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REVIEW 3 major objections 5 minor 57 references

Clifford-Dressed Variational Principles for Precise Loschmidt Echoes

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper claims that Clifford-dressed TDVP extends the reliable time horizon of Loschmidt-echo simulations by roughly one to two time units at fixed bond dimension, by reducing the echo to an overlap between a stabilizer state and a…

desk verdict A genuinely new estimator for MPS–stabilizer overlaps, applied to Loschmidt echoes, but the accuracy claim needs an exact baseline before it becomes convincing. read the letter →

arxiv 2502.01872 v1 pith:XFTQUPKH submitted 2025-02-03 quant-ph

Antonio Francesco Mello , Alessandro Santini , Mario Collura This is my paper · ORCID
classification quant-ph
keywords Clifford-dressedTDVPLoschmidtechomatrixproductstatesstabilizerentanglementcontroltime-dependentvariationalprincipleoverlapestimationIsingmodeldynamics
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 is trying to establish that the Clifford-dressed time-dependent variational principle (TDVP) — a time-evolution method that applies classically simulable Clifford circuits to slow down entanglement growth in a matrix product state (MPS) — can compute Loschmidt echoes, the squared overlap between an evolved state and its initial state, and not just expectation values. The essential reduction is that when the initial state is a stabilizer state (a state generated from $|0\rangle^{\otimes N}$ by Clifford circuits), the Clifford gates used to dress the evolution can be pulled back onto that initial state, so the Loschmidt amplitude becomes the overlap between a stabilizer state and an MPS. The authors give two ways to evaluate that overlap: a stochastic estimator that samples configurations from the stabilizer state with variance $1-|\langle s|\phi\rangle|^2$, independent of system size, and a deterministic projection onto the stabilizer group represented by bond-dimension-two controlled-Pauli operators. On the critical one-dimensional transverse-field Ising model and a non-integrable next-to-nearest-neighbor Ising chain, the dressed method keeps the echo accurate for about two time units longer than bare TDVP at the same bond dimension, and a $5\times5$ two-dimensional Ising model gains about one time unit in the ordered phase. If correct, this extends classical simulation of quench dynamics into regimes where entanglement growth would otherwise force a larger bond dimension.

What carries the argument

The central object is the Clifford-enhanced MPS: an MPS dressed by a sequence of two-site Clifford gates chosen to lower its entanglement, together with the two stabilizer-state fidelity estimators. The key identity is the rewriting of the Loschmidt amplitude as $\langle s_m|\phi_m\rangle$, because the Clifford gates can be pulled back onto the initial stabilizer state at the cost of updating its tableau, while the MPS evolves at a controlled bond dimension. The stochastic estimator uses the identity $\langle s|\phi\rangle = \mathbb{E}_{x\sim|\langle x|s\rangle|^2}[\langle x|\phi\rangle/\langle x|s\rangle]$ and achieves variance $1-|\langle s|\phi\rangle|^2$, independent of system size. The projective estimator uses the stabilizer-group projector $\prod_j (1+g_j)/2$, where each factor is a bond-dimension-two MPO with a controlled-Pauli form, applied to the MPS followed by truncation.

What would settle it

For a small lattice, say a $4\times4$ system with $N=16$, obtain the exact Loschmidt echo by exact diagonalization of the Ising Hamiltonian and compare $L(t)$ with the Clifford-dressed TDVP result and with bare TDVP at the same bond dimension. If the dressed result departs from the exact curve at the same time as bare TDVP, the claimed extension in time reach is an artifact of the benchmark scale rather than of the disentangling routine.

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Extended reading notes

Core claim

Within Clifford-dressed TDVP, the evolved state is represented as a Clifford-enhanced MPS (CMPS) with fixed maximum bond dimension. Because the dynamics is generated by dressing the Hamiltonian with two-site Clifford gates chosen to reduce entanglement, and because the initial state is a stabilizer state, the Loschmidt amplitude at evolution step $m$ reduces to $\langle s_m|\phi_m\rangle$, where $|s_m\rangle$ is a stabilizer state obtained by composing all Clifford gates on the initial state and $|\phi_m\rangle$ is the low-entanglement MPS. The stabilizer group of $|s_m\rangle$ is given by $N$ Pauli generators, so the overlap can be computed by importance sampling over the Born distribution of the stabilizer state with per-sample cost $O(N\chi^2)$ and variance $1-|\langle s|\phi\rangle|^2$, or by inserting the projector $\prod_j (1+g_j)/2$ as a network of bond-dimension-two controlled-Pauli MPOs and contracting with truncation. The paper reports that both methods outperform standard TDVP at fixed $\chi=32$ on the critical one-dimensional Ising model, and that the projective method shows similar or better accuracy on the next-to-nearest-neighbor chain and on the two-dimensional lattice, matching a higher-bond-dimension TDVP reference for a longer simulation time.

Load-bearing premise

The load-bearing premise is that the Clifford disentangling circuits chosen by the heuristic routine do not bias the overlap: the truncated dressed MPS still reproduces the true overlap with the dressed initial state, with no error bound and with validation only against higher-bond-dimension TDVP.

