-body correction.

pith-pipeline@v0.9.1-grok · 5804 in / 1327 out tokens · 40402 ms · 2026-06-29T04:59:28.535961+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

56 extracted references · 16 canonical work pages · 2 internal anchors

  1. [1]

    To this end, we define the walk operator ˆQW = PREP·SEL·PREP † ,(F2) following the prepare–select framework of Ref

    At theΓ-point At the Γ-point we consider only the modeQ=k= 0, so the density operator in the LCU representation reduces to ˆρq Γ ≡ˆρ= 1 2 X J=0,1 Nb/2X i=1 sign(f(J) i )|f(J) i | ˆU(J) Z(J) i ˆU(J)† .(F1) Our goal is to encode this operator as a block of a larger unitary matrix. To this end, we define the walk operator ˆQW = PREP·SEL·PREP † ,(F2) followin...

  2. [2]

    Away from theΓ-point We now turn to the general density operator for fixed nonzero momentum transferq. The corresponding walk operator is ˆQ(q) W = PREPq ·SEL q ·PREP † q .(F19) Relative to the Γ-point case, the key changes are that the LCU superposition must now include an explicit crystal- momentum label, and the select operation must route the controll...

  3. [3]

    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 Communications14, 10.1038/s41467-023- 37587-6 (2023)

    work page doi:10.1038/s41467-023- 2023

  4. [4]

    Gundlach, K

    H. Gundlach, K. Sharkey, J. Lynch, V. Hazoglou, K.-C. Hsu, C. Dukatz, E. Crane, K. Walczyk, M. Bodziak, J. Galatsanos- Dueck, and N. Thompson, Quantum advantage in computational chemistry? (2025), arXiv:2508.20972 [quant-ph]

    work page arXiv 2025

  5. [5]

    S. N. Genin, O. Kwon, S. M. H. Jenab, S.-J. Lim, T. Kim, T.-G. Kim, R. Gherib, A. F. Harper, I. G. Ryabinkin, and M. G. Helander, Towards quantum advantage in chemistry (2026), arXiv:2512.13657 [physics.chem-ph]

    work page arXiv 2026

  6. [6]

    Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferov´ a, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. D. Sawaya, S. Sim, L. Veis, and A. Aspuru-Guzik, Quantum chemistry in the age of quantum computing, Chemical 30 Reviews119, 10856–10915 (2019)

    2019

  7. [7]

    Schleich, L

    P. Schleich, L. M. Calder´ on, C. Sun, M. Bagherimehrab, A. Aldossary, J. S. Kottmann, and A. Aspuru-Guzik,Quantum Computing for Quantum Chemistry(American Chemical Society, Washington, DC, USA, 2025)

    2025

  8. [8]

    Babbush, C

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear t complexity, Phys. Rev. X8, 041015 (2018)

    2018

  9. [9]

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

    2021

  10. [10]

    G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum3, 163 (2019)

    2019

  11. [11]

    N. C. Rubin, D. W. Berry, F. D. Malone, A. F. White, T. Khattar, A. E. DePrince, S. Sicolo, M. K¨ uehn, M. Kaicher, J. Lee, and R. Babbush, Fault-tolerant quantum simulation of materials using bloch orbitals, PRX Quantum4, 040303 (2023)

    2023

  12. [12]

    A. V. Ivanov, A. Patterson, M. Bothe, C. S¨ underhauf, B. K. Berntson, J. J. Mortensen, M. Kuisma, E. Campbell, and R. Izs´ ak, Quantum computation of electronic structure with projector augmented-wave method and plane wave basis set (2025), arXiv:2408.03159 [quant-ph]

    work page arXiv 2025

  13. [13]

    Fault-tolerant simulation of the electronic structure using Projector Augmented-Waves and Bloch orbitals

    R. Bhardwaj, A. R. Mu˜ noz, T. E. Jones, and J. Golden, Fault-tolerant simulation of the electronic structure using Projector Augmented-Waves and Bloch orbitals, arXiv preprint arXiv:2604.12142 (2026), arXiv:2604.12142 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  14. [14]

    Poilblanc, Twisted boundary conditions in cluster calculations of the optical conductivity in two-dimensional lattice models, Physical Review B44, 9562 (1991)

    D. Poilblanc, Twisted boundary conditions in cluster calculations of the optical conductivity in two-dimensional lattice models, Physical Review B44, 9562 (1991)

    1991

  15. [15]

    C. Lin, F. H. Zong, and D. M. Ceperley, Twist-averaged boundary conditions in continuum quantum monte carlo algorithms, Physical Review E64, 10.1103/physreve.64.016702 (2001)

    work page doi:10.1103/physreve.64.016702 2001

  16. [16]

    Shimazaki and S

    T. Shimazaki and S. Hirata, On the brillouin-zone integrations in second-order many-body perturbation calculations for extended systems of one-dimensional periodicity, International Journal of Quantum Chemistry109, 2953 (2009)

    2009

  17. [17]

    A. A. Shukri, F. Bruneval, and L. Reining, Ab initio electronic stopping power of protons in bulk materials, Phys. Rev. B 93, 035128 (2016)

