Natural super-orbitals representation of many-body operators

Maxime Debertolis

REVIEW 4 major objections 5 minor 5 cited by

Natural super-orbitals representation of many-body operators

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Natural super-orbitals reveal that in quantum impurity models, time-evolution and local operators have exponentially decaying occupation spectra, enabling compact matrix-product-operator representations and reduced non-stabilizerness.

desk verdict Clean framework, credible numerics, but the compact-MPO payoff is inferred, not shown. read the letter →

arxiv 2507.10690 v2 pith:MHPCWDDE submitted 2025-07-14 cond-mat.str-el quant-ph

Maxime Debertolis This is my paper

classification cond-mat.str-elquant-ph

keywords naturalsuper-orbitalsone-bodysuper-densitymatrixproductoperatorsnon-stabilizernessoperatornon-Gaussianitycorrelationentropyinteractingresonantlevelmodelt-V

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

Many-body operators are hard to simulate because their entanglement grows rapidly when written as matrix product operators. This paper introduces natural super-orbitals—the eigenvectors of a one-body super-density matrix built from a vectorized operator—and asks whether a change of one-particle basis can make an operator simple. The central finding is that in quantum impurity models, the occupation spectrum of these super-orbitals decays exponentially at all times for both the time-evolution operator and an evolved local operator, whereas in the bulk $t$-$V$ chain no preferred basis exists. Exponential decay means that only a small number of super-orbitals contribute to quantum correlations, which the paper argues enables a compact matrix-product-operator representation and a reduced non-stabilizerness (how non-Clifford, and therefore how hard to simulate classically, an operator is). If correct, this gives a basis-rotation route to reaching longer times and larger systems in impurity-model simulations, for example in computations of out-of-time-order correlators.

What carries the argument

The load-bearing object is the one-body super-density matrix $R[\hat O]_{m,n} = \langle\langle \hat O | \hat{\hat d}_m^\dagger \hat{\hat d}_n | \hat O \rangle\rangle / \mathrm{Tr}[\hat O^\dagger \hat O]$, built from the $2L$ super-operators $\hat{\hat d}_m$ acting on the doubled Hilbert space of a vectorized operator. Its eigenvectors are the natural super-orbitals and its eigenvalues $n_\alpha$ form the correlation spectrum of the operator. For a unitary $\hat O$, the matrix $R$ has diagonal blocks $\tfrac12 \mathbb{1}_L$ and an off-diagonal block $A_{mn}=\mathrm{Tr}[\hat O^\dagger \hat c_m^\dagger \hat O \hat c_n]/\mathrm{Tr}[\mathbb{1}]$, so its eigenvalues come in pairs $\tfrac12 \pm \sqrt{\mu_\alpha}$ where $\mu_\alpha$ are the eigenvalues of $A^\dagger A$, and the corresponding rotation acts as an ordinary orbital rotation. The correlation entropy $S_{\rm corr} = -\sum_\alpha n_\alpha \log n_\alpha$ turns the spectrum into a single non-Gaussianity measure, while the exponential decay of $n_\alpha$ is what allows orbitals to be truncated and bounds the operator's non-stabilizerness.

What would settle it

For the interacting resonant level model, rotate the numerically obtained matrix-product-operator representation of $\hat U(t)$ at $L=50$ into the natural super-orbital basis at times past the real-space truncation time $t=8$ and measure the minimal bond dimension needed at fixed precision: if the bond dimension grows with time or system size while the occupation spectrum stays exponentially decaying, the compression claim is contradicted.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the natural super-orbital basis of a unitary operator is just an ordinary orbital rotation, so vectorizing an operator does not destroy its physical meaning. Applied to the interacting resonant level model, it finds that the eigenvalues of the one-body super-density matrix—the natural super-orbital occupations—decay exponentially with orbital index at all simulated times for both $\hat U(t)$ and the time-evolved impurity operator $\hat n_1(t)-1/2$. This exponential decay is the signature the paper uses to conclude that only a few orbitals carry the correlations, that the matrix-product-operator representation in this basis becomes compact, and that non-stabilizerness grows only linearly in time with a light-cone-like structure despite the operator's non-locality in real space. For the local impurity operator the correlation entropy saturates at finite time, indicating that its complexity in the natural super-orbital basis does not grow indefinitely. In the $t$-$V$ chain, by contrast, the spectrum fills in uniformly and the correlation entropy saturates, so no one-particle basis reduces the operator complexity. The paper also shows the framework reproduces the expected limits: free-system time-evolution operators factorize completely in the super-orbital basis, while Haar-random unitaries have a flat spectrum with no preferred basis.

