Autonomous Lindblad Realizability of Nonunitary Linear Dynamics with a Carleman Lattice Boltzmann Application

Muhammad Idrees Khan

REVIEW 2 major objections 5 minor 19 references

Autonomous Lindblad Realizability of Nonunitary Linear Dynamics with a Carleman Lattice Boltzmann Application

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A finite linear map is autonomous-GKSL-realizable exactly when it is invertible and power bounded, and the construction gives the Hamiltonian, jump operators, and one-shot encoding-decoding scheme.

desk verdict The iff theorem is solid and new; the D2Q9 application is a valid but numerically hinged demonstration that deserves peer review with a request for the code or an analytic semisimplicity proof. read the letter →

arxiv 2608.09808 v1 pith:CMBCMLXX submitted 2026-08-10 physics.comp-ph quant-ph

Muhammad Idrees Khan This is my paper · ORCID

classification physics.comp-phquant-ph MSC 81S2215A1676M28

keywords autonomousGKSLrealizationpower-boundedlineardynamicsCarlemanlinearizationlatticeBoltzmannmethodopenquantumsystemsvacuumcoherenceencodingnonunitarymapsmultiple-relaxation-timecollision

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Carleman lifting converts nonlinear polynomial dynamics into a finite linear system, but the resulting finite endpoint is generally nonunitary and therefore cannot be run directly on a closed quantum computer. The paper proves that such an endpoint admits an autonomous open-system realization exactly when it is invertible and power bounded: spectral radius at most one, with every unit-modulus eigenvalue semisimple. The proof is constructive, producing a time-independent Hamiltonian, finitely many jump operators, a Kraus family, and a Stinespring dilation, so the nonunitary map can be evolved over repeated timesteps with one encoding and one decoding and no postselection. The criterion is applied to the complete D2Q9 multiple-relaxation-time lattice Boltzmann collide-and-stream step, whose order-two Carleman endpoint on a $3\times3$ periodic lattice is verified admissible; the GKSL evolution matches the classical Carleman trajectory to about $10^{-14}$ over ten timesteps while the deviation from the nonlinear LB dynamics is the expected degree-three truncation error. If correct, the result replaces a structural obstruction with a precise matrix-level criterion for when finite nonunitary dynamics are physically realizable as autonomous open quantum evolution.

What carries the argument

The load-bearing object is the pair consisting of a positivity metric $P$ for the generator logarithm and the vacuum-coherence encoding. Given $G=\tau^{-1}\log A$, the condition $PG+G^*P\preceq0$ makes $e^{tG}$ a contraction in the $P$-metric, and after the congruence $D=\Sigma G\Sigma^{-1}$ with $\Sigma=P^{1/2}$, the dissipative part $\Gamma=-(D+D^*)\succeq0$ is factored into row vectors that become jump operators $L_\ell=|0\rangle r_\ell$. The logical state $z$ lives in the coherence $|v(z)\rangle\langle0|+|0\rangle\langle v(z)|$ on $\mathcal H=\mathrm{span}\{|0\rangle\}\oplus\mathbb C^m$, with $v(z)=\Sigma Jz/\kappa$; the positivity budget $\|v(z)\|^2\le1/(m+1)$ fixes the required scale $\kappa$. These pieces make dissipation a physical contraction of the logical amplitude rather than a postselection of a unitary block, which is what allows the same time-independent generator to be re-used for every timestep.

What would settle it

Recompute the order-two Carleman endpoint $A_2$ on the $3\times3$ periodic lattice with the paper's exact D2Q9 parameters and inspect its Jordan form; finding any unit-modulus eigenvalue with a nontrivial Jordan block, or any eigenvalue above $1$ in modulus beyond roundoff, would falsify the claimed power boundedness and hence the existence of the autonomous GKSL realization.

