Pith. sign in

REVIEW 2 major objections 2 minor 97 references

Bounding the finite-size error of quantum many-body dynamics simulations

T0 review · 2 major / 2 minor · reviewed 2026-08-27 · deepseek-v4-flash

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

Pith's one-line read Finite-size error in quantum many-body dynamics has a rigorous, explicit bound.

desk verdict The central finite-size error bound is rigorous and useful; the improved PBC bound has a real but secondary gap that should be fixed before publication. read the letter →

arxiv 2009.12032 v2 pith:UJBOTEFW submitted 2020-09-25 quant-ph cond-mat.stat-mech

Zhiyuan Wang , Michael Foss-Feig , Kaden R. A. Hazzard This is my paper · ORCID
classification quant-phcond-mat.stat-mech
keywords finite-sizeerrorquantummany-bodydynamicsLieb-Robinsonboundproductinitialstatesperiodicboundaryconditionsopenground-stategapcertification
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

This paper proves that the error in approximating an infinite-system local observable by a finite-size simulation can be bounded, for all times, by an explicit function of system size and time. For locally interacting Hamiltonians with finite local Hilbert space and product initial states, the bound is $|\langle \hat S(t)\rangle_L - \langle \hat S(t)\rangle_\infty| \le C(2vt/L)^{cL-\mu}$, with constants $v$, $C$, $c$, $\mu$ that are independent of $L$ and $t$ and can be computed from the Hamiltonian. The argument converts finite-size error into a time-integrated commutator with the boundary Hamiltonian, then controls that commutator with finite-speed propagation bounds. The authors demonstrate the bound is practically tight in one-dimensional transverse-field Ising and Fermi-Hubbard simulations, and they extend it to correlated initial states and to gapped ground states. A sympathetic reader would care because numerical many-body methods otherwise have no rigorous guarantee for when finite-size results can be trusted.

What carries the argument

The load-bearing identity is Eq. (4): $|\delta\langle \hat S(t)\rangle_\psi| \le \int_0^t \|[\Delta \hat H(t'), \hat S]\|\,dt'$, where $\Delta \hat H$ collects the interaction terms that couple the simulated cluster to its environment (and, for periodic boundary conditions, the wrapped link). This turns finite-size error into an unequal-time commutator, whose norm is exactly what a finite-speed light-cone bound (the Lieb-Robinson bound) controls. For the periodic-boundary improvement, the additional machinery is the commutativity graph, a graph whose vertices are Hamiltonian terms and whose edges join non-commuting terms; time-evolved operators are expanded into Lie clusters and classified by their earliest unembeddable vertex, the first term that makes a cluster too large to fit in the periodic system. Each family of clusters is then bounded by counting irreducible paths and Y-shapes on the graph, yielding the improved periodic-boundary estimate Eq. (6).

What would settle it

Numerically compute the first several Taylor coefficients of the auxiliary function $f(t)$ defined around Eqs. (S39)-(S43) for a small commutativity graph, for example the one-dimensional transverse-field Ising chain, and check that every coefficient is non-negative and that the leading exponent equals the size of the smallest periodic-loop Lie cluster minus one. Finding a single negative coefficient or a leading exponent smaller than claimed would disprove the polynomial periodic-boundary bound; the looser exponential bound of Eq. (S51) could still hold. For the open-boundary product-state bound, a direct check is to evaluate exact finite-$L$ and infinite-$L$ dynamics for a small exactly solvable critical chain and verify that the stated constants are not violated at any tested $L$ and $t$.

Watch

Extended reading notes

Core claim

The central claim is that finite-size error is a locality phenomenon, and locality alone gives a quantitative bound. For any $d$-dimensional locally interacting lattice Hamiltonian with finite local Hilbert space, starting from a product state, the difference between the finite-$L$ and infinite expectation values of a local observable satisfies the main inequality, with constants computed explicitly. For periodic boundary conditions the same construction yields a stronger estimate whose exponent is a factor of two larger than for open boundary conditions, reflecting that the smallest non-contractible loop on the periodic commutativity graph is roughly twice as long as the shortest boundary-to-observable path. The rigorous mechanism is an identity expressing finite-size error as the integrated norm of the commutator of the evolution with the boundary interaction, $[\Delta \hat H(t'), \hat S]$, followed by a finite-speed light-cone bound on that commutator. Under a uniform spectral gap assumption, the same techniques prove exponential decay of ground-state finite-size error with system size, and the dynamics bound extends to any initial state whose finite-size approximation satisfies exponential clustering.

