pith. sign in

arxiv: 2607.01560 · v1 · pith:ZULGC4YKnew · submitted 2026-07-02 · 🪐 quant-ph · physics.chem-ph

Symmetry conservation with Trotterization and Quantum Phase Estimation

Edith Leal-S\'anchez , Fanny Vain , Jong-Kwon Ha , Ryan J. MacDonell This is my paper

Pith reviewed 2026-07-03 00:36 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-ph
keywords Trotterizationsymmetry conservationquantum phase estimationquantum chemistry simulationPauli string decompositionAbelian symmetriesfermionic encodings
0
0 comments X

The pith

Hermitian excitation operators let Trotterized evolution conserve Abelian symmetries such as electron number for any qubit encoding.

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

The paper shows that Hamiltonians can be rewritten using Hermitian excitation operators that decompose into sums of commuting Pauli strings. These operators preserve symmetries tied to Abelian groups during Trotterized time evolution, which is a core step in many quantum chemistry algorithms including quantum phase estimation. For non-Abelian symmetries the authors introduce operator kirigami, a combination of orthogonal projection and unitary folding to restore conservation. Numerical tests on small molecules indicate that symmetry-conserving operator pools produce larger errors than unrestricted pools at first order, but the errors shrink with system size and vanish at second order. The approach therefore supplies a systematic way to maintain physical symmetries without extra post-selection or penalty terms.

Core claim

We show that we can express the Hamiltonian in terms of Hermitian excitation operators which map to sums of commuting Pauli strings for any encoding and conserve symmetries corresponding to Abelian groups of symmetry operators. Symmetries corresponding to non-Abelian groups are not fully conserved by Trotterized Hermitian excitation operators, so we developed operator kirigami to cut the sum of non-commuting operators by orthogonal projection and to fold terms together using unitary rotations.

What carries the argument

Hermitian excitation operators that map to sums of commuting Pauli strings while spanning the relevant space and conserving Abelian symmetries, together with operator kirigami (orthogonal projection plus unitary rotations) for non-Abelian cases.

If this is right

  • Trotterized time evolution and quantum phase estimation preserve electron number and total spin without additional enforcement steps.
  • Symmetry-breaking errors decrease with molecular size and are eliminated by second-order Trotterization.
  • Operator pools built from these excitations can be validated on classical computers using existing electronic-structure symmetry tools.
  • The same construction applies to any qubit encoding of fermionic systems that respect Abelian symmetries.

Where Pith is reading between the lines

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

  • The method may reduce the need for symmetry-projected ansatzes in variational algorithms that also use Trotterized operators.
  • Operator kirigami could be applied to other Trotter decompositions outside chemistry, such as lattice gauge theories with non-Abelian groups.
  • If the commuting-Pauli property holds for larger bases, the approach scales to systems where full symmetry projection becomes costly.

Load-bearing premise

Hermitian excitation operators can always be chosen to map to commuting Pauli sums, span the necessary space, and conserve Abelian symmetries for arbitrary encodings without further restrictions on basis or molecule size.

What would settle it

A first-order Trotterized simulation on a molecule and encoding where the chosen Hermitian excitation operators produce observable breaking of electron number or spin symmetry.

Figures

Figures reproduced from arXiv: 2607.01560 by Edith Leal-S\'anchez, Fanny Vain, Jong-Kwon Ha, Ryan J. MacDonell.

Figure 1
Figure 1. Figure 1: Summary of Hamiltonian fragmentation schemes and their conservation of symmetries with Trotterization. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of Trotter errors for molecular simulations with the different operator pools. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Energy differences between the full CI electronic ground state energy and the Trotter energy for different molecules [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time evolution of Trotter errors for molecular simulations with the different operator pools. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of Nˆ and Sˆz for molecular simulations with the Qubit operator pool. a–c, The average electron number, ⟨Nˆ⟩, and its standard deviation for 10 random orders of operators. d–f, The average spin z-projection expectation values, ⟨Sˆz⟩, and their standard deviations for 10 random orders of operators. Columns represent different molecules and basis sets: H2 with a cc-pVDZ basis (a, d), LiH with … view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution of Nˆ and Sˆz for molecular simulations with the Qubit operator pool. a–c, The average electron number, ⟨Nˆ⟩, and its standard deviation for 10 random orders of operators. d–f, The average spin z-projection expectation values, ⟨Sˆz⟩, and their standard deviations for 10 random orders of operators. Columns represent different molecules and basis sets: H2 with a STO-3G basis (a, d), H2 with a … view at source ↗
Figure 7
Figure 7. Figure 7: Energy differences between the full CI electronic ground state energy and the Trotter energy for different molecules [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
read the original abstract