Watch

Extended reading notes

Core claim

The paper's Theorem 4 states that, for a real $m\times m$ matrix $A$ and a fixed timestep $\tau>0$, there exist a generator $G$, a full-rank embedding $J$, and a Hermitian $P\succ0$ with $e^{\tau G}J=JA$ and $PG+G^*P\preceq0$ if and only if $A$ is invertible and power bounded. In the forward direction, power boundedness implies a branch of $\tau^{-1}\log A$ has spectrum in the closed left half-plane with semisimple imaginary-axis part, so a Schur separation and Lyapunov equations supply the metric $P$. The logical vector is stored in the off-diagonal vacuum coherences of a density matrix, and the generator's dissipative part is factored into jump operators $L_\ell=|0\rangle r_\ell$; the coherence block then evolves by $\dot v = Dv$ with $D=\Sigma G\Sigma^{-1}$, while the full state remains normalized and positive. The identity $\mathrm{decode}(e^{n\tau\mathcal L}E(z))=A^n z$ holds for every $n$ with one encoding and one decoding. For the compiled endpoint $A_K=C_K(P_S)C_K(F_C)$ on the $3\times3$ periodic lattice at order two, the numerical audit finds spectral radius $1$ and nine unit-modulus eigenvalues, all semisimple, so the construction applies; the realized channel agrees with $A_K^n z$ to $1.1\times10^{-14}$ over ten steps.

Load-bearing premise

The load-bearing premise of the application is the numerical audit, not a proof, that all nine unit-modulus eigenvalues of the compiled $3\times3$ endpoint are semisimple; the analytic argument covers only the conserved $+1$ sector.

Editorial extensions

If this is right

Any finite nonunitary linear endpoint that is invertible and power bounded can be implemented as autonomous open-system dynamics with one encoding, one decoding, and no intermediate measurements or feedback.
The criterion depends only on the endpoint matrix, so it applies to any source of finite linear dynamics, not only Carleman sections or lattice Boltzmann methods.
The parameter choice $\tau_\nu=1$ for the LB relaxation makes the endpoint singular and therefore non-realizable; endpoints with spectral radius above one or with nontrivial Jordan blocks on the unit circle are excluded for the same reason.
For admissible endpoints the construction also yields the equivalent CPTP channel, Kraus family, and Stinespring dilation, so the realization is not just a semigroup identity but a full physical dilation.
The GKSL realization exactly reproduces the Carleman trajectory, so all observed discrepancy with the true LB dynamics is Carleman truncation error, which scales cubically in perturbation amplitude at order two.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The endpoint-only character of Theorem 4 suggests the same realizability test can be applied to other numerical procedures that produce finite linear propagators, such as moment hierarchies or coarse-grained linearizations, without re-deriving a Lindblad construction in each case.
Because the compiled generator costs $O(d_K^3)$ classical work and readout scales with the coherence amplitude and the condition number of $\Sigma J$, practical quantum advantage would have to come from sparse or structured logarithms and improved readout; the paper itself disclaims advantage but its construction identifies exactly where such improvements would enter.
A fully analytic proof of semisimplicity for all unit-modulus eigenvalues of the LB endpoint would remove the numerical audit from the load-bearing path and extend the result to larger lattices where the dense audit becomes expensive.
The complex-drift construction suggests that a real-valued open-system realization would require a larger state space, and comparing Hilbert dimensions might reveal whether realification ever buys a smaller physical footprint.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this paper's real contribution is Theorem 4, and it holds up. The iff characterization—invertible and power bounded—for autonomous GKSL realization on vacuum coherences is new, and the proof is self-contained. The (i)⇒(ii) direction via metric contraction is straightforward; the (ii)⇒(i) construction via Schur decomposition, Lyapunov equation for the stable block, and a semisimple neutral block is clean and explicit. The GKSL, Kraus, and Stinespring data all follow without hand-waving. I checked the key steps in the logic and they're sound.

The Carleman LB application is a genuine concrete test: compiling collision and periodic streaming into a single endpoint and verifying the theorem's hypotheses is exactly the kind of demonstration the field needs. The numerical residuals in Table I are at the 1e-14 level, and the truncation error scaling with amplitude and order is consistent with a degree-two truncation. The paper is also honest about what it doesn't claim: no quantum advantage, no analytic truncation bound, and the principal logarithm unavailable due to negative real eigenvalues.