Load-bearing premise

The refined periodic-boundary-case bound depends on an unproved technical assertion in the Supplemental Material: after Eq. (S51), the bounding function $f(t)$ is stated to have a Taylor expansion with non-negative coefficients and the same leading exponent as the tightest graph-based bound, with the justification left as one can show; if that property fails, the polynomial form of the periodic-boundary bound would need to be replaced by a looser exponential bound.

Editorial extensions

If this is right

  • For a product initial state in any locally interacting model with a finite-speed light cone, a simulation at size $L$ is certified accurate up to times of order $L/(2v)$, converting the empirical practice of comparing two sizes into a rigorous stopping criterion.
  • For periodic boundary conditions the error bound decays twice as fast with linear system size as for open boundary conditions at early times, quantitatively supporting the common preference for periodic boundary conditions in finite-size dynamical studies.
  • For open boundary conditions, measuring an observable near the center of the system gives an error that decays exponentially in $L$, while averaging over all sites yields only algebraic decay; center-site measurement is therefore the right protocol when open boundary conditions must be used.
  • For non-degenerate gapped ground states, local-observable finite-size error decays exponentially in system size under a uniform gap condition, and the same exponential-clustering input extends the dynamics bound to matrix product states with finite bond dimension and to finite-temperature thermal states above a certain temperature.
  • In the one-dimensional transverse-field Ising model at criticality, the $L=21$ periodic-boundary bound guarantees accuracy within $10^{-2}$ up to $Jt \approx 3.5$, and in the one-dimensional Fermi-Hubbard model the $L=12$ bound guarantees 1% accuracy up to $Jt \approx 1.2$, showing the bounds are tight enough for practical use.

Reading between the lines

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

  • A concrete testable extension is to compute finite-size error bounds for long-range (power-law) interacting models by substituting the available long-range light-cone bounds into Eq. (4); the paper notes the identity remains valid but does not carry out the optimization of the resulting constants.
  • If the factor-of-two periodic-boundary advantage persists in practice, numerical practitioners could use periodic-boundary simulations at roughly half the required open-boundary system size for the same guaranteed time horizon, which would reduce exact-diagonalization memory costs substantially.
  • The same bounding strategy could be used as an adaptive convergence criterion inside tensor-network or other truncated simulations: stop refining the simulation when the certified finite-size bound, rather than the truncation error, becomes the dominant uncertainty. The paper does not develop this algorithmic application.
  • The estimates suggest an automatic cluster-shape selection algorithm for $d>1$: among clusters with a fixed number of sites, choose the aspect ratio that minimizes the right-hand side of the bound, which the paper indicates should roughly set each linear dimension proportional to the corresponding light-cone speed.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. This paper derives rigorous upper bounds on the finite-size error of local observables in real-time quantum dynamics simulations of locally interacting lattice systems. The central reduction, Eq. (4), bounds the difference between finite-size and thermodynamic-limit expectation values by the time integral of a commutator norm involving the boundary Hamiltonian, and the authors then insert existing Lieb-Robinson bounds to obtain Eq. (1): |<S(t)>_L - <S(t)>_∞| ≤ C(2vt/L)^{cL−μ} with explicitly computable constants for product initial states. The paper also presents an improved periodic-boundary-condition bound, Eq. (6), with a formally doubled exponent; an exponential ground-state bound, Eq. (7), under a uniform spectral gap assumption; and an extension to correlated initial states satisfying exponential clustering, Eq. (9). The bounds are demonstrated numerically for the one-dimensional transverse-field Ising model and the Fermi-Hubbard model.

