pith. sign in

arxiv: 2605.28040 · v1 · pith:6VE2PJ7Xnew · submitted 2026-05-27 · 🪐 quant-ph · physics.comp-ph

Filter-assisted quantum subspace diagonalization via wavefunction sparsity engineering

Han Xu , Tomonori Shirakawa , Seiji Yunoki This is my paper

Pith reviewed 2026-06-29 11:43 UTC · model grok-4.3

classification 🪐 quant-ph physics.comp-ph
keywords quantum subspace diagonalizationSQDwavefunction sparsityquantum filterGini coefficientquantum Ising modelmany-body systemstensor-network encoding
0
0 comments X

The pith

A quantum filter concentrates ground-state weight onto fewer basis states to improve SQD accuracy and cut sampling costs.

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

The paper introduces a filter-assisted version of sample-based quantum diagonalization that applies a unitary transformation to the Hamiltonian. This transformation is chosen to concentrate the ground-state amplitude on a small number of computational basis states, raising the sparsity of the wavefunction as quantified by the Gini coefficient. Higher sparsity directly lowers the subspace dimension and the number of samples needed for a given energy accuracy. The filter is implemented by encoding the target state into a quantum circuit via a tensor-network algorithm. Benchmarks on the quantum Ising model with transverse and longitudinal fields show orders-of-magnitude smaller energy errors and reduced sampling overhead relative to ordinary SQD.

Core claim

By applying a unitary transformation to the Hamiltonian that concentrates the ground-state weight onto fewer computational basis states, the filter-assisted SQD protocol enhances wavefunction sparsity as measured by the Gini coefficient, which in turn reduces the required subspace dimension and sampling overhead for accurate energy estimation.

What carries the argument

The quantum filter: a unitary transformation of the Hamiltonian designed to concentrate ground-state weight onto a small number of computational basis states.

If this is right

  • The subspace dimension needed for a target accuracy becomes smaller.
  • Sampling overhead for ground-state energy estimation is substantially reduced.
  • The method applies to strongly correlated regimes where standard SQD suffers from low sparsity.
  • Tensor-network circuit encoding supplies the filter with adjustable fidelity.

Where Pith is reading between the lines

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

  • The same sparsity-engineering step could be inserted into other sampling-based quantum algorithms that rely on configuration interaction.
  • If the tensor-network encoder scales, the approach might extend to system sizes beyond those reachable by direct state preparation.
  • Gini-coefficient bounds on sampling cost may generalize to other sparsity measures used in quantum chemistry.

Load-bearing premise

A tensor-network-based circuit-encoding algorithm can map the target filtered states to quantum circuits with controllable fidelity while remaining implementable on current hardware.

What would settle it

Executing the filter-assisted protocol on the quantum Ising model and measuring that the ground-state energy error does not drop by orders of magnitude relative to standard SQD would falsify the performance improvement.

Figures

Figures reproduced from arXiv: 2605.28040 by Han Xu, Seiji Yunoki, Tomonori Shirakawa.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic comparison between the standard SQD workflow and the proposed filter-assisted SQD (FSQD) protocol. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic diagram of the Lorenz curve [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic illustration of the forward and back [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Matrix representations of the original Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Deviation of the Gini coefficient from unity, 1 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Ground-state energy-estimation error [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a)–(c) Ground-state energy-estimation error [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) shows the ground-state energy-estimation error ϵ/n versus NS, in analogy with the numerical re￾sults shown in Figs. 6(a)–6(c). The colored markers de￾note the experimental results obtained directly from raw bitstring samples. The solid curves show the correspond￾ing noise-free numerical references based on the ideal sampler states |ψg⟩ and Pˆ |0¯⟩Uˆ † MPS,I |ψg⟩, i.e., the same nu￾merical results alrea… view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Optimization trajectories of the MPS-based circuit [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a) Schematic relation among the quantum [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a) Optimization trajectories of the MPS-based [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Optimization trajectories of the MPS-based circuit [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. (a) Qubit layouts and coupling paths for the [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
read the original abstract