Quantum algorithms for quantum chemistry and other many-body fermionic systems work by expressing the Hamiltonian in a basis of qubits and fragmenting the Hamiltonian into a sum of products of Pauli operators whose exponentials are easily encoded on a quantum device. Applying the product of exponentials, known as Trotterization, leads to an error associated with the non-commutativity of operators. This error can lead to breaking the symmetries of the Hamiltonian because the fragments are not symmetry conserving in general. Nonetheless, many algorithms for time evolution rely on Trotterization, including time evolution and quantum phase estimation. We show that we can express the Hamiltonian in terms of Hermitian excitation operators which map to sums of commuting Pauli strings for any encoding and conserve symmetries corresponding to Abelian groups of symmetry operators. Symmetries corresponding to non-Abelian groups, on the other hand, are not fully conserved by Trotterized Hermitian excitation operators, so we developed ``operator kirigami'' to cut the sum of non-commuting operators by orthogonal projection and to fold terms together using unitary rotations. We tested pools of operators for small molecules and basis sets, and found that electron number and spin symmetry conserving pools led to greater errors that decreased for larger molecules and were negated with second-order Trotterization. Our work shows the potential for testing quantum computing algorithms on classical computers by adapting tools used in electronic structure theory with conserved symmetries.

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 paper claims that the molecular Hamiltonian can be expressed using Hermitian excitation operators mapping to sums of commuting Pauli strings for any qubit encoding, thereby conserving Abelian symmetries under Trotterization for time evolution and QPE. For non-Abelian symmetries it introduces 'operator kirigami' (orthogonal projection plus unitary rotations) to restore conservation. Tests on small molecules with symmetry-conserving operator pools show larger Trotter errors that decrease with system size and vanish under second-order Trotterization.

Significance. If the general construction is valid, the method would allow symmetry-preserving Trotterized evolution without sacrificing the ability to represent the full Hamiltonian, which is valuable for quantum chemistry simulations on near-term devices. The kirigami technique is a concrete contribution for handling non-Abelian cases.

major comments (2)
  1. [Abstract / construction sections] Abstract and main construction (likely §2–3): the central claim that Hermitian excitation operators exist 'for any encoding' such that each is a sum of mutually commuting Paulis, the set spans the full encoded Hamiltonian, and Abelian symmetries are conserved, is asserted without a general proof or counter-example search. Only numerical tests on small molecules with standard encodings are mentioned; if the commuting-Pauli constraint restricts the linear span for encodings that mix parity sectors or introduce non-local strings, the exact representation fails.
  2. [Results / testing paragraph] Results / testing paragraph: the statement that 'electron number and spin symmetry conserving pools led to greater errors that decreased for larger molecules' is given only qualitatively, with no error metrics, baselines, molecule sizes, or tables of Trotter errors versus order or pool size. This prevents quantitative assessment of whether the symmetry benefit outweighs the reported error increase.
minor comments (2)
  1. [Method / kirigami subsection] The term 'operator kirigami' is introduced without a precise algorithmic definition or pseudocode; a short subsection or appendix spelling out the projection and folding steps would improve reproducibility.
  2. [Introduction / notation] Notation for the Hermitian excitation operators E and their mapping to Pauli sums should be introduced with an explicit equation early in the text rather than only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive feedback on our manuscript. We address each major comment below and outline revisions that will strengthen the presentation while preserving the core claims.

read point-by-point responses

  1. Referee: [Abstract / construction sections] Abstract and main construction (likely §2–3): the central claim that Hermitian excitation operators exist 'for any encoding' such that each is a sum of mutually commuting Paulis, the set spans the full encoded Hamiltonian, and Abelian symmetries are conserved, is asserted without a general proof or counter-example search. Only numerical tests on small molecules with standard encodings are mentioned; if the commuting-Pauli constraint restricts the linear span for encodings that mix parity sectors or introduce non-local strings, the exact representation fails.

    Authors: The construction proceeds by defining Hermitian excitation operators that change particle number by an even integer (or preserve other Abelian quantum numbers) and then mapping each such operator to its Pauli-string representation under an arbitrary encoding. Because the operator is Hermitian and the encoding is a linear isometry from the fermionic Fock space, the resulting Pauli strings within each operator necessarily commute; their sum therefore generates a symmetry-preserving unitary. The full Hamiltonian is recovered because the chosen pool is complete by construction—it is the image of the standard fermionic excitation basis under the encoding map. We acknowledge that the manuscript presents this argument at a high level rather than as a formal theorem. In revision we will add a concise general argument (including why the linear span remains full for any encoding that respects the underlying algebra) together with an explicit check on one non-standard encoding that mixes parity sectors. revision: yes

  2. Referee: [Results / testing paragraph] Results / testing paragraph: the statement that 'electron number and spin symmetry conserving pools led to greater errors that decreased for larger molecules' is given only qualitatively, with no error metrics, baselines, molecule sizes, or tables of Trotter errors versus order or pool size. This prevents quantitative assessment of whether the symmetry benefit outweighs the reported error increase.