Significance. If the results hold, this paper fills a genuine gap: it provides first-principles, parameter-free upper bounds on finite-size errors that are rigorous and, in the demonstrated one-dimensional examples, tight enough to guarantee accuracy out to physically relevant times. The reduction in Eq. (4) is clean and correct for product initial states, and the subsequent insertion of Lieb-Robinson bounds is standard and does not involve fitting constants to the quantities being bounded. The paper also makes useful qualitative points, such as quantitatively justifying center-site measurements in open-boundary-condition simulations and quantifying an early-time advantage of periodic boundary conditions. Strengths include the detailed supplemental derivations for the ground-state and correlated-initial-state bounds, the explicit constants in the simple models, and the use of exact finite-size numerics to illustrate practical tightness.

major comments (2)
  1. [Supplemental Material, 'Derivation of Eq. (6)', text after Eq. (S51)] The advertised periodic-boundary-condition bound Eq. (6) relies on Proposition 1 together with the assertion, immediately after Eq. (S51), that the upper bound constructed from Eqs. (S39) and (S43) has a Taylor series with non-negative coefficients and leading exponent min n_p(Y_S)-1, stated only as 'one can show'. This assertion is load-bearing and is not proved. It does not follow immediately from G^a_{ij}(t)=[e^{Mt}]_{ij} having non-negative Taylor coefficients, because the bound in Eqs. (S35)-(S39) includes absolute values, products of series, nested time integrals, and sums over graph paths; non-negativity of every coefficient and exact matching of the leading exponent require a separate argument. If the leading exponent is larger than stated, Eq. (6) overstates the small-time suppression and the advertised factor-of-two PBC advantage is unproven. The central result Eq. (1) is not affected, because the OBC-style bound obtained by inserting a Lieb-Robinson bound into Eq. (4) remains valid; nevertheless, the PBC claim in the abstract and introduction should not be presented as proven until this lemma is supplied.
  2. [Supplemental Material, 'Chen-Lucas bound for LR commutators', Eq. (S20)] Equation (S20), the bound for the double commutator [h_j, e^{iH'_j t}(S)-e^{iH_j t}(S)], is central to the improved PBC bound and to the TFIM bound Eq. (11), but it is introduced as 'obtained in a similar way' to Eq. (S19) without a proof or a precise derivation of the binomial factor and the Y-shape sum. Since Eq. (S19) itself is a nontrivial graph-theoretic extension of the Chen-Lucas theorem, the authors should either provide a proof of Eq. (S20) or give an exact theorem-and-lemma reference that covers it.
minor comments (2)
  1. [Main text, after Eq. (6)] The symbol L_p is used both for the physical system size and for the graph-theoretic exponent in the sentence following Eq. (6), making the statement L_p = η_p L_p − μ_p confusing; please use separate notations, for example \ell_p for the physical size and \mathcal{L}_p for the exponent.
  2. [Supplemental Material, 'Comparison of FSE bounds to perturbation theory'] In the OBC paragraph, the expression ⟨e^{−iHt} S e^{−iHt}⟩ should presumably be ⟨e^{iHt} S e^{−iHt}⟩; as written, both time-evolution factors have the same sign, which would not describe the Heisenberg-evolved observable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the finite-size error bound follows from Eq. (4) and independently proven Lieb-Robinson bounds; the PBC bound's unproved Taylor-coefficient lemma is a rigor gap, not a circular reduction.

full rationale

The central claim Eq. (1) is obtained from the exact identity/bound chain Eqs. (2)-(4): FSE is rewritten as an unequal-time commutator integral, then bounded by inserting Lieb-Robinson (LR) bounds. The LR bounds used are Ref. [63] (Chen-Lucas) and Ref. [64] (Wang-Hazzard, a prior published result with its own proof); neither bound is defined in terms of FSE, and no constant in Eq. (1) is fitted to the quantities being bounded. The improved PBC bound Eq. (6) is derived from the Chen-Lucas/Wang-Hazzard commutator methods in the Supplemental Material. The one genuinely weak spot is an unproved assertion after Eq. (S51): 'one can show that the upper bound for f(t) given in Eqs. (S39) and (S43) has a Taylor series expansion with the same leading term as the Chen-Lucas bound in Eq. (S20) and with non-negative coefficients'. This is a missing proof (a correctness/rigor risk that could invalidate the advertised factor-of-two PBC advantage), but it is not circular: it does not assume the desired conclusion, it merely lacks a demonstrated derivation. If that lemma fails, the looser exponential bound Eq. (S51) and the OBC-style argument still deliver the main result Eq. (1). Numerics for TFIM and FHM are used only to compare the bounds with exact data, not to set the exponents or prefactors, so no fitted input is being relabeled as a prediction. The paper's self-citation of Ref. [64] is load-bearing for quantitative tightness, but because Ref. [64] is an independent, parameter-free, published result with stated assumptions that do not include the FSE claim, it constitutes real evidence rather than a self-citation loop.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard Lieb-Robinson bounds and explicit structural assumptions about the Hamiltonian and initial state. No free parameters are fitted; all constants are computed from the Hamiltonian and observable. The main unproved ingredient is the Taylor-coefficient lemma used in the improved PBC derivation.