Load-bearing premise

The load-bearing premise is that exponential decay of the occupations in the one-body super-density spectrum implies a compact matrix-product-operator representation in the rotated basis; the paper does not directly demonstrate the resulting bond dimension.

Editorial extensions

If this is right

In quantum impurity models, $\hat U(t)$ should be representable as a compact matrix product operator in the natural super-orbital basis, extending reachable simulation times beyond the real-space bond-dimension ceiling.
The non-stabilizerness of the impurity time-evolution operator is bounded by a linear-in-time growth $2vt\log 2 + \tilde c$ in this basis, mirroring the known light-cone bound for local operators.
The evolved impurity local operator has a correlation entropy that saturates at finite time, meaning its natural-super-orbital complexity does not grow forever.
For bulk interacting chains like the $t$-$V$ model, one-body rotations cannot simplify operators, so different compression strategies are needed for that class of systems.
The compressed representation opens a route to computing out-of-time-order correlators in large impurity systems at long times.

Reading between the lines

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

A testable extension is to apply the same construction to spinful or multi-impurity versions of the interacting resonant level model; if the exponential occupation decay persists, the mechanism is robust rather than a quirk of the spinless single-impurity geometry.
Because compression is demonstrated at the operator level rather than for a particular state, the same natural super-orbitals should also compress the family of states generated by these operators, potentially strengthening state-level natural-orbital methods for impurity quenches.
For open systems, the natural super-orbitals of mixed-state density matrices could track how one-body correlations degrade under environment coupling, a direction the paper mentions but does not develop.
The absence of a preferred basis in the bulk $t$-$V$ chain suggests that operator compression there, if any, would require higher-body or nonlinear rotations beyond one-body super-density matrices.

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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

Short version: the paper has a clean core and an honest numerical observation, but the headline claim about compact matrix-product-operator representation is an inference, not a demonstrated result. The one-body super-density matrix R was already around in the entanglement literature; the new contribution is treating its eigenbasis as a resource measure and working out the unitary-operator block structure. That algebra is solid, and the short-time expansions in Sec. V.C are a nice check. The numerical contrast between the t-V chain (no preferred basis) and the IRLM (exponentially decaying occupations) is credible; the data are independent TEBD runs, not a fit to the framework.

The soft spot is exactly the step from the spectrum to the practical advantage. Exponential decay of the one-body occupation spectrum of a vectorized operator does not, by itself, control the Schmidt spectrum across a spatial cut. Two states can have nearly identical one-body occupations and very different multipartite entanglement because R is blind to higher-body correlations. The paper never constructs an MPO in the natural super-orbital basis, and the conclusion phrase 'suggests' is doing real work. Eq. (35) is also more of a scaling hypothesis: the linear growth rate v is an undetermined parameter and the connection to non-stabilizerness via Pauli coefficients a_S is not derived. The author is upfront that the t≈8 plateau in the IRLM time-evolution data is a truncation artifact at χ=1024, which is honest but means the late-time claim rests on that plateau. The atomic-limit comparison needing a hand-rescaled envelope is a minor issue, and the author acknowledges that the toy model is missing bath physics.

None of this kills the paper. The main observation—that the IRLM operator has a hidden one-particle structure while the t-V operator does not—is new and worth publishing. But the abstract and conclusion overstate what has been shown. A serious referee should ask for either a small-system MPO in the rotated basis (even L=20 would demonstrate the compression) or a reformulated claim that treats the compact-MPO statement as a conjecture.

Referee Report

4 major / 5 minor

Summary. This manuscript introduces natural super-orbitals for many-body operators, defined as the eigenvectors of the one-body super-density matrix R[O] associated with a vectorized operator. The author derives general properties for unitary operators, noninteracting time-evolution operators, and Haar-random unitaries, and defines correlation entropy together with operator non-Gaussianity and non-stabilizerness measures. Numerical tensor-network results are presented for the t-V chain and the interacting resonant level model (IRLM), with the central claim being that in the IRLM the exponential decay of natural-super-orbital occupations enables a compact MPO representation and reduced non-stabilizerness in the natural-orbital basis. Short-time analytic scalings for the correlation entropy are also derived and checked numerically.