The soft spot is the one the stress-test flags. The power boundedness of A2 on the 3x3 periodic lattice—specifically the semisimplicity of the nine unit-modulus eigenvalues—is verified numerically, with a structural argument only for the conserved +1 sector. Since Theorem 4 requires exactly this condition to invoke the GKSL realization, the application's advertised result sits on that numerical audit. It's a finite matrix check, and the paper provides complete conventions in Appendix A to reproduce it, so I don't see this as a fatal flaw. But without shipped code or an analytic argument for the remaining unit-modulus eigenvalues, the example is one missing Jordan block away from being outside the theorem's scope. The paper itself acknowledges this by calling it an audit, so the authors aren't hiding it.

For whom: anyone working on quantum simulation of nonlinear dynamics, Carleman linearization, or open quantum realizations of nonunitary maps. A serious referee should engage. My recommendation: send to peer review. The theorem is the payload and deserves publication; the application should be accepted conditionally, with a request for either the code or an analytic proof of the unit-modulus semisimplicity.

Referee Report

2 major / 5 minor

Summary. The paper proves a necessary and sufficient condition for a finite real linear endpoint A to be realized as the stroboscopic map of an autonomous Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) semigroup acting on vacuum coherences. Theorem 4 states that such a realization exists if and only if A is invertible and power bounded; the constructive direction builds a dissipative generator through Schur and Lyapunov techniques and then derives the Hamiltonian, jump operators, Kraus family, and Stinespring isometry. The authors apply this criterion to the D2Q9 multiple-relaxation-time lattice Boltzmann collide-and-stream map on a 3x3 periodic lattice, compiling collision and streaming into a Carleman endpoint A2 of dimension 3402. Numerical residuals in Table I are at the 1e-14 level, and the truncation error against the nonlinear LB reference is reported to scale cubically with perturbation amplitude.

Significance. If correct, Theorem 4 is a clean and useful structural result: it reduces a quantum-dynamical realizability question to a matrix-level condition and provides an explicit no-postselection construction with the same time-independent generator repeated over timesteps. The proof is largely self-contained, the numerical residual checks are strong, and the limitations are stated honestly, including the absence of an analytic truncation bound and the lack of a quantum-advantage claim. The principal caveat is that the D2Q9 application hinges on a numerically audited semisimplicity condition rather than on a proof, and the audit is not fully reproducible from the manuscript as written.

major comments (2)

[IV.B and V.A] The claim that the compiled endpoint A2 is power bounded is load-bearing for the D2Q9 application, because Theorem 4 requires every unit-modulus eigenvalue to be semisimple. The paper proves structural semisimplicity only for the conserved +1 sector; for the remaining unit-modulus eigenvalues it relies on a numerical audit, with no algorithm, error bounds, or shipped implementation. A defective unit-modulus eigenvalue would make ||A2^n|| grow polynomially and would exclude the advertised GKSL realization. Please provide a reproducible certification of this finite matrix property, for example a high-precision Schur factorization with interval arithmetic or the code and data used for the eigenvalue and Jordan-structure audit.
[Theorem 4 proof] In the proof of (ii) implies (i), the assertion that the imaginary-axis part of the chosen logarithm G is semisimple because A has semisimple unit-modulus eigenvalues is correct but terse, and it is important since the construction claims insensitivity to the logarithm branch. A short justification, namely that a nontrivial Jordan block of G at a pure-imaginary eigenvalue would exponentiate to a nontrivial Jordan block of A on the unit circle, would make the proof fully rigorous and would remove any doubt about branch choices.

minor comments (5)

[IV.B] The text says 'Combining Theorems 9 and 10 with Theorem 1', but the manuscript contains Propositions 9 and 10 and Proposition 1, not Theorems by those numbers; please correct the cross-references.
[Remark 6 and Section V] The cross-references are inconsistent: Remark 6 cites a nonexistent 'Theorem 6', and Section V refers to 'Theorem 11' where Remark 11 is meant. The proof of Theorem 4 and Section VI also call Lemma 3 'Theorem 3'. Please harmonize all numbered references.
[Remark 2] The final sentence 'leaves the hypotheses of Theorem 4' is incomplete; it should say 'violates the hypotheses of Theorem 4' or 'falls outside the hypotheses of Theorem 4'.
[V.A] The statement that 'the largest dyadic-power norm sampled in the numerical audit is 40.68' should be defined precisely: over which set of exponents was the maximum taken, and why does checking dyadic powers suffice for the semisimplicity audit?
[Figure 1 and Table IV] Please clarify the numerical tolerance used to declare the nine unit-modulus eigenvalues semisimple, and rephrase 'dense density matrix' in Table IV as 'density-matrix storage requirement' to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the GKSL realizability theorem is derived from first principles, and the D2Q9 application verifies its hypotheses rather than fitting them.