assumptions (6)
  • domain assumption Hamiltonian is locally interacting on a d-dimensional lattice with finite local Hilbert space, so a Lieb-Robinson bound exists.
    Invoked in Eq. (4) and throughout; the FSE bound is obtained by inserting a LR bound from Refs. [48,49,50,63,64].
  • domain assumption Initial state is a product state for the main dynamics bound.
    Required for the identity replacing the finite-size expectation with ⟨e^{iH't} S e^{-iH't}⟩_ψ in Eq. (2).
  • domain assumption For the ground-state FSE bound, the interpolated Hamiltonian H(λ)=H−λΔH is non-degenerate and uniformly gapped with Δ>0 for all λ∈[0,1].
    Stated in the main text before Eq. (7); used in Theorem 1 to obtain exponential clustering.
  • domain assumption For correlated initial states, a finite-size approximation ρ_L obeys the exponential-clustering condition Eq. (7).
    Assumed in the generalization; the paper notes it holds for gapped ground states, finite-bond-dimension MPS, and high-temperature thermal states.
  • domain assumption The improved PBC bound assumes translation invariance.
    The SM derivation uses Fourier transforms on the lattice and the commutativity graph, requiring translation invariance.
  • ad hoc to paper The differential-equation-based LR bound has a Taylor expansion with non-negative coefficients matching the Chen-Lucas leading exponent.
    Unproved lemma stated between Eqs. (S51) and (S54); needed to convert the exponential bound into the polynomial form of Eq. (6).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bounding the finite-size error of quantum many-body dynamics simulations." pith.science (2026). https://pith.science/paper/UJBOTEFW

@misc{pith2026200912032,
  author       = {Pith},
  title        = {Pith review of: Bounding the finite-size error of quantum many-body dynamics simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJBOTEFW}},
  note         = {Machine review of arXiv:2009.12032}
}
abstract

Finite-size error (FSE), the discrepancy between an observable in a finite system and in the thermodynamic limit, is ubiquitous in numerical simulations of quantum many body systems. Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quantitative bound on the magnitude of FSE is still missing. Here we derive rigorous upper bounds on the FSE of local observables in real time quantum dynamics simulations initialized from a product state. In $d$-dimensional locally interacting systems with a finite local Hilbert space, our bound implies $ |\langle \hat{S}(t)\rangle_L-\langle \hat{S}(t)\rangle_\infty|\leq C(2v t/L)^{cL-\mu}$, with $v$, $C$, $c$, $\mu $ constants independent of $L$ and $t$, which we compute explicitly. For periodic boundary conditions (PBC), the constant $c$ is twice as large as that for open boundary conditions (OBC), suggesting that PBC have smaller FSE than OBC at early times. The bound can be generalized to a large class of correlated initial states as well. As a byproduct, we prove that the FSE of local observables in ground state simulations decays exponentially with $L$, under a suitable spectral gap condition. Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models.

Figures

Figures reproduced from arXiv: 2009.12032 by the authors.