Significance. If the advertised compression is established, the framework would extend the natural-orbital idea from states to operators and connect it to fermionic non-Gaussianity and magic. The algebraic core of the paper is clean: the block structure of R for unitary operators, the factorization in the noninteracting case, and the Haar-random calculation are valuable and clearly presented. The numerical data are produced by an independent TEBD simulation rather than by fitting to the proposed mechanism, and the qualitative contrast between the delocalized t-V spectrum and the exponentially decaying IRLM spectrum is a meaningful observation. The short-time scaling predictions (quadratic for time-evolution operators, quartic/sixth-power for local operators in the two models) are falsifiable and add substance. The main weakness is that the headline practical claim, compact MPO representation in the natural-super-orbital basis, is asserted without constructing such an MPO or proving a bound connecting one-body occupations to operator entanglement across a spatial cut.

major comments (4)

[Abstract; Sec. V.B; Conclusion] The claim that exponential decay of the natural-super-orbital occupations n_alpha 'enables a compact matrix-product-operator representation' is not supported by any construction or proof. The spectrum of the one-body super-density matrix R[O] is not a sufficient statistic for the Schmidt spectrum of the vectorized operator across a spatial partition: operators with identical natural occupations can differ by higher-body correlations that are invisible to R. No MPO in the rotated basis is ever built, and no algorithm is given for applying the orbital rotation W to the real-space MPO used in the TEBD simulations. To make the advertised result load-bearing, the manuscript should either explicitly construct the rotated-basis MPO for the IRLM at modest sizes, or provide a rigorous bound linking the tail of n_alpha to the required MPO bond dimension; otherwise the Abstract and Conclusion should be softened to claim only a one-body spectral signature.
[Sec. V.B, Eq. (35)] Equation (35) is stated as a bound on the non-stabilizerness, M_NSO(U(t)) <= 2vt log(2) + c~, but it is not derived from the exponential decay of occupations. The number of natural super-orbitals with n_alpha > delta does not control the number or magnitude of the many-body Pauli coefficients a_S in the Pauli expansion; non-stabilizerness depends on the full coefficient distribution, not only on the one-body occupation spectrum. The velocity v is left unspecified, and the constants c and c~ are not quantified or related to delta and the Hamiltonian parameters. As written, Eq. (35) is a heuristic scaling statement rather than a bound; it should either be derived from explicit coefficient estimates or clearly relabeled as a conjecture.
[Sec. III, Correlation entropy and non-Gaussianity] The statement that Scorr = 0 if and only if the operator can be written as exp(M_mn c†_m c_n) with antisymmetric M is inconsistent with the creation-operator example presented just above. The operator c†_0 has Scorr = 0 according to Eqs. (23)-(24), but c†_0 changes the fermion number by one and is not of the even-parity exponential form exp(M_mn c†_m c_n). Thus either the class of 'Gaussian' or 'product' operators must be broadened (for example to include matchgate-type operations with linear terms), or the asserted iff characterization of Scorr is false. The claim that Scorr is a faithful measure of operator non-Gaussianity depends on resolving this inconsistency.
[Sec. III, Non-stabilizerness paragraph] The assertion that the natural-super-orbital basis 'maximizes the weight of the coefficients a_S' for a fixed number r of included super-orbitals and thereby minimizes non-stabilizerness is not proven, and the cited reference [59] concerns natural orbitals for quantum states, not Pauli decompositions of operators. This assertion underlies the Conclusion's statement that non-stabilizerness is reduced in the natural-orbital basis. A proof of this optimization property, or a removal of the claim, is needed.

minor comments (5)

[Sec. I] The sentence 'which has also can also be extended to fermionic systems' is grammatically broken; it should read 'which can also be extended to fermionic systems'.
[Sec. III, Eq. (24)] The notation in Eq. (24) is confusing: expressions such as '1 - <n2> = 1' are listed together with a final array '[0,0,1,1,1,1]' without clarifying whether these are occupations or eigenvalues of R. Please rewrite this display to make the convention explicit.
[Sec. V.B, Eq. (35)] The phrase 'c22vt' should be typeset as 'c 2^{2vt}', and the symbol c is used later for the coefficients c_alpha in Sec. V.C; please use distinct notation for the two quantities.
[Sec. V.B, Fig. 4] The statement that the slow oscillating envelope had to be rescaled by hand to match the simulation should be stated more prominently in the main text, since it indicates that the analytical first-order-in-1/U expression is not quantitatively predictive for the slow component.
[Sec. III, Eq. (12)] The derivation of the diagonal blocks of R for unitary operators via 'cyclicity of the trace' is very terse; a short explanation using the reduced density matrix of the maximally entangled vectorized state would make the result easier to verify.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the IRLM exponential-decay observation is an independent TEBD computation, self-citations are motivational, and the compact-MPO claim is an unproven inference rather than a circular step.