    Authors: We agree that the results paragraph is too terse. The revised manuscript will include a new table (or expanded figure) reporting Trotter errors for first- and second-order decompositions, explicit molecule sizes and basis sets (H_{2}, LiH, BeH_{2}, etc.), direct comparisons against non-symmetry-conserving pools, and the dependence on pool size. These quantitative data will allow readers to evaluate the trade-off between symmetry preservation and Trotter error. revision: yes

Circularity Check

0 steps flagged

No significant circularity: central claims rest on explicit operator constructions and tests, not self-definition or fitted inputs

full rationale

The paper asserts existence of Hermitian excitation operators that are sums of commuting Paulis, conserve Abelian symmetries, and span the Hamiltonian for arbitrary encodings. This is presented as a direct algebraic construction rather than a fit or renaming; the non-Abelian handling via 'operator kirigami' is introduced as a new projection/rotation technique. No equations reduce a claimed prediction to a fitted parameter by construction, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation. Empirical tests on small molecules serve as verification, not the derivation itself. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that Hermitian excitation operators map to commuting Pauli strings while preserving Abelian symmetries, plus the new kirigami construction for non-Abelian cases. No free parameters or invented physical entities are indicated.

axioms (1)
  • domain assumption Hermitian excitation operators can be expressed as sums of commuting Pauli strings for any encoding while conserving Abelian symmetries.
    Directly stated as the basis for the symmetry conservation result in the abstract.
invented entities (1)
  • operator kirigami no independent evidence
    purpose: Technique to cut sums of non-commuting operators via orthogonal projection and fold terms using unitary rotations to address non-Abelian symmetry breaking.
    New method introduced to handle cases where Trotterized Hermitian operators do not fully conserve non-Abelian symmetries.

pith-pipeline@v0.9.1-grok · 5788 in / 1316 out tokens · 30743 ms · 2026-07-03T00:36:40.859107+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

51 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    Each fermion-to-qubit encoding adopts different sets of indices forµ(p), π(p)and ϕ(p)

    of occupations of orbitals with indices less thanp [40]. Each fermion-to-qubit encoding adopts different sets of indices forµ(p), π(p)and ϕ(p). If we write two Hermitian excitation operators, ˆΞp q = ˆΞq p = ˆa† pˆaq + ˆa† qˆap = 1 2 ˆγ+ p ˆγ+ q + ˆγ− p ˆγ− q ,(5) ˆΣp q =− ˆΣq p =i(ˆa† pˆaq −ˆa† qˆap) = 1 2 ˆγ− p ˆγ+ q −ˆγ+ p ˆγ− q ,(6) 3 then the operato...

  2. [2]

    operator kirigami

    Without spin-orbit coupling or an external magnetic field, the electronic Hamiltonian can be written as ˆH= 1 2 X pq ˜kp q X σ ˆΞpσ qσ + 1 16 X pqrs ˜vpq rs X σσ ′ {ˆΞpσ rσ , ˆΞqσ ′ sσ′ },(7) where ˜kp q and ˜vpq rs are coefficients given by integrals over spatial orbitals (provided in Appendix A). The Hamil- tonian has several symmetries corresponding to...

  3. [3]

    V. Murg, F. Verstraete, R. Schneider, P. R. Nagy, and Ö. Legeza, J. of Chem. Theory Comput.11, 1027 (2015), pMID: 25844072

    2015

  4. [4]

    Schollwöck, Rev

    U. Schollwöck, Rev. Mod. Phys.77, 259 (2005)

    2005

  5. [5]

    Schirmer, Phys

    J. Schirmer, Phys. Chem. Chem. Phys.27, 4992 (2025)

    2025

  6. [6]

    Pederson, B

    R. Pederson, B. Kalita, and K. Burke, Nat. Rev. Phys. 4, 357 (2022)

    2022

  7. [7]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2011)

    2011

  8. [8]

    García-Pérez, M

    G. García-Pérez, M. A. Rossi, B. Sokolov, F. Tacchino, P. K. Barkoutsos, G. Mazzola, I. Tavernelli, and S. Man- iscalco, PRX Quantum2, 040342 (2021)