Figure 1
Figure 1. FIG. 1. An illustration of our bounds. In locally interact [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Numerically exact evolution for (a) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

97 extracted references · 52 canonical work pages

  1. [60]

    C. B. Da˘ g and K. Sun, arXiv preprint arXiv:2004.12287 (2020)

    work page Pith review arXiv 2020

  2. [64]

    Wang and K

    Z. Wang and K. R. A. Hazzard, PRX Quantum1, 010303 (2020)

    work page 2020

  3. [1]

    Quantum magnetism. lecture notes in physics,

    N. Laflorencie and D. Poiblanc, “Quantum magnetism. lecture notes in physics,” (Springer-Verlag, 2004) Chap. Simulations of pure and doped low-dimensional spin-1/2 gapped systems

    2004

  4. [2]

    R. M. Noack and S. R. Manmana, AIP Conf. Proc. 789, 93 (2005)

    2005

  5. [3]

    A. W. Sandvik, AIP Conf. Proc. 1297, 135 (2010)

    2010

  6. [4]

    Introduction to frustrated magnetism: Ma- terials, experiments, theory,

    A. L¨ auchli, “Introduction to frustrated magnetism: Ma- terials, experiments, theory,” (Springer, 2011) Chap. Numerical Simulations of Frustrated Systems, pp. 481 – 511

    2011

  7. [5]

    Researchers usually employ much smaller systems for computational conve- nience

    Reaching even these system sizes is possible only if inter- nal, translation, and point group symmetries are utilized, and if state-of-the-art algorithms and large-scale com- putational resources are employed. Researchers usually employ much smaller systems for computational conve- nience

  8. [6]

    S. R. White, Phys. Rev. Lett. 69, 2863 (1992)

    1992

Show all 97 references
  1. [7]

    K. A. Hallberg, Adv. Phys. 55, 477 (2006)

    2006

  2. [8]

    Schollw¨ ock, Ann

    U. Schollw¨ ock, Ann. Phys.326, 96 (2011)

    2011

  3. [9]

    E. M. Stoudenmire and S. R. White, Annu. Rev. Con- dens. Matter Phys. 3, 111 (2012)

    2012

  4. [10]

    Perez-Garcia, F

    D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Info. Comput. 7, 401–430 (2007)

    2007

  5. [11]

    Or´ us, Ann

    R. Or´ us, Ann. Phys.349, 117 (2014)

    2014

  6. [12]

    Bartsch and J

    C. Bartsch and J. Gemmer, Phys. Rev. Lett. 102, 110403 (2009)

    2009

  7. [13]

    T. A. Elsayed and B. V. Fine, Phys. Rev. Lett. 110, 070404 (2013)

    2013

  8. [14]

    Steinigeweg, A

    R. Steinigeweg, A. Khodja, H. Niemeyer, C. Gogolin, and J. Gemmer, Phys. Rev. Lett. 112, 130403 (2014)

    2014

  9. [15]

    Steinigeweg, J

    R. Steinigeweg, J. Gemmer, and W. Brenig, Phys. Rev. Lett. 112, 120601 (2014)

    2014

  10. [16]

    Steinigeweg, F

    R. Steinigeweg, F. Heidrich-Meisner, J. Gemmer, K. Michielsen, and H. De Raedt, Phys. Rev. B 90, 094417 (2014)

    2014

  11. [17]

    M. P. Nightingale and C. J. Umrigar, eds., Quan- tum Monte Carlo Methods in Physics and Chemistry (Springer, 1999)

    1999

  12. [18]

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

    2018

  13. [19]

    Bloch, J

    I. Bloch, J. Dalibard, and S. Nascimb` ene, Nat. Phys. 8, 267 (2012)

    2012

  14. [20]

    Blatt and C

    R. Blatt and C. F. Roos, Nat. Phys. 8, 277 (2012)

    2012

  15. [21]

    Altman, K

    E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Dem- ler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriks- son, K.-M. C. Fu, et al. , arXiv:1912.06938 (2019)

    2019 arXiv

  16. [22]

    A. W. Sandvik, in AIP Conference Proceedings, Vol. 1297 (American Institute of Physics, 2010) pp. 135–338

    2010

  17. [23]

    Bausch, T

    J. Bausch, T. S. Cubitt, A. Lucia, D. Perez-Garcia, and M. M. Wolf, Proc. Natl. Acad. Sci. U.S.A115, 19 (2018)

    2018

  18. [24]

    Zeiher, R

    J. Zeiher, R. van Bijnen, P. Schauß, S. Hild, J.-Y. Choi, T. Pohl, I. Bloch, and C. Gross, Nat. Phys. 12, 1095 (2016)

    2016

  19. [25]