Subspace diagonalization techniques based on quantum sampling, such as quantum selected configuration interaction (QSCI) and sample-based quantum diagonalization (SQD), have recently emerged as promising quantum-centric approaches for approximating ground-state energies of many-body systems. However, their performance is fundamentally limited by an intrinsic trade-off between sampling efficiency and the sparsity of the ground-state wavefunction, which becomes particularly severe in strongly correlated systems. Here, we introduce a filter-assisted SQD protocol that engineers wavefunction sparsity via a quantum filter, i.e., a unitary transformation of the Hamiltonian designed to concentrate the ground-state weight onto a small number of computational basis states. Using the Gini coefficient as a robust sparsity measure, we establish a quantitative relationship between wavefunction sparsity and the resource requirements of SQD, providing theoretical bounds on the required subspace dimension and sampling cost. To realize the quantum filter, we employ a tensor-network-based circuit-encoding algorithm that maps target states to quantum circuits with controllable fidelity. We benchmark our approach on the quantum Ising model with transverse and longitudinal fields using both numerical simulations and quantum hardware experiments. Our results demonstrate that, compared with standard SQD, the proposed protocol significantly enhances wavefunction sparsity, reduces ground-state energy estimation errors by orders of magnitude, and substantially lowers sampling overhead. These findings establish filter-assisted subspace diagonalization as a powerful and scalable framework for quantum many-body calculations in the strongly correlated regime.

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 / 0 minor

Summary. The paper proposes a filter-assisted sample-based quantum diagonalization (SQD) protocol that applies a unitary quantum filter, realized via a tensor-network circuit-encoding algorithm, to concentrate ground-state weight and thereby increase wavefunction sparsity as measured by the Gini coefficient. Theoretical bounds are derived relating this sparsity to reduced subspace dimension and sampling overhead in SQD; the approach is benchmarked on the transverse-plus-longitudinal-field quantum Ising model via both classical simulations and quantum hardware runs, with claims of orders-of-magnitude lower ground-state energy errors and sampling costs relative to unfiltered SQD.

Significance. If the tensor-network encoding can be shown to deliver the required controllable fidelity on hardware without eroding the engineered sparsity, the protocol would offer a concrete route to mitigating the sparsity-sampling trade-off that currently limits quantum subspace methods in strongly correlated regimes, potentially enabling larger-scale quantum-centric calculations.

major comments (1)
  1. [Hardware experiments section (Ising-model benchmarks)] The central claim of orders-of-magnitude error reduction and overhead savings is conditional on the tensor-network-based circuit-encoding algorithm producing shallow, high-fidelity circuits for the filtered states. The hardware experiments on the Ising model must explicitly quantify how the reported gate counts and circuit depths affect the achieved sparsity (Gini coefficient) and energy accuracy; without such data the theoretical bounds do not translate to practical advantage.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on the hardware experiments. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Hardware experiments section (Ising-model benchmarks)] The central claim of orders-of-magnitude error reduction and overhead savings is conditional on the tensor-network-based circuit-encoding algorithm producing shallow, high-fidelity circuits for the filtered states. The hardware experiments on the Ising model must explicitly quantify how the reported gate counts and circuit depths affect the achieved sparsity (Gini coefficient) and energy accuracy; without such data the theoretical bounds do not translate to practical advantage.

    Authors: We agree that an explicit quantification of how gate counts and circuit depths influence the achieved sparsity and energy accuracy is necessary to connect the hardware results to the theoretical bounds. The current manuscript reports gate counts, depths, Gini coefficients, and energy errors from the hardware runs, but does not include a dedicated analysis or visualization of their interdependencies. In the revised version we will add a supplementary table and accompanying discussion that tabulates these quantities for each Ising instance, together with the tensor-network encoding fidelity, and will comment on how deviations from ideal sparsity arise from finite circuit depth and noise. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain is self-contained and independent of fitted inputs or self-referential definitions.

full rationale

The paper defines a new filter-assisted SQD protocol, introduces the Gini coefficient as an external sparsity metric, derives theoretical bounds on subspace dimension and sampling cost from that metric, and specifies a tensor-network circuit-encoding algorithm to implement the filter. None of these elements reduce by construction to parameters fitted from the target data, self-citations that carry the central claim, or ansatzes smuggled from prior author work. The Ising-model benchmarks are presented as separate numerical and hardware validation. The derivation therefore stands on its own stated assumptions and external measures without the forbidden patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The subspace dimension and sampling cost appear as derived quantities rather than fitted inputs.

axioms (1)
  • domain assumption The Gini coefficient provides a robust quantitative measure of wavefunction sparsity that directly bounds SQD resource requirements.
    Invoked to establish the relationship between sparsity and sampling cost.