    2021

  9. [9]

    Noh and D

    C. Noh and D. G. Angelakis, Rep. Prog. Phys.80, 016401 (2016)

    2016

  10. [10]

    A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature607, 667 (2022)

    2022

  11. [11]

    A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Proc. Natl. Acad. Sci.115, 9456 (2018)

    2018

  12. [12]

    S. Lee, J. Lee, H. Zhai, Y. Tong, A. M. Dalzell, A. Kumar, P. Helms, J. Gray, Z.-H. Cui, W. Liua, M. Kastoryano, R. Babbush, J. Preskill, D. R. Reichman, E. T. Campbell, E. F. Valeev, L. Lin, and G. K.-L. Chan, Nat. Commun. 14, 1952 (2023)

    1952

  13. [13]

    Miessen, P

    A. Miessen, P. J. Ollitrault, F. Tacchino, and I. Tavernelli, Nat. Comput. Sci.3, 25 (2023)

    2023

  14. [14]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, Nat. Commun.5, 4213 (2014)

    2014

  15. [15]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Nat. Rev. Phys.3, 625 (2021)

    2021

  16. [16]

    H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall, Nat. Commun.10, 3007 (2019)

    2019

  17. [17]

    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, Phys. Rep.986, 1 (2022), the Variational Quantum Eigensolver: a review of methods and best practices

    2022

  18. [18]

    J. F. Gonthier, M. D. Radin, C. Buda, E. J. Doskocil, C. M. Abuan, and J. Romero, Phys. Rev. Res.4, 033154 (2022)

    2022

  19. [19]

    W. J. Huggins, J. R. McClean, N. C. Rubin, Z. Jiang, N. Wiebe, K. B. Whaley, and R. Babbush, npj Quantum Inf.7, 23 (2021)

    2021

  20. [20]

    Verteletskyi, T.-C

    V. Verteletskyi, T.-C. Yen, and A. F. Izmaylov, J. Chem. Phys.152(2020). 8

    2020

  21. [21]

    Aspuru-Guzik, A

    A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head- Gordon, Science309, 1704 (2005)

    2005

  22. [22]

    McArdle, S

    S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Rev. Mod. Phys.92, 015003 (2020)

    2020

  23. [23]

    Babbush, N

    R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, Phys. Rev. X8, 011044 (2018)

    2018

  24. [24]

    H. F. Trotter, Proc. Amer. Math. Soc.10, 545 (1959)

    1959

  25. [25]

    D. S. Abrams and S. Lloyd, Phys. Rev. Lett.79, 2586 (1997)

    1997

  26. [26]

    Lloyd, Science273, 1073 (1996)

    S. Lloyd, Science273, 1073 (1996)

    1996

  27. [27]

    Jordan and E

    P. Jordan and E. Wigner, Z. Phys.47, 631 (1928)

    1928

  28. [28]

    S. B. Bravyi and A. Y. Kitaev, Ann. Phys.298, 210 (2002)

    2002

  29. [29]

    J. T. Seeley, M. J. Richard, and P. J. Love, J. Chem. Phys.137, 224109 (2012)

    2012

  30. [30]

    S. B. Bravyi and A. Y. Kitaev, Annals of Physics298, 210 (2002)

    2002

  31. [31]

    Setia and J

    K. Setia and J. D. Whitfield, J. Chem. Phys.148, 164104 (2018)

    2018

  32. [32]

    Helgaker, P

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

    2000

  33. [33]

    Szabo and N

    A. Szabo and N. S. Ostlund,Modern Quantum Chemistry. Introduction to Advanced Electronic Structure Theory, 2nd ed. (Dover, New York, USA, 1989)

    1989

  34. [34]

    P. Jain, A. F. Izmaylov, and E. R. Kjellgren, J. Chem. Phys.164(2026)

    2026

  35. [35]

    Magoulas, M

    I. Magoulas, M. Zhang, and F. A. Evangelista, J. Chem. Phys.164(2026)

    2026

  36. [36]

    K. J. Sung, I. Choi, M. Amico, B. Andrews, E. Ayan- tuna, Y. Kawashima, W.-H. Lin, D. Omanovic, S. Pic- cinelli, J. R. Moreno, A. A. Saki, J. Shee, S. Shin, M. C. Tran, K. Ueda, H. Zhang, and M. Motta, arXiv (2026), 2605.03123 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  37. [37]

    Picozzi and J

    D. Picozzi and J. Tennyson, Quantum Sci. Technol.8, 035026 (2023)

    2023

  38. [38]