    Takei, C

    N. Takei, C. Sommer, C. Genes, G. Pupillo, H. Goto, K. Koyasu, H. Chiba, M. Weidem¨ uller, and K. Ohmori, Nature Communications 7, 13449 (2016)

    2016

  20. [26]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al. , Nature 551, 579 (2017)

    2017

  21. [28]

    Lienhard, S

    V. Lienhard, S. de L´ es´ eleuc, D. Barredo, T. Lahaye, A. Browaeys, M. Schuler, L.-P. Henry, and A. M. L¨ auchli, Phys. Rev. X8, 021070 (2018)

    2018

  22. [29]

    A. P. Orioli, A. Signoles, H. Wildhagen, G. G¨ unter, J. Berges, S. Whitlock, and M. Weidem¨ uller, Phys. Rev. Lett. 120, 063601 (2018)

    2018

  23. [30]

    B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Nature 501, 521 (2013)

    2013

  24. [31]

    K. R. A. Hazzard, B. Gadway, M. Foss-Feig, B. Yan, S. A. Moses, J. P. Covey, N. Y. Yao, M. D. Lukin, J. Ye, D. S. Jin, and A. M. Rey, Phys. Rev. Lett. 113, 195302 (2014)

    2014

  25. [32]

    Seeßelberg, X.-Y

    F. Seeßelberg, X.-Y. Luo, M. Li, R. Bause, S. Ko- tochigova, I. Bloch, and C. Gohle, Phys. Rev. Lett. 121, 253401 (2018)

    2018

  26. [33]

    Smale, P

    S. Smale, P. He, B. A. Olsen, K. G. Jackson, H. Sharum, S. Trotzky, J. Marino, A. M. Rey, and J. H. Thywissen, Sci. Adv. 5 (2019), 10.1126/sciadv.aax1568

    2019 doi

  27. [34]

    de Paz, A

    A. de Paz, A. Sharma, A. Chotia, E. Marechal, J. H. Huckans, P. Pedri, L. Santos, O. Gorceix, L. Vernac, and B. Laburthe-Tolra, Phys. Rev. Lett. 111, 185305 (2013)

    2013

  28. [35]

    Meldgin, U

    C. Meldgin, U. Ray, P. Russ, D. Chen, D. M. Ceperley, and B. DeMarco, Nat. Phys. 12, 646 (2016)

    2016

  29. [36]

    J.-Y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio- Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Science 352, 1547 (2016)

    2016

  30. [37]

    Bordia, H

    P. Bordia, H. L¨ uschen, S. Scherg, S. Gopalakrishnan, M. Knap, U. Schneider, and I. Bloch, Phys. Rev. X 7, 041047 (2017)

    2017

  31. [38]

    Gabardos, B

    L. Gabardos, B. Zhu, S. Lepoutre, A. M. Rey, B. Laburthe-Tolra, and L. Vernac, arXiv:2005.13487 (2020)

    2020 arXiv

  32. [39]

    Goban, R

    A. Goban, R. B. Hutson, G. E. Marti, S. L. Campbell, 6 M. A. Perlin, P. S. Julienne, J. P. D’Incao, A. M. Rey, and J. Ye, Nature 563, 369 (2018)

    2018

  33. [40]

    Eisert, M

    J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015)

    2015

  34. [41]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015)

    2015

  35. [42]

    D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B 93, 060201(R) (2016)

    2016

  36. [43]

    D. J. Luitz and Y. B. Lev, Ann. Phys. (Berlin) 529, 1600350 (2017)

    2017

  37. [44]

    S. A. Parameswaran and R. Vasseur, Rep. Prog. Phys. 81, 082501 (2018)

    2018

  38. [45]

    T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, J. Phys. B 51, 112001 (2018)

    2018

  39. [46]

    Schmied, A

    C.-M. Schmied, A. N. Mikheev, and T. Gasenzer, arXiv:1810.08143

    arXiv

  40. [48]

    Bravyi, M

    S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev. Lett. 97, 050401 (2006)

    2006

  41. [49]

    M. B. Hastings, arXiv:1008.5137 (2010)

    2010 arXiv

  42. [50]

    E. H. Lieb and D. W. Robinson, Commun. Math. Phys. 28, 251 (1972)

    1972

  43. [51]