full rationale

Theorem 4 is not circular: its proof derives (i)->(ii) from contractivity of e^{tG} in a metric norm and standard power-boundedness facts, and derives (ii)->(i) by an explicit construction G = tau^{-1} log A, a Schur decomposition, Lyapunov and Sylvester solutions, and explicit GKSL data. The equivalence is not assumed in the conditions. The encoding/decoding identity decode(E(z)) = z is definitional (Eq. (8) and the Moore-Penrose identity), not a prediction, and the semigroup identity decode(e^{n tau L} E(z)) = A^n z follows by construction from e^{tau D} Sigma = Sigma A. In the LB application, the endpoint A_K is compiled from stated D2Q9/MRT conventions, not inferred from the later results; the power-boundedness audit is a hypothesis check for Theorem 4, and the truncation-error scaling is measured against an independently defined nonlinear MRT reference, with no fitted constants. The only self-citation, [15], is a related-work comparison for the MRT relaxation block and is not load-bearing for Theorem 4 or for the application. Concerns about the numerical semisimplicity audit are correctness risks, not circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The main theorem rests on standard linear algebra and the GKSL framework. The application introduces no new entities; the only assumptions are the modeling choice of fixed-reference-density equilibrium and the numerical verification of power boundedness. The free parameters are standard LB inputs, not fitted to data.

free parameters (2)

MRT relaxation rates = se=1.19, s_epsilon=1.40, s_q=1.20, tau_nu=0.508
Standard D2Q9 MRT parameters chosen from the literature (Lallemand and Luo); not fitted to any output. They affect the specific endpoint but not the theorem.
Perturbation amplitude = 0.020
Test input for the simulation; not a model parameter. Used to set the initial condition and to test truncation error scaling.

assumptions (5)

standard math Standard linear algebra facts: Schur decomposition, Lyapunov equation solvability, matrix logarithm existence for invertible matrices over C
Used in the proof of Theorem 4, Section III.B.
standard math GKSL semigroup generates a CPTP dynamical map
Relies on the Gorini-Kossakowski-Sudarshan-Lindblad theorem; cited [3,4].
domain assumption Carleman truncation gives an exact finite linear operator on the jet algebra
Proposition 1, Section II; the truncation error versus the original nonlinear map is measured separately.
domain assumption Fixed-reference-density equilibrium makes the LB collision exactly quadratic
Proposition 9, Section IV; this is a modeling choice specific to the D2Q9 MRT model.
domain assumption Power boundedness of the compiled LB endpoint (all unit-modulus eigenvalues semisimple)
Numerically verified in Sections IV.B and V.A; not proven analytically except for the conserved +1 sector.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Autonomous Lindblad Realizability of Nonunitary Linear Dynamics with a Carleman Lattice Boltzmann Application." pith.science (2026). https://pith.science/paper/CMBCMLXX

@misc{pith2026260809808,
  author       = {Pith},
  title        = {Pith review of: Autonomous Lindblad Realizability of Nonunitary Linear Dynamics with a Carleman Lattice Boltzmann Application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMBCMLXX}},
  note         = {Machine review of arXiv:2608.09808}
}

read the original abstract

Carleman lifting converts nonlinear polynomial dynamics into finite linear systems, but the resulting truncations are generally nonunitary and need not correspond to physical quantum evolution. We prove that a finite linear endpoint admits an autonomous Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) realization on vacuum coherences if and only if it is invertible and power bounded. The construction is explicit and realizes the nonunitary map directly as open-system dynamics, with no endpoint postselection and with one encoding and one decoding over repeated timesteps. We apply the result to the complete D2Q9 multiple-relaxation-time lattice Boltzmann (LB) timestep by compiling collision and periodic streaming into a single Carleman endpoint. The resulting GKSL evolution reproduces the classical Carleman trajectory over multiple timesteps, while the remaining discrepancy from nonlinear LB dynamics is the expected Carleman truncation error. The result establishes a general criterion for autonomous open-quantum realization of finite nonunitary dynamics, with Carleman--LB dynamics as a concrete example.