pith-pipeline@v0.9.1-grok · 5781 in / 1285 out tokens · 26968 ms · 2026-06-29T11:43:42.281456+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

52 extracted references · 14 canonical work pages · 3 internal anchors

  1. [1]

    Sachdev,Quantum Phase Transitions, 2nd ed

    S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cam- bridge University Press, 2011)

    2011

  2. [2]

    J. Wang, J. Surace, I. Fr´ erot, B. Legat, M.-O. Renou, V. Magron, and A. Ac´ ın, Certifying ground-state prop- erties of many-body systems, Phys. Rev. X14, 031006 (2024)

    2024

  3. [3]

    B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri,et al., Towards quantum chemistry on a quantum computer, Nature Chem.2, 106 (2010)

    2010

  4. [4]

    S. Lee, J. Lee, H. Zhai, Y. Tong, A. M. Dalzell, A. Ku- mar, P. Helms, J. Gray, Z.-H. Cui, W. Liu,et al., Eval- uating the evidence for exponential quantum advantage in ground-state quantum chemistry, Nat. Commun.14, 1952 (2023)

    1952

  5. [5]

    W. J. Huggins, J. Lee, U. Baek, B. O’Gorman, and K. B. Whaley, A non-orthogonal variational quantum eigen- solver, New J. Phys.22, 073009 (2020)

    2020

  6. [6]

    Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral op- erators, J

    C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral op- erators, J. Res. Natl. Bur. Stand.45, 255 (1950)

    1950

  7. [7]

    Koch, The Lanczos method, The LDA+DMFT ap- proach to strongly correlated materials (2011)

    E. Koch, The Lanczos method, The LDA+DMFT ap- proach to strongly correlated materials (2011)

    2011

  8. [8]

    Seki and S

    K. Seki and S. Yunoki, Quantum power method by a superposition of time-evolved states, PRX Quantum2, 010333 (2021)

    2021

  9. [9]

    N. H. Stair, R. Huang, and F. A. Evangelista, A multiref- erence quantum Krylov algorithm for strongly correlated electrons, J. Chem. Theory Comput.16, 2236 (2020)

    2020

  10. [10]

    C. L. Cortes and S. K. Gray, Quantum Krylov subspace algorithms for ground- and excited-state energy estima- tion, Phys. Rev. A105, 022417 (2022)

    2022

  11. [11]

    R. M. Parrish and P. L. McMahon, Quantum filter diagonalization: Quantum eigendecomposition without full quantum phase estimation (2019), arXiv:1909.08925 23 TABLE II. Single- and two-qubit properties ofibm kobeused in this study, including the qubit relaxation timeT 1, qubit dephasing timeT 2, readout error, and CZ error. Property Min. Max. Median Avg. n...

    work page arXiv 2019

  12. [12]

    J. Cohn, M. Motta, and R. M. Parrish, Quantum filter di- agonalization with compressed double-factorized hamil- tonians, PRX Quantum2, 040352 (2021)

    2021

  13. [13]

    T. A. Bespalova and O. Kyriienko, Hamiltonian operator approximation for energy measurement and ground-state preparation, PRX Quantum2, 030318 (2021)

    2021

  14. [14]

    Motta, C

    M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. Brandao, and G. K.-L. Chan, De- termining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nat. Phys.16, 205 (2020)

    2020

  15. [15]

    Yeter-Aydeniz, R

    K. Yeter-Aydeniz, R. C. Pooser, and G. Siopsis, Practi- cal quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and lanc- zos algorithms, npj Quantum Inf.6, 63 (2020)

    2020

  16. [16]

    Frame, R

    D. Frame, R. He, I. Ipsen, D. Lee, D. Lee, and E. Rrapaj, Eigenvector continuation with subspace learning, Phys. Rev. Lett.121, 032501 (2018)

    2018

  17. [17]

    Francis, A

    A. Francis, A. A. Agrawal, J. H. Howard, E. K¨ okc¨ u, and A. F. Kemper, Subspace diagonalization on quan- tum computers using eigenvector continuation (2022), arXiv:2209.10571 [quant-ph]

    work page arXiv 2022

  18. [18]

    Quantum-Selected Configuration Interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum computers