full rationale

The central numerical result, the exponential decay of the natural super-orbital occupations in the IRLM, is produced by an independent TEBD simulation and is not obtained by fitting a parameter and then relabeling that fit as a prediction. The self-citations (Refs. [24,30,31]) are used to motivate the analogy with natural orbitals of impurity ground states and to point toward a future few-body algorithm, but they do not carry the derivation of the operator-level correlation spectrum, which is computed directly. The abstract's claim that the exponential decay enables a compact MPO representation and reduced non-stabilizerness is an inference from one-body occupation data to operator entanglement and Pauli-weight concentration; the paper itself acknowledges in the Conclusion that a concrete algorithm would be needed to validate and extend the findings. This is a support gap, not a circular reduction: no equation defines the MPO bond dimension as an output of the occupation spectrum, and Eq. (35) is an upper bound obtained by taking the logarithm of the observed linear growth of significant orbitals, not a fitted prediction of a separately measured non-stabilizerness. The score is set to 2 only to reflect the presence of minor, non-load-bearing self-citations; the central derivation is self-contained and not circular.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The main numerical observation is not fitted, but the paper relies on unproved structural assumptions: faithfulness of Scorr as non-Gaussianity, linear active-orbital growth in IRLM, and the implication from one-body spectrum decay to MPO compression. The atomic-limit comparison contains an admitted rescale-to-match. No new physical entities are introduced; natural super-orbitals are eigenvectors of a previously defined matrix.

free parameters (5)

Linear growth rate v of active natural super-orbitals
Introduced in Eq. (35) to bound non-stabilizerness growth as M_NSO <= 2vt log(2) + c~; no microscopic derivation or fitted value is provided.
Rescaled envelope factor for U->infinity slow oscillation = Not specified
In Section V.B and Fig. 4, the atomic-limit envelope had to be rescaled by hand to match the numerical data because the toy model overestimates the slow component.
Occupation threshold delta for negligible super-orbitals = Unspecified
Used to define which orbitals are active in the non-stabilizerness bound; the paper says 'arbitrarily small' but does not fix a value.
Maximum TEBD bond dimension chi = 1024
Numerical truncation parameter; the IRLM time-evolution MPO reaches chi=1024 at t≈8, so later data are truncation-limited.
Trotter time step dt = 0.01
Fourth-order Trotter decomposition with dt=1e-2 is used to control time-discretization error.

assumptions (7)

ad hoc to paper Correlation entropy Scorr is a faithful measure of operator non-Gaussianity: Scorr=0 iff the operator is Gaussian.
Asserted in Section III without a proof of the converse direction; the claim that a one-body conjugation matrix with all singular values 1/2 forces the operator to be quadratic is nontrivial.
ad hoc to paper In the IRLM, the number of natural super-orbitals with occupation above a threshold delta grows linearly as 2vt.
Underlies Eq. (35); presented as an observation but v is unspecified and no mechanism for the linear growth is derived.
ad hoc to paper Exponential decay of natural-super-orbital occupations implies a compact MPO representation in the rotated basis.
The central practical claim; no formal relation between one-body super-density spectrum and operator entanglement across a fixed spatial partition is given.
ad hoc to paper The natural-super-orbital basis maximizes the weight of the largest Pauli-like coefficients and minimizes non-stabilizerness.
Asserted in Section III; the cited Lowdin reference [59] concerns natural orbitals for states, not operator stabilizer entropy.
standard math Trace identities in Eq. (11) correctly express R elements as MPO traces.
Straightforward algebraic manipulation of the vectorized operator and cyclicity of the trace; treated as exact.
standard math Weingarten calculus large-d asymptotics used for Haar random unitaries.
Standard result from Refs. [66,70] applied in Appendix A.
domain assumption Conclusions hold for U(1)-symmetric, particle-hole-symmetric spinless fermion chains in one dimension with open boundaries.
The numerical study is restricted to the t-V chain and IRLM at L=50-100; transfer to higher dimensions or other symmetry sectors is not established.