    2016

  18. [18]

    Taheridehkordi, M

    A. Taheridehkordi, M. Schlipf, Z. Sukurma, M. Humer, A. Gr¨ uneis, and G. Kresse, Phaseless auxiliary field quantum monte carlo with projector-augmented wave method for solids, The Journal of Chemical Physics159, 044109 (2023)

    2023

  19. [19]

    Holzmann, R

    M. Holzmann, R. C. Clay, M. A. Morales, N. M. Tubman, D. M. Ceperley, and C. Pierleoni, Theory of finite size effects for electronic quantum monte carlo calculations of liquids and solids, Physical Review B94, 10.1103/physrevb.94.035126 (2016)

    work page doi:10.1103/physrevb.94.035126 2016

  20. [20]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Computational complexity and fundamental limitations to fermionic quantum monte carlo simulations, Physical Review Letters94, 170201 (2005)

    2005

  21. [21]

    E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, Sign problem in the numerical simulation of many-electron systems, Physical Review B41, 9301 (1990)

    1990

  22. [22]

    W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Quantum monte carlo simulations of solids, Reviews of Modern Physics73, 33 (2001)

    2001

  23. [23]

    F. F. Assaad and H. G. Evertz, World-line and determinantal quantum monte carlo methods for spins, phonons and electrons, inComputational Many-Particle Physics, Lecture Notes in Physics, Vol. 739 (Springer, 2008) pp. 277–356

    2008

  24. [24]

    A. V. Ivanov, C. S¨ underhauf, N. Holzmann, T. Ellaby, R. N. Kerber, G. Jones, and J. Camps, Quantum computation for periodic solids in second quantization, Physical Review Research5, 10.1103/physrevresearch.5.013200 (2023)

    work page doi:10.1103/physrevresearch.5.013200 2023

  25. [25]

    J. J. Shepherd, G. E. Scuseria, and J. S. Spencer, Sign problem in full configuration interaction quantum monte carlo: Linear and sublinear representation regimes for the exact wave function, Phys. Rev. B90, 155130 (2014)

    2014

  26. [26]

    Liao and A

    K. Liao and A. Gr¨ uneis, Communication: Finite size correction in periodic coupled cluster theory calculations of solids, The Journal of Chemical Physics145, 141102 (2016)

    2016

  27. [27]

    Shi and S

    H. Shi and S. Zhang, Some recent developments in auxiliary-field quantum monte carlo for real materials, The Journal of Chemical Physics154, 024107 (2021)

    2021

  28. [28]

    Azadi and W

    S. Azadi and W. M. C. Foulkes, Efficient method for grand-canonical twist averaging in quantum monte carlo calculations, Phys. Rev. B100, 245142 (2019)

    2019

  29. [29]

    Annaberdiyev, P

    A. Annaberdiyev, P. Ganesh, and J. T. Krogel, Enhanced twist-averaging technique for magnetic metals: Applications using quantum monte carlo, Journal of Chemical Theory and Computation20, 2786–2797 (2024)

    2024

  30. [30]

    Dobardˇ zi´ c, M

    E. Dobardˇ zi´ c, M. V. Milovanovi´ c, and N. Regnault, Geometrical description of fractional chern insulators based on static structure factor calculations, Phys. Rev. B88, 115117 (2013)

    2013

  31. [31]

    R. G. Pereira, J. Sirker, J.-S. Caux, R. Hagemans, J. M. Maillet, S. R. White, and I. Affleck, Dynamical structure factor at small q for the xxz spin-1/2 chain, Journal of Statistical Mechanics: Theory and Experiment2007, P08022 (2007)

    2007

  32. [32]

    Shukla, Ab initio hartree-fock computation of the electronic static structure factor for crystalline insulators: Benchmark results on lif, Phys

    A. Shukla, Ab initio hartree-fock computation of the electronic static structure factor for crystalline insulators: Benchmark results on lif, Phys. Rev. B60, 4539 (1999)

    1999

  33. [33]

    N. D. Drummond, R. J. Needs, A. Sorouri, and W. M. C. Foulkes, Finite-size errors in continuum quantum monte carlo calculations, Phys. Rev. B78, 125106 (2008)

    2008

  34. [34]

    Chiesa, D

    S. Chiesa, D. M. Ceperley, R. M. Martin, and M. Holzmann, Random phase approximation and the finite size errors in many body simulations, AIP Conference Proceedings918, 284 (2007)

    2007

  35. [35]

    Dornheim and J

    T. Dornheim and J. Vorberger, Overcoming finite-size effects in electronic structure simulations at extreme conditions, The Journal of Chemical Physics154, 10.1063/5.0045634 (2021)

    work page doi:10.1063/5.0045634 2021

  36. [36]

    Chiesa, D

    S. Chiesa, D. M. Ceperley, R. M. Martin, and M. Holzmann, Finite-size error in many-body simulations with long-range interactions, Phys. Rev. Lett.97, 076404 (2006)

    2006

  37. [37]

    Gaudoin, I

    R. Gaudoin, I. G. Gurtubay, and J. M. Pitarke, Momentum-space finite-size corrections for quantum monte carlo calcula- 31 tions, Phys. Rev. B85, 125125 (2012)