    K. Kanno, M. Kohda, R. Imai, S. Koh, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, Quantum-selected configuration interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum comput- ers (2023), arXiv:2302.11320 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  19. [19]

    Robledo-Moreno, M

    J. Robledo-Moreno, M. Motta, H. Haas, A. Javadi- Abhari, P. Jurcevic, W. Kirby, S. Martiel, K. Sharma, S. Sharma, T. Shirakawa, I. Sitdikov, R.-Y. Sun, K. J. Sung, M. Takita, M. C. Tran, S. Yunoki, and A. Mezza- capo, Chemistry beyond the scale of exact diagonaliza- tion on a quantum-centric supercomputer, Sci. Adv.11, eadu9991 (2025)

    2025

  20. [20]

    J. Yu, J. R. Moreno, J. T. Iosue, L. Bertels, D. Claudino, B. Fuller, P. Groszkowski, T. S. Humble, P. Jurce- vic, W. Kirby, T. A. Maier, M. Motta, B. Pokharel, A. Seif, A. Shehata, K. J. Sung, M. C. Tran, V. Tripathi, A. Mezzacapo, and K. Sharma, Quantum-centric algo- rithm for sample-based Krylov diagonalization (2025), arXiv:2501.09702 [quant-ph]

    work page arXiv 2025

  21. [21]

    Sugisaki, S

    K. Sugisaki, S. Kanno, T. Itoko, R. Sakuma, and N. Yamamoto, Hamiltonian simulation-based quantum- selected configuration interaction for large-scale elec- tronic structure calculations with a quantum computer (2025), arXiv:2412.07218 [quant-ph]

    work page arXiv 2025

  22. [22]

    Mikkelsen and Y

    M. Mikkelsen and Y. O. Nakagawa, Quantum-selected configuration interaction with time-evolved state (2024), arXiv:2412.13839 [quant-ph]

    work page arXiv 2024

  23. [23]

    Nogaki, S

    K. Nogaki, S. Backes, T. Shirakawa, S. Yunoki, and R. Arita, Symmetry-adapted sample-based quantum di- agonalization: Application to lattice model (2025), arXiv:2505.00914 [quant-ph]

    work page arXiv 2025

  24. [24]

    Danilov, J

    D. Danilov, J. Robledo-Moreno, K. J. Sung, M. Motta, and J. Shee, Enhancing the accuracy and effi- ciency of sample-based quantum diagonalization with phaseless auxiliary-field quantum Monte Carlo (2025), arXiv:2503.05967 [quant-ph]

    work page arXiv 2025

  25. [25]

    Barison, J

    S. Barison, J. R. Moreno, and M. Motta, Quantum- centric computation of molecular excited states with ex- tended sample-based quantum diagonalization, QST10, 025034 (2025)

    2025

  26. [26]

    Duriez, P

    A. Duriez, P. C. Carvalho, M. A. Barroca, F. Zipoli, B. Jaderberg, R. N. B. Ferreira, K. Sharma, A. Mezza- capo, B. Wunsch, and M. Steiner, Computing band gaps of periodic materials via sample-based quantum diago- nalization (2025), arXiv:2503.10901 [quant-ph]

    work page arXiv 2025

  27. [27]

    Ohgoe, H

    T. Ohgoe, H. Iwakiri, K. Ichikawa, S. Koh, and M. Ko- hda, Quantum computation of a quasiparticle band struc- ture with the quantum-selected configuration interaction (2025), arXiv:2504.00309 [quant-ph]

    work page arXiv 2025

  28. [28]

    Shirakawa, J

    T. Shirakawa, J. Robledo-Moreno, T. Itoko, V. Tripathi, K. Ueda, Y. Kawashima, L. Broers, W. Kirby, H. Pathak, H. Paik, M. Tsuji, Y. Kodama, M. Sato, C. Evangelinos, S. Seelam, R. Walkup, S. Yunoki, M. Motta, P. Jurce- vic, H. Horii, and A. Mezzacapo, Closed-loop calcula- tions of electronic structure on a quantum processor and a classical supercomputer a...

    work page arXiv 2025

  29. [29]