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Cite this review

Pith. "Pith review of Natural super-orbitals representation of many-body operators." pith.science (2026). https://pith.science/paper/MHPCWDDE

@misc{pith2026250710690,
  author       = {Pith},
  title        = {Pith review of: Natural super-orbitals representation of many-body operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MHPCWDDE}},
  note         = {Machine review of arXiv:2507.10690}
}

abstract

We introduce the concept of natural super-orbitals for many-body operators, defined as the eigenvectors of the one-body super-density matrix associated with a vectorized operator. We relate these objects to measures of non-Gaussianity of operators associated to the occupations of the natural super-orbitals, and define how the non-stabilizerness of operators can be affected by such a basis rotation. We first analyze the general analytical properties of these objects in various contexts, including the time-evolution operator of non-interacting Hamiltonians and Haar-random unitaries. We then perform a numerical investigation of the natural super-orbitals corresponding to both the time-evolution operator and a time-evolved local operator, focusing on two many-body systems: the fermionic $t\text{-}V$ chain and an impurity model, using tensor network simulations. Our results reveal that the $t\text{-}V$ model lacks a preferred super-orbital basis, while in the impurity model, the occupations of the natural orbitals for both operators decay exponentially at all times. This indicates that only a small number of orbitals contribute significantly to quantum correlations, enabling a compact matrix-product-operator representation and a reduced non-stabilizerness in the natural orbital basis. Finally, we examine the spatial spread of the natural orbitals for time-evolved local operators in the impurity model and show that the complexity of this operator in the natural orbital basis saturates over time. This new framework opens the door to future research that leverages the compressed structure of operators in their natural super-orbital basis, enabling for instance the computation of out-of-time-order correlators in large interacting systems over extended time scales.

Figures

Figures reproduced from arXiv: 2507.10690 by the authors.

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** FIG. 4. Correlation entropy of the time-evolution operator [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: FIG. 6. Time evolution of the occupations of the natural [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. Spatial dispersion of the natural super-orbitals [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 9.** Figure 9: FIG. 9. Weingarten diagram for the trace of operators [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

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Reference graph

Works this paper leans on

70 extracted references · 46 canonical work pages · cited by 4 Pith papers

[59]

L¨ owdin, Quantum theory of many-particle sys- tems

P.-O. L¨ owdin, Quantum theory of many-particle sys- tems. i. physical interpretations by means of den- sity matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction, Phys. Rev. 97, 1474 (1955)

work page 1955
[1]

Keimer, S

B. Keimer, S. Kivelson, M. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature 518, 179–186 (2015)

work page 2015
[2]

Savary and L

L. Savary and L. Balents, Quantum spin liquids: a review, Reports on Progress in Physics 80, 016502 (2016)

work page 2016
[3]

Nandkishore and D

R. Nandkishore and D. A. Huse, Many-body local- ization and thermalization in quantum statistical me- chanics, Annual Review of Condensed Matter Physics 6, 15 (2015)

work page 2015
[4]

D. A. Abanin, E. Altman, I. Bloch, and M. Ser- byn, Colloquium: Many-body localization, thermal- ization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019)

work page 2019
[5]

Lanczos, Solution of systems of linear equations by minimized iterations, Journal of research of the National Bureau of Standards 49, 33 (1952)

C. Lanczos, Solution of systems of linear equations by minimized iterations, Journal of research of the National Bureau of Standards 49, 33 (1952)

work page 1952
[6]

Benner and H

P. Benner and H. Faßbender, An implicitly restarted symplectic lanczos method for the hamiltonian eigen- value problem, Linear Algebra and its Applications 263, 75 (1997). 14

work page 1997
[7]

A. W. Sandvik and J. Kurkij¨ arvi, Quantum monte carlo simulation method for spin systems, Phys. Rev. B 43, 5950 (1991)

work page 1991

Show all 70 references

[8]

W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Ra- jagopal, Quantum monte carlo simulations of solids, Rev. Mod. Phys. 73, 33 (2001)

2001
[9]

Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011)

U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011)

2011
[10]