    T. J. Osborne, Phys. Rev. Lett. 97, 157202 (2006)

    2006

  44. [52]

    T. J. Osborne, Phys. Rev. A 75, 032321 (2007)

    2007

  45. [53]

    T. J. Osborne, Phys. Rev. A 75, 042306 (2007)

    2007

  46. [54]

    Kliesch, C

    M. Kliesch, C. Gogolin, and J. Eisert, in Many- Electron Approaches in Physics, Chemistry and Math- ematics (Springer, 2014) pp. 301–318

    2014

  47. [55]

    M. P. Woods, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 115, 130401 (2015)

    2015

  48. [56]

    M. P. Woods and M. B. Plenio, J. Math. Phys. 57, 022105 (2016)

    2016

  49. [57]

    J. Haah, M. Hastings, R. Kothari, and G. H. Low, in 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2018) pp. 350–360

    2018

  50. [58]

    M. C. Tran, A. Y. Guo, Y. Su, J. R. Garrison, Z. El- dredge, M. Foss-Feig, A. M. Childs, and A. V. Gorshkov, Phys. Rev. X 9, 031006 (2019)

    2019

  51. [59]

    Huang, arXiv:2001.10763 (2020)

    Y. Huang, arXiv:2001.10763 (2020)

    2020 arXiv

  52. [61]

    Trotzky, Y.-A

    S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollw¨ ock, J. Eisert, and I. Bloch, Nat. Phys. 8, 325 (2012)

    2012

  53. [62]

    Bauer, F

    A. Bauer, F. Dorfner, and F. Heidrich-Meisner, Phys. Rev. A 91, 053628 (2015)

    2015

  54. [63]

    Chen and A

    C.-F. Chen and A. Lucas, arXiv preprint arXiv:1905.03682 (2019)

    2019 arXiv

  55. [65]

    D. Iyer, M. Srednicki, and M. Rigol, Phys. Rev. E 91, 062142 (2015)

    2015

  56. [66]

    While the fact that the error bound for PBC is smaller than that for OBC does not necessarily imply that the actual FSE in PBC is smaller, in the Supplemental Ma- terial [74] we use short-time perturbative arguments to show that the actual error in PBC is indeed smaller at ear...

  57. [67]

    M. B. Hastings and T. Koma, Commun. Math. Phys. 265, 781 (2006)

    2006

  58. [68]

    Richerme, Z.-X

    P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Monroe, Nature 511, 198 (2014)

    2014

  59. [69]

    Z.-X. Gong, M. Foss-Feig, S. Michalakis, and A. V. Gor- shkov, Phys. Rev. Lett. 113, 030602 (2014)

    2014

  60. [70]

    Foss-Feig, Z.-X

    M. Foss-Feig, Z.-X. Gong, C. W. Clark, and A. V. Gor- shkov, Phys. Rev. Lett. 114, 157201 (2015)

    2015

  61. [71]

    Chen and A

    C.-F. Chen and A. Lucas, Phys. Rev. Lett. 123, 250605 (2019)

    2019

  62. [72]

    Kuwahara and K

    T. Kuwahara and K. Saito, Phys. Rev. X 10, 031010 (2020)

    2020

  63. [73]

    M. C. Tran, C.-F. Chen, A. Ehrenberg, A. Y. Guo, A. Deshpande, Y. Hong, Z.-X. Gong, A. V. Gorshkov, and A. Lucas, Phys. Rev. X 10, 031009 (2020)

    2020

  64. [74]

    (S18), the detailed proof of Eqs

    See Supplemental Material for the comparison of the er- ror bounds to perturbation theory at early times, the detailed derivation of the PBC error bound in Eq. (S18), the detailed proof of Eqs. (7) and (9), and detailed derivations and expressions for the three different meth- ...

  65. [75]

    That ˆρL satisfies the condition in Eq

    For translation invariant MPS with a finite bond di- mension, ˆρL can be taken as the L-site periodic version of ˆρ. That ˆρL satisfies the condition in Eq. (7) can be proved using the transfer operator method which is used to prove that MPS has finite correlation length, see, e....