    K. M. Merz, Jr., A. Shajan, D. Kaliakin, F. Liang, Y. Otsuka, T. Shirakawa, L. Broers, H. Xu, M. Tsuji, M. Sato, S. Yunoki, R. Wakizaka, Y. Kawashima, J. Doi, T. Itoko, H. Horii, T. Pellegrini, J. Robledo Moreno, K. J. Sung, E. Fejer, R. Walkup, S. Seelam, and M. Motta, Crossing the 12,000-atom barrier with het- erogeneous quantum-classical supercomputing...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  30. [30]

    Reinholdt, K

    P. Reinholdt, K. M. Ziems, E. R. Kjellgren, S. Coriani, S. P. A. Sauer, and J. Kongsted, Critical limitations in quantum-selected configuration interaction methods, J. Chem. Theory Comput.21, 6811 (2025)

    2025

  31. [31]

    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, arXiv:2201.08309 (2022)

    work page arXiv 2022

  32. [32]

    M. O. Lorenz, Methods of measuring the concentration of wealth, J. Am. Stat. Assoc.9, 209 (1905)

    1905

  33. [33]

    J. L. Gastwirth, A general definition of the lorenz curve, Econometrica39, 1037 (1971)

    1971

  34. [34]

    Gini, Measurement of inequality of incomes, Econ

    C. Gini, Measurement of inequality of incomes, Econ. J. 31, 124 (1921). 24

    1921

  35. [35]

    Dalton, The measurement of the inequality of incomes, Econ

    H. Dalton, The measurement of the inequality of incomes, Econ. J.30, 348 (1920)

    1920

  36. [36]

    Hurley and S

    N. Hurley and S. Rickard, Comparing measures of spar- sity, IEEE Trans. Inf. Theory55, 4723 (2009)

    2009

  37. [37]

    A note on the coupon - collector's problem with multiple arrivals and the random sampling

    M. Ferrante and N. Frigo, A note on the coupon - col- lector’s problem with multiple arrivals and the random sampling (2012), arXiv:1209.2667 [math.PR]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  38. [38]

    Shirakawa, H

    T. Shirakawa, H. Ueda, and S. Yunoki, Automatic quan- tum circuit encoding of a given arbitrary quantum state, Phys. Rev. Res.6, 043008 (2024)

    2024

  39. [39]

    Ran, Encoding of matrix product states into quan- tum circuits of one- and two-qubit gates, Phys

    S.-J. Ran, Encoding of matrix product states into quan- tum circuits of one- and two-qubit gates, Phys. Rev. A 101, 032310 (2020)

    2020

  40. [40]

    D. Malz, G. Styliaris, Z.-Y. Wei, and J. I. Cirac, Prepa- ration of matrix product states with log-depth quantum circuits, Phys. Rev. Lett.132, 040404 (2024)

    2024

  41. [41]

    K. C. Smith, A. Khan, B. K. Clark, S. Girvin, and T.- C. Wei, Constant-depth preparation of matrix product states with adaptive quantum circuits, PRX Quantum 5, 030344 (2024)

    2024

  42. [42]

    Pfeuty, The one-dimensional ising model with a trans- verse field, Ann

    P. Pfeuty, The one-dimensional ising model with a trans- verse field, Ann. Phys.57, 79 (1970)

    1970

  43. [43]

    E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys.16, 407 (1961)

    1961

  44. [44]

    Sachdev, Quantum phase transitions, Physics world 12, 33 (1999)

    S. Sachdev, Quantum phase transitions, Physics world 12, 33 (1999)

    1999

  45. [45]

    Coldea, D

    R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzyn- ska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Quantum criticality in an ising chain: Ex- perimental evidence for emergent e8 symmetry, Science 327, 177 (2010)

    2010

  46. [46]

    Bezanson, A

    J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Review59, 65 (2017)

    2017

  47. [47]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calcula- tions, SciPost Phys. Codebases , 4 (2022)

    2022

  48. [48]

    Sorella, Generalized lanczos algorithm for variational quantum monte carlo, Phys

    S. Sorella, Generalized lanczos algorithm for variational quantum monte carlo, Phys. Rev. B64, 024512 (2001)

    2001

  49. [49]

    Mizusaki and M

    T. Mizusaki and M. Imada, Precise estimation of shell model energy by second-order extrapolation method, Phys. Rev. C67, 041301 (2003)

    2003

  50. [50]

    Qiskit contributors, Qiskit: An open-source framework for quantum computing (2023)

    2023

  51. [51]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, 2010)

    2010

  52. [52]

    Li and S

    Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X7, 021050 (2017)

    2017