M. C. Ba˜ nuls, Tensor network algorithms: A route map, Annual Review of Condensed Matter Physics 14, 173 (2023)

2023
[11]

S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992)

1992
[12]

¨Ostlund and S

S. ¨Ostlund and S. Rommer, Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett. 75, 3537 (1995)

1995
[13]

A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Journal of Experimental and Theoretical Physics (1969)

1969
[14]

Maldacena, S

J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics 2016, 106 (2016)

2016
[15]

Bohrdt, C

A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap, Scrambling and thermalization in a diffusive quan- tum many-body system, New Journal of Physics 19, 063001 (2017)

2017
[16]

H´ emery, F

K. H´ emery, F. Pollmann, and D. J. Luitz, Matrix product states approaches to operator spreading in ergodic quantum systems, Phys. Rev. B 100, 104303 (2019)

2019
[17]

Xu and B

S. Xu and B. Swingle, Accessing scrambling using ma- trix product operators, Nature Physics 16, 199–204 (2020)

2020
[18]

Gisti, D

M. Gisti, D. J. Luitz, and M. Debertolis, Symme- try resolved out-of-time-order correlators of heisen- berg spin chains using projected matrix product op- erators, arXiv:2503.20327 (2025)

2025
[19]

Calabrese and J

P. Calabrese and J. Cardy, Evolution of entangle- ment entropy in one-dimensional systems, Journal of Statistical Mechanics: Theory and Experiment 2005, P04010 (2005)

2005
[20]

Fagotti and P

M. Fagotti and P. Calabrese, Evolution of entangle- ment entropy following a quantum quench: Analytic results for the xy chain in a transverse magnetic field, Phys. Rev. A 78, 010306 (2008)

2008
[21]

Eisert, M

J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010)

2010
[22]

V. Alba, More on the operator space entanglement (ose): R´ enyi ose, revivals, and integrability breaking, Journal of Physics A: Mathematical and Theoretical 10.1088/1751-8121/adc9e6 (2025)

2025 doi
[23]

Murciano, J

S. Murciano, J. Dubail, and P. Calabrese, More on symmetry resolved operator entanglement, Journal of Physics A: Mathematical and Theoretical 57, 145002 (2024)

2024
[24]

N´ u˜ nez-Fern´ andez, M

Y. N´ u˜ nez-Fern´ andez, M. Debertolis, and S. Florens, Resolving space-time structures of quantum impuri- ties with a numerically-exact algorithm using few- body revealing, arXiv:2503.13706 (2025)

2025
[25]

Zgid and G

D. Zgid and G. K.-L. Chan, Dynamical mean-field theory from a quantum chemical perspective, The Journal of Chemical Physics 134, 094115 (2011)

2011
[26]

He and Z.-Y

R.-Q. He and Z.-Y. Lu, Quantum renormalization groups based on natural orbitals, Phys. Rev. B 89, 085108 (2014)

2014
[27]

Y. Lu, M. H¨ oppner, O. Gunnarsson, and M. W. Haverkort, Efficient real-frequency solver for dynami- cal mean-field theory, Phys. Rev. B90, 085102 (2014)

2014
[28]

Bravyi and D

S. Bravyi and D. Gosset, Complexity of quantum im- purity problems, Communications in Mathematical Physics 356, 451–500 (2017)

2017
[29]

Boutin and B

S. Boutin and B. Bauer, Quantum impurity mod- els using superpositions of fermionic gaussian states: Practical methods and applications, Phys. Rev. Res. 3, 033188 (2021)

2021
[30]

Debertolis, S

M. Debertolis, S. Florens, and I. Snyman, Few-body nature of kondo correlated ground states, Phys. Rev. B 103, 235166 (2021)

2021
[31]

Debertolis, I

M. Debertolis, I. Snyman, and S. Florens, Simulating realistic screening clouds around quantum impurities: Role of spatial anisotropy and disorder, Phys. Rev. B 106, 125115 (2022)

2022
[32]

S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Many-body localization character- ized from a one-particle perspective, Phys. Rev. Lett. 115, 046603 (2015)

2015
[33]

S. Bera, T. Martynec, H. Schomerus, F. Heidrich- Meisner, and J. H. Bardarson, One-particle density matrix characterization of many-body localization, Annalen der Physik 529, 1600356 (2017)

2017
[34]