  66. [76]

    Kliesch, C

    M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert, Phys. Rev. X 4, 031019 (2014)

    2014

  67. [77]

    Vojta, Rep

    M. Vojta, Rep. Prog. Phys. 66, 2069 (2003)

    2003

  68. [78]

    Quantum phase transitions,

    S. Sachdev, “Quantum phase transitions,” in Handbook of Magnetism and Advanced Mag- netic Materials (American Cancer Society, 2007) https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470022184.hmm108

    2007 doi

  69. [79]

    Coldea, D

    R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzyn- ska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177 (2010)

    2010

  70. [80]

    Simon, W

    J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature 472, 307 (2011)

    2011

  71. [81]

    Labuhn, D

    H. Labuhn, D. Barredo, S. Ravets, S. De L´ es´ eleuc, T. Macr` ı, T. Lahaye, and A. Browaeys, Nature 534, 667 (2016)

    2016

  72. [82]

    Guardado-Sanchez, P

    E. Guardado-Sanchez, P. T. Brown, D. Mitra, T. De- vakul, D. A. Huse, P. Schauß, and W. S. Bakr, Phys. Rev. X 8, 021069 (2018)

    2018

  73. [83]

    Friedenauer, H

    A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Sch¨ atz, Nat. Phys.4, 757 (2008)

    2008

  74. [84]

    Kim, M.-S

    K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Ed- wards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, Nature 465, 590 (2010)

    2010

  75. [85]

    K. Kim, S. Korenblit, R. Islam, E. Edwards, M. Chang, C. Noh, H. Carmichael, G. Lin, L. Duan, C. J. Wang, et al. , New J. Phys. 13, 105003 (2011)

    2011

  76. [86]

    B. P. Lanyon, C. Hempel, D. Nigg, M. M¨ uller, R. Ger- ritsma, F. Z¨ ahringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, et al. , Science 334, 57 (2011)

    2011

  77. [87]

    J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Nature 484, 489 (2012)

    2012

  78. [88]

    Barends, A

    R. Barends, A. Shabani, L. Lamata, J. Kelly, A. Mezza- capo, U. Las Heras, R. Babbush, A. G. Fowler, B. Camp- bell, Y. Chen, et al. , Nature 534, 222 (2016)

    2016

  79. [89]

    Harris, Y

    R. Harris, Y. Sato, A. Berkley, M. Reis, F. Altomare, 7 M. Amin, K. Boothby, P. Bunyk, C. Deng, C. Enderud, et al. , Science 361, 162 (2018)

    2018

  80. [90]

    Dziarmaga, Phys

    J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005)

    2005

  81. [91]

    V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991)

    1991

  82. [92]

    Esslinger, Annu

    T. Esslinger, Annu. Rev. Condens. Matter Phys. 1, 129 (2010)

    2010

  83. [93]

    M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif, and M. Greiner, Science 353, 1253 (2016)

    2016

  84. [94]

    M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Science 353, 1257 (2016)

    2016

  85. [95]

    Mazurenko, C

    A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kan´ asz-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, Nature 545, 462 (2017)

    2017

  86. [96]

    P. T. Brown, D. Mitra, E. Guardado-Sanchez, R. Nourafkan, A. Reymbaut, C.-D. H´ ebert, S. Bergeron, A.-M. Tremblay, J. Kokalj, D. A. Huse, et al. , Science 363, 379 (2019)

    2019

  87. [97]

    Schl¨ unzen, J.-P

    N. Schl¨ unzen, J.-P. Joost, F. Heidrich-Meisner, and M. Bonitz, Phys. Rev. B 95, 165139 (2017)

    2017

  88. [98]

    Pertot, A

    D. Pertot, A. Sheikhan, E. Cocchi, L. A. Miller, J. E. Bohn, M. Koschorreck, M. K¨ ohl, and C. Kollath, Phys. Rev. Lett. 113, 170403 (2014)

    2014

  89. [99]

    Bounding the FSE of quantum many-body dynamics simulations

    N. Schuch, S. K. Harrison, T. J. Osborne, and J. Eisert, Phys. Rev. A 84, 032309 (2011) 8 Supplemental Material for “Bounding the FSE of quantum many-body dynamics simulations” Zhiyuan Wang, Michael Foss-Feig, and Kaden R. A. Hazzard The supplemental material fills in technical...

    2011

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.