    Harrison, D

    B. Harrison, D. Nelson, D. Adamiak, and J. Whitfield, arXiv (2022), 2211.04501 [quant-ph]

    work page arXiv 2022

  39. [39]

    S. G. Mehendale, L. A. Martínez-Martínez, P. D. Kamath, and A. F. Izmaylov, Digit. Discov.4, 3540 (2025)

    2025

  40. [40]

    A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Phys. Rev. X11, 011020 (2021)

    2021

  41. [41]

    Campbell, Phys

    E. Campbell, Phys. Rev. Lett.123, 070503 (2019)

    2019

  42. [42]

    Loaiza, A

    I. Loaiza, A. Sankar Brahmachari, and A. F. Izmaylov, Quantum Science and Technology10, 035035 (2025)

    2025

  43. [43]

    Q. Sun, X. Zhang, S. Banerjee, P. Bao, M. Barbry, N. S. Blunt, N. A. Bogdanov, G. H. Booth, J. Chen, Z.-H. Cui, J. J. Eriksen, Y. Gao, S. Guo, J. Hermann, M. R. Hermes, K. Koh, P. Koval, S. Lehtola, Z. Li, J. Liu, N. Mardirossian, J. D. McClain, M. Motta, B. Mus- sard, H. Q. Pham, A. Pulkin, W. Purwanto, P. J. Robin- son, E. Ronca, E. R. Sayfutyarova, M. ...

    2020

  44. [44]

    Wang and C

    L.-P. Wang and C. Song, J. Chem. Phys.144(2016)

    2016

  45. [45]

    operator kirigami

    J. R. McClean, N. C. Rubin, K. J. Sung, I. D. Kivlichan, X. Bonet-Monroig, Y. Cao, C. Dai, E. S. Fried, C. Gidney, B. Gimby, P. Gokhale, T. Häner, T. Hardikar, V. Havlíček, O. Higgott, C. Huang, J. Izaac, Z. Jiang, X. Liu, S. McAr- dle, M. Neeley, T. O’Brien, B. O’Gorman, I. Ozfidan, M. D. Radin, J. Romero, N. P. D. Sawaya, B. Senjean, K. Setia, S. Sim, D...

    2020

  46. [46]

    Evaluate the Trotter product at a given−i∆t

  47. [47]

    Diagonalizeexp(−i ˆHeff∆t)to obtainUexp(−iθ k(∆t))U †

  48. [48]

    When∆ t→ 0the eigenvalue of the exact Hamiltonian,|φk⟩ can be used as reference

    Select the eigenstate |ϕk,∆t⟩ as the one with the highest overlap with the k-th eigenvalue ofexp(−i ˆHeff∆t′), |ϕk,∆t′⟩, where∆ t′ < ∆t. When∆ t→ 0the eigenvalue of the exact Hamiltonian,|φk⟩ can be used as reference

  49. [49]

    13 0.00 0.01 0.02 0.03 0.04 Infidelity / 10 4 H2 STO-3Ga 0 500 H2 6-31Gb Qubit Slater CSF 0 500 1000 1500 LiH STO-3Gc 0 100 200 t / a.u

    Multiplyexp(−iθ k(∆t)) exp(i∆tEk(∆t′))whereE k(∆t′)is the eigenvalue corresponding to|ϕk,∆t′⟩. 13 0.00 0.01 0.02 0.03 0.04 Infidelity / 10 4 H2 STO-3Ga 0 500 H2 6-31Gb Qubit Slater CSF 0 500 1000 1500 LiH STO-3Gc 0 100 200 t / a.u. 0.050 0.025 0.000 0.025 0.050 S2 / 10 3 a.u. H2 STO-3Gd 0 50 100 150 200 t / a.u. 0 10 20 30 40 H2 6-31Ge 0 100 200 t / a.u. ...

  50. [50]

    In Python it is suggested to usenumpy.angle to find the argument in the correct quadrant

    Calculate Ek(∆t) = Ek(∆t′) − 1 ∆tArg(e−iθk(∆t)ei∆tEk(∆t′)). In Python it is suggested to usenumpy.angle to find the argument in the correct quadrant

  51. [51]

    It is always necessary to have a reference energy and eigenstate, since the value∆tEk determines the branch of the logarithm

    Repeat the procedure until the desired time interval is covered. It is always necessary to have a reference energy and eigenstate, since the value∆tEk determines the branch of the logarithm. For small time steps, thek-th eigenstate of the exact Hamiltonian can be used for the steps 3-5 of the procedure. However, for larger time steps, this value can lead ...