Buijsman, V

W. Buijsman, V. Gritsev, and V. Cheianov, Many- body localization in the fock space of natural orbitals, SciPost Phys. 4, 038 (2018)

2018
[35]

Hopjan and F

M. Hopjan and F. Heidrich-Meisner, Many-body lo- calization from a one-particle perspective in the dis- ordered one-dimensional bose-hubbard model, Phys. Rev. A 101, 063617 (2020)

2020
[36]

Chitambar and G

E. Chitambar and G. Gour, Quantum resource theo- ries, Rev. Mod. Phys. 91, 025001 (2019)

2019
[37]

Gottesman, The heisenberg representation of quantum computers, arXiv:quant-ph/9807006 (1998)

D. Gottesman, The heisenberg representation of quantum computers, arXiv:quant-ph/9807006 (1998)

1998 arXiv
[38]

Aaronson and D

S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004)

2004
[39]

Barenco, C

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVin- cenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995)

1995
[40]

Bravyi and A

S. Bravyi and A. Kitaev, Universal quantum com- putation with ideal clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005)

2005
[41]

Veitch, S

V. Veitch, S. A. Hamed Mousavian, D. Gottesman, and J. Emerson, The resource theory of stabilizer quantum computation, New Journal of Physics 16, 013009 (2014)

2014
[42]

Bravyi and D

S. Bravyi and D. Gosset, Improved classical simula- tion of quantum circuits dominated by clifford gates, Phys. Rev. Lett. 116, 250501 (2016)

2016
[43]

Bravyi, G

S. Bravyi, G. Smith, and J. A. Smolin, Trading classi- cal and quantum computational resources, Phys. Rev. X 6, 021043 (2016)

2016
[44]

Hamaguchi, K

H. Hamaguchi, K. Hamada, and N. Yoshioka, Hand- book for Quantifying Robustness of Magic, Quantum 8, 1461 (2024)

2024
[45]

Collura, J

M. Collura, J. D. Nardis, V. Alba, and G. Lami, The quantum magic of fermionic gaussian states, arXiv:2412.05367 (2025)

2025
[46]

Sierant, P

P. Sierant, P. Stornati, and X. Turkeshi, Fermionic magic resources of quantum many-body systems, arXiv:2506.00116 (2025)

2025
[47]

Leone, S

L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer r´ enyi entropy, Phys. Rev. Lett.128, 050402 (2022)

2022
[48]

Haug and L

T. Haug and L. Piroli, Quantifying nonstabilizerness of matrix product states, Phys. Rev. B 107, 035148 (2023). 15

2023
[49]

Lami and M

G. Lami and M. Collura, Nonstabilizerness via perfect pauli sampling of matrix product states, Phys. Rev. Lett. 131, 180401 (2023)

2023
[50]

Leone and L

L. Leone and L. Bittel, Stabilizer entropies are mono- tones for magic-state resource theory, Phys. Rev. A 110, L040403 (2024)

2024
[51]

P. S. Tarabunga, E. Tirrito, M. C. Ba˜ nuls, and M. Dalmonte, Nonstabilizerness via matrix product states in the pauli basis, Phys. Rev. Lett. 133, 010601 (2024)

2024
[52]

B. M. Terhal and D. P. DiVincenzo, Classical sim- ulation of noninteracting-fermion quantum circuits, Phys. Rev. A 65, 032325 (2002)

2002
[53]

Bravyi, Lagrangian representation for fermionic lin- ear optics, arXiv:quant-ph/0404180 (2004)

S. Bravyi, Lagrangian representation for fermionic lin- ear optics, arXiv:quant-ph/0404180 (2004)

2004 arXiv
[54]

Reardon-Smith, M

O. Reardon-Smith, M. Oszmaniec, and K. Korzekwa, Improved simulation of quantum circuits dominated by free fermionic operations, Quantum8, 1549 (2024)

2024
[55]

Dias and R

B. Dias and R. Koenig, Classical simulation of non- Gaussian fermionic circuits, Quantum 8, 1350 (2024)

2024
[56]

Lyu and K

X. Lyu and K. Bu, Fermionic gaussian test- ing and non-gaussian measures via convolution, arXiv:2409.08180 (2024)

2024 arXiv
[57]

Coffman, G

L. Coffman, G. Smith, and X. Gao, Measuring non- gaussian magic in fermions: Convolution, entropy, and the violation of wick’s theorem and the match- gate identity, arXiv:2501.06179 (2025)