Figures

Figures reproduced from arXiv: 2608.09808 by the authors.

**Figure 2.** FIG. 2. Truncation error against the nonlinear reference [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 14 canonical work pages

[1]

Population order, velocities and moment transform At each lattice site the populations are ordered as q= (0,1,2,3,4,5,6,7,8) = (rest,E,N,W,S,NE,NW,SW,SE), (A1) with velocities and weights q 0 1 2 3 4 5 6 7 8 cqx 0 1 0−1 0 1−1−1 1 cqy 0 0 1 0−1 1 1−1−1 wq 4 9 1 9 1 9 1 9 1 9 1 36 1 36 1 36 1 36 .(A2) The moment vector is ordered as m= (ρ, e, ϵ, jx, qx, jy,...

work page
[2]

Its poly- nomial degree is therefore two

Fixed-reference-density MR T collision The collision step is m+ = (I−S)m+Sm eq(m), f + =M −1m+,(A5) with the fixed-reference-density equilibrium meq =   ρ −2ρ+ 3(j 2 x +j 2 y )/ρ0 ρ−3(j 2 x +j 2 y )/ρ0 jx −jx jy −jy (j2 x −j 2 y )/ρ0 jxjy/ρ0   .(A6) Thus the equilibrium consists exactly of linear and homogeneous quadratic terms i...

work page
[3]

(A12) be the flat population index

Periodic streaming and flat indexing For ann x ×n y periodic lattice let α(x, y, q) = 9(xny +y) +q, 0≤x < nx,0≤y < n y. (A12) be the flat population index. Streaming sends a popula- tion at ( x, y, q) to ( x + cqx, y+ cqy , q) modulo the lattice extents, equivalently (PS)α((x+cqx) modn x,(y+cqy ) modn y,q), α(x,y,q)= 1,(A13) with all other entries in that...

work page
[4]

For an off-diagonal monomial δfiδfj, the coefficient includes the two symmetric bilinear contributions before being stored once

Degree-two monomial convention and endpoint assembly For d = 9 nxny population variables, the degree-two monomials are stored once, without √ 2 normalization, in lexicographic order δf [2] = δfiδfj 0≤i≤j<d,Φ 2(δf) = δf δf [2] .(A14) The symmetric power L⊙2 is understood in this same unnormalized monomial basis and is defined by (Lx)[2] =L ⊙2x[2].(A15) 10 ...

work page
[5]

Carleman, Application de la theorie des equations integrales lineaires aux systemes d equations differentielles non lineaires, Acta Mathematica59, 63 (1932)

T. Carleman, Application de la theorie des equations integrales lineaires aux systemes d equations differentielles non lineaires, Acta Mathematica59, 63 (1932)

work page 1932
[6]

Forets and A

M. Forets and A. Pouly, Explicit error bounds for carleman linearization, arXiv preprint 1711.02552 10.48550/arXiv.1711.02552 (2017)

arXiv doi:10.48550/arxiv.1711.02552 2017
[7]

Gorini, A

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Com- pletely positive dynamical semigroups of n level systems, Journal of Mathematical Physics17, 821 (1976)

work page 1976
[8]

Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

1976

Show all 19 references

[9]

E. Choi, J. E. Kim, X. Lu, and Y. Wang, Lindbladian homotopy analysis method to solve nonlinear partial differential equations, arXiv preprint arXiv:2604.18924 10.48550/arXiv.2604.18924 (2026)

2026 doi arXiv
[10]

R. A. Horn and C. R. Johnson,Matrix Analysis, 2nd ed. (Cambridge University Press, 2013)

2013
[11]

W. F. Stinespring, Positive functions on c star algebras, Proceedings of the American Mathematical Society6, 211 (1955)

1955
[12]

Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications10, 285 (1975)

M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications10, 285 (1975)

1975
[13]

Kraus,States Effects and Operations(Springer, 1983)

K. Kraus,States Effects and Operations(Springer, 1983)

1983
[14]

Lallemand and L.-S

P. Lallemand and L.-S. Luo, Theory of the lattice boltz- mann method. dispersion dissipation isotropy galilean in- variance and stability, Physical Review E61, 6546 (2000)