    2012

  38. [38]

    Y. Tong, D. An, N. Wiebe, and L. Lin, Fast inversion, preconditioned quantum linear system solvers, fast green’s-function computation, and fast evaluation of matrix functions, Phys. Rev. A104, 032422 (2021)

    2021

  39. [39]

    Lin, Lecture notes on quantum algorithms for scientific computation (2022), arXiv:2201.08309 [quant-ph]

    L. Lin, Lecture notes on quantum algorithms for scientific computation (2022), arXiv:2201.08309 [quant-ph]

    work page arXiv 2022

  40. [40]

    Marteau, Ground state preparation via qubitization (2023), arXiv:2306.14993 [quant-ph]

    C. Marteau, Ground state preparation via qubitization (2023), arXiv:2306.14993 [quant-ph]

    work page arXiv 2023

  41. [41]

    Y. Dong, L. Lin, and Y. Tong, Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices, PRX Quantum3, 040305 (2022)

    2022

  42. [42]

    Fomichev, K

    S. Fomichev, K. Hejazi, M. S. Zini, M. Kiser, J. Fraxanet, P. A. M. Casares, A. Delgado, J. Huh, A.-C. Voigt, J. E. Mueller, and J. M. Arrazola, Initial state preparation for quantum chemistry on quantum computers, PRX Quantum5, 10.1103/prxquantum.5.040339 (2024)

    work page doi:10.1103/prxquantum.5.040339 2024

  43. [43]

    Pathak, A

    S. Pathak, A. E. Russo, S. K. Seritan, and A. D. Baczewski, Quantifyingt-gate-count improvements for ground-state-energy estimation with near-optimal state preparation, Phys. Rev. A107, L040601 (2023)

    2023

  44. [44]

    Gidney, A classical-quantum adder with constant workspace and linear gates (2025), arXiv:2507.23079 [quant-ph]

    C. Gidney, A classical-quantum adder with constant workspace and linear gates (2025), arXiv:2507.23079 [quant-ph]

    work page arXiv 2025

  45. [45]

    Dagrada, S

    M. Dagrada, S. Karakuzu, V. L. Vildosola, M. Casula, and S. Sorella, Exact special twist method for quantum monte carlo simulations, Physical Review B94, 245108 (2016)

    2016

  46. [46]

    T. N. Mihm, A. R. McIsaac, and J. J. Shepherd, An optimized twist angle to find the twist-averaged correlation energy applied to the uniform electron gas, The Journal of Chemical Physics150, 191101 (2019)

    2019

  47. [47]

    T. N. Mihm, W. Z. Van Benschoten, and J. J. Shepherd, Accelerating convergence to the thermodynamic limit with twist angle selection applied to methods beyond many-body perturbation theory, The Journal of Chemical Physics154, 024113 (2021)

    2021

  48. [48]

    T. N. Mihm, T. Sch¨ afer, S. K. Ramadugu, L. Weiler, A. Gr¨ uneis, and J. J. Shepherd, A shortcut to the thermodynamic limit for quantum many-body calculations of metals, Nature Computational Science1, 801 (2021)

    2021

  49. [49]

    Zaklama, D

    T. Zaklama, D. Luo, and L. Fu, Structure factor and topological bound of twisted bilayer semiconductors at fractional fillings, Phys. Rev. B112, L041115 (2025)

    2025

  50. [50]

    Watanabe, H

    N. Watanabe, H. Hayashi, Y. Udagawa, S. L. Ten-no, and S. Iwata, Static structure factor and electron correlation effects studied by inelastic x-ray scattering spectroscopy, Journal of Chemical Physics108, 4545 (1998)

    1998

  51. [51]

    Mazzone, F

    G. Mazzone, F. Sacchetti, and V. Contini, Static structure factor and electron-electron correlations in be. a comparison of experiment with electron-gas theory, Phys. Rev. B28, 1772 (1983)

    1983

  52. [52]

    E. C. Svensson, V. F. Sears, A. D. B. Woods, and P. Martel, Neutron-diffraction study of the static structure factor and pair correlations in liquid 4He, Phys. Rev. B21, 3638 (1980)

    1980

  53. [53]

    L. N. Trefethen and J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Review56, 385 (2014), https://doi.org/10.1137/130932132

    work page doi:10.1137/130932132 2014

  54. [54]

    Kazashi, Y

    Y. Kazashi, Y. Suzuki, and T. Goda, Suboptimality of gauss–hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness, SIAM Journal on Numerical Analysis61, 1426 (2023), https://doi.org/10.1137/22M1480276

    work page doi:10.1137/22m1480276 2023

  55. [55]

    J. Dick, C. Irrgeher, G. Leobacher, and F. Pillichshammer, On the optimal order of integration in hermite spaces with finite smoothness (2017), arXiv:1608.06061 [math.NA]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  56. [56]

    Lindhard, On the properties of a gas of charged particles, Kgl

    J. Lindhard, On the properties of a gas of charged particles, Kgl. Danske Videnskab. Selskab Mat.-fys. Medd.Vol: 28, No. 8(1954)

    1954