2025 arXiv
[58]

Dowling, P

N. Dowling, P. Kos, and X. Turkeshi, Magic of the heisenberg picture, arXiv:2408.16047 (2025)

2025 arXiv
[60]

E. R. Davidson, Natural orbitals, Advances in Quan- tum Chemistry 6, 235 (1972)

1972
[61]

B. O. Roos, P. R. Taylor, and P. E. Sigbahn, A com- plete active space scf method (casscf) using a den- sity matrix formulated super-ci approach, Chemical Physics 48, 157 (1980)

1980
[62]

A. Rath, V. Vitale, S. Murciano, M. Votto, J. Dubail, R. Kueng, C. Branciard, P. Calabrese, and B. Ver- mersch, Entanglement barrier and its symmetry res- olution: Theory and experimental observation, PRX Quantum 4, 010318 (2023)

2023
[63]

K. Siva, Y. Zou, T. Soejima, R. S. K. Mong, and M. P. Zaletel, Universal tripartite entanglement signature of ungappable edge states, Phys. Rev. B 106, L041107 (2022)

2022
[64]

Gottesman, Stabilizer codes and quantum error correction, arXiv:quant-ph/9705052 (1997)

D. Gottesman, Stabilizer codes and quantum error correction, arXiv:quant-ph/9705052 (1997)

1997 arXiv
[65]

Thouless, Stability conditions and nuclear rota- tions in the hartree-fock theory, Nuclear Physics 21, 225 (1960)

D. Thouless, Stability conditions and nuclear rota- tions in the hartree-fock theory, Nuclear Physics 21, 225 (1960)

1960
[66]

Collins and I

B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Communications in Mathematical Physics 297, 345 (2010)

2010
[67]

Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys

G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004)

2004
[68]

Jordan and E

P. Jordan and E. Wigner, ¨Uber das paulische ¨ aquivalenzverbot, Zeitschrift f¨ ur Physik 47, 631 (1928)

1928
[69]

Khemani, D

V. Khemani, D. A. Huse, and A. Nahum, Velocity- dependent lyapunov exponents in many-body quan- tum, semiclassical, and classical chaos, Phys. Rev. B 98, 144304 (2018)

2018
[70]

Fukuda, R

M. Fukuda, R. K¨ onig, and I. Nechita, Rtni—a sym- bolic integrator for haar-random tensor networks, Journal of Physics A: Mathematical and Theoretical 52, 425303 (2019). A. WEINGAR TEN CALCULA TION FOR HAAR RANDOM UNIT ARIES In this appendix, we detail the derivation of the c...

2019

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [59]

L¨ owdin, Quantum theory of many-particle sys- tems

P.-O. L¨ owdin, Quantum theory of many-particle sys- tems. i. physical interpretations by means of den- sity matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction, Phys. Rev. 97, 1474 (1955)

work page 1955

[2] [1]

Keimer, S

B. Keimer, S. Kivelson, M. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature 518, 179–186 (2015)

work page 2015

[3] [2]

Savary and L

L. Savary and L. Balents, Quantum spin liquids: a review, Reports on Progress in Physics 80, 016502 (2016)

work page 2016

[4] [3]

Nandkishore and D

R. Nandkishore and D. A. Huse, Many-body local- ization and thermalization in quantum statistical me- chanics, Annual Review of Condensed Matter Physics 6, 15 (2015)

work page 2015

[5] [4]

D. A. Abanin, E. Altman, I. Bloch, and M. Ser- byn, Colloquium: Many-body localization, thermal- ization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019)

work page 2019

[6] [5]

Lanczos, Solution of systems of linear equations by minimized iterations, Journal of research of the National Bureau of Standards 49, 33 (1952)

C. Lanczos, Solution of systems of linear equations by minimized iterations, Journal of research of the National Bureau of Standards 49, 33 (1952)

work page 1952

[7] [6]

Benner and H

P. Benner and H. Faßbender, An implicitly restarted symplectic lanczos method for the hamiltonian eigen- value problem, Linear Algebra and its Applications 263, 75 (1997). 14

work page 1997

[8] [7]

A. W. Sandvik and J. Kurkij¨ arvi, Quantum monte carlo simulation method for spin systems, Phys. Rev. B 43, 5950 (1991)

work page 1991