2000
[15]

Succi,The Lattice Boltzmann Equation for Complex States of Flowing Matter(Oxford University Press, 2018)

S. Succi,The Lattice Boltzmann Equation for Complex States of Flowing Matter(Oxford University Press, 2018)

2018
[16]

J.-P. Liu, D. An, D. Fang, J. Wang, G. H. Low, and S. P. Jordan, Efficient quantum algorithm for nonlinear reaction diffusion equations and energy estimation, Com- munications in Mathematical Physics404, 963 (2023)

2023
[17]

Sanavio and S

C. Sanavio and S. Succi, Lattice boltzmann carleman quantum algorithm and circuit for fluid flows at moderate reynolds number, A VS Quantum Science6, 023802 (2024)

2024
[18]

Sanavio, W

C. Sanavio, W. A. Simon, A. Ralli, P. Love, and S. Succi, Carleman lattice boltzmann quantum circuit with matrix access oracles, Physics of Fluids37, 037123 (2025)

2025
[19]

M. I. Khan, S. Succi, and H.-D. Yao, Deterministic realiza- tion of classical dissipation on quantum computers, arXiv preprint arXiv:2604.25429 10.48550/arXiv.2604.25429 (2026)

2026 doi arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Population order, velocities and moment transform At each lattice site the populations are ordered as q= (0,1,2,3,4,5,6,7,8) = (rest,E,N,W,S,NE,NW,SW,SE), (A1) with velocities and weights q 0 1 2 3 4 5 6 7 8 cqx 0 1 0−1 0 1−1−1 1 cqy 0 0 1 0−1 1 1−1−1 wq 4 9 1 9 1 9 1 9 1 9 1 36 1 36 1 36 1 36 .(A2) The moment vector is ordered as m= (ρ, e, ϵ, jx, qx, jy,...

work page

[2] [2]

Its poly- nomial degree is therefore two

Fixed-reference-density MR T collision The collision step is m+ = (I−S)m+Sm eq(m), f + =M −1m+,(A5) with the fixed-reference-density equilibrium meq =   ρ −2ρ+ 3(j 2 x +j 2 y )/ρ0 ρ−3(j 2 x +j 2 y )/ρ0 jx −jx jy −jy (j2 x −j 2 y )/ρ0 jxjy/ρ0   .(A6) Thus the equilibrium consists exactly of linear and homogeneous quadratic terms i...

work page

[3] [3]

(A12) be the flat population index

Periodic streaming and flat indexing For ann x ×n y periodic lattice let α(x, y, q) = 9(xny +y) +q, 0≤x < nx,0≤y < n y. (A12) be the flat population index. Streaming sends a popula- tion at ( x, y, q) to ( x + cqx, y+ cqy , q) modulo the lattice extents, equivalently (PS)α((x+cqx) modn x,(y+cqy ) modn y,q), α(x,y,q)= 1,(A13) with all other entries in that...

work page

[4] [4]

For an off-diagonal monomial δfiδfj, the coefficient includes the two symmetric bilinear contributions before being stored once

Degree-two monomial convention and endpoint assembly For d = 9 nxny population variables, the degree-two monomials are stored once, without √ 2 normalization, in lexicographic order δf [2] = δfiδfj 0≤i≤j<d,Φ 2(δf) = δf δf [2] .(A14) The symmetric power L⊙2 is understood in this same unnormalized monomial basis and is defined by (Lx)[2] =L ⊙2x[2].(A15) 10 ...

work page

[5] [5]

Carleman, Application de la theorie des equations integrales lineaires aux systemes d equations differentielles non lineaires, Acta Mathematica59, 63 (1932)

T. Carleman, Application de la theorie des equations integrales lineaires aux systemes d equations differentielles non lineaires, Acta Mathematica59, 63 (1932)

work page 1932

[6] [6]

Forets and A

M. Forets and A. Pouly, Explicit error bounds for carleman linearization, arXiv preprint 1711.02552 10.48550/arXiv.1711.02552 (2017)

arXiv doi:10.48550/arxiv.1711.02552 2017

[7] [7]

Gorini, A

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Com- pletely positive dynamical semigroups of n level systems, Journal of Mathematical Physics17, 821 (1976)

work page 1976

[8] [8]

Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

1976