Editorial extensions

If this is right

  • At fixed bond dimension $\chi=32$, Clifford-dressed TDVP produces Loschmidt echoes on the critical one-dimensional Ising chain and the next-to-nearest-neighbor chain that follow the $\chi=256$ standard-TDVP reference for about two time units longer than bare TDVP at the same $\chi$.
  • The variance identity $\mathrm{Var}(S)=1-|\langle s|\phi\rangle|^2$ means the stochastic estimator's fluctuation does not grow with the number of qubits, making the cost of the overlap evaluation predictable as $N$ increases.
  • The projective estimator remains accurate even when the bond dimension used during the stabilizer-group projection $\chi_P$ is smaller than the evolution bond dimension $\chi$, reducing the cost of the final contraction.
  • The protocol applies to every initial state of the form $C|0\rangle^{\otimes N}$ with $C$ Clifford, which includes the usual product-state quench initial states used in Loschmidt-echo studies.
  • Clifford-dressed TDVP is thereby extended from observables to fidelities, making quantities that are direct functions of the Loschmidt amplitude, such as work statistics and dynamical phase transitions, accessible under dressed evolution.

Reading between the lines

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

  • The paper does not test the stochastic estimator on the next-to-nearest-neighbor or two-dimensional models, but because its variance is bounded independently of $N$, it should remain competitive there; this is an extrapolation beyond the benchmarks shown.
  • The tolerance of a smaller projection bond dimension $\chi_P$ suggests the stabilizer-group projection can be read as a variational truncation; a systematic study of the error versus $\chi_P$ would show whether this robustness is generic or specific to the ordered-phase two-dimensional case.
  • The same stabilizer-state–MPS overlap estimator applies beyond Hamiltonian dynamics: combined with computational-basis sampling of Clifford-augmented MPS, it gives a route to state fidelities for circuit evolution, connecting to the classical-simulation program for Clifford+T circuits.
  • Because the paper plots the rescaled echo $(1/N)\log L(t)$ but does not extract rate-function zeros, a natural next step would be to use the dressed evolution to probe dynamical phase transitions in two-dimensional models where exact results are absent.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript extends the Clifford-dressed TDVP (C-TDVP) framework to the evaluation of Loschmidt echoes. For a stabilizer initial state, the evolved C-TDVP state is a Clifford-augmented MPS, and the Loschmidt amplitude reduces to an overlap between a stabilizer state and an MPS. Two estimators are proposed: a Monte Carlo estimator based on perfect sampling from the stabilizer state, and a projective estimator built from stabilizer-group projectors represented as MPOs. The methods are benchmarked on the 1D transverse-field Ising chain at criticality and applied to a non-integrable next-to-nearest-neighbor Ising chain and a 5x5 2D Ising model. The central claim is that Clifford disentangling extends the accurate time reach at fixed bond dimension, with reported gains of about two time units in 1D and about one time unit in the 2D ordered phase.

Significance. If the central claim holds, the paper is a useful methodological advance: it extends the applicability of Clifford-augmented tensor networks from expectation values to a non-local observable, the Loschmidt amplitude, which is relevant for dynamical phase transitions, work statistics, and decoherence studies. The overlap identities in Eqs. (1) and (3)-(4) are mathematically sound, the stabilizer-state sampling is standard, and the comparison against plain TDVP at matched bond dimension is an appropriate baseline. The authors correctly attribute the Clifford-dressing heuristic to their earlier work [19] and note the overlap with the independent work of Liu and Clark [24]. The main weakness is evidential: the accuracy claim is anchored only to higher-bond-dimension TDVP rather than to an exact reference, and the two estimators have convergence properties that are not controlled in the regime where Loschmidt echoes are exponentially small.

major comments (3)
  1. [Fig. 2 and 'Loschmidt Echo with CMPS'] The central two-time-unit advantage is not anchored to an exact value. The 1D transverse-field Ising chain at h=hc=1 is free-fermion integrable, so exact N=20 Loschmidt amplitudes are obtainable at negligible cost via Jordan-Wigner. Without this comparison, Fig. 2 only demonstrates that C-TDVP with chi=32 is closer to TDVP with chi=256 than plain TDVP with chi=32 is; both approximations could share a common truncation bias, so the stated advantage does not by itself establish accuracy. The same caveat applies to Figs. 3 and 4, where the only reference is higher-chi TDVP.
  2. [Eqs. (1)-(2) and Fig. 2] The variance bound Var(S)=1-|⟨s|phi⟩|^2 is correct, but it bounds the absolute error of a single sample. For the sample-mean estimator with M samples the standard deviation is sqrt((1-|⟨s|phi⟩|^2)/M) ≈ 1/sqrt(M), independent of the overlap; hence resolving L(t)=|⟨s|phi⟩|^2 when it is exponentially small requires M ~ 1/L(t). With the stated 10^4 samples the stochastic estimator is adequate only while the overlap is not much smaller than 10^-2, and no sample-size convergence or error bars are shown. This is a load-bearing limitation for method 1 in the regime the paper targets.
  3. [Framework, method 2, and Figs. 2-4] The projective estimator is uncontrolled. Applying the product of stabilizer-group projectors, Eq. (3), requires repeated truncation of the MPS bond dimension, and the resulting error depends on the chosen maximum projector bond dimension chi_P. The paper states that 'both methods introduce a certain degree of approximation' but provides no chi_P-convergence data and no estimate of the truncation error. Since the plotted results use chi_P = 32 or 64 in different cases, the reader cannot judge whether the projected overlap is converged or whether the observed time reach is limited by the projection step.
minor comments (5)
  1. [Framework, method 2] The sentence 'calculating ⟨x|s⟩ ⟨s|phi⟩ and then dividing by the amplitude ⟨x|s⟩' is circular as written; it should read that one evaluates ⟨x|phi'⟩ for the projected MPS |phi'⟩=|s⟩⟨s|phi⟩ and divides by ⟨x|s⟩.
  2. [Eq. (2)] The notation |S|^2 is used for both E|S|^2 and |E S|^2, which is confusing; using distinct symbols for the estimator and its mean would clarify the variance derivation.
  3. [Eq. (5)] The definition of the controlled-Pauli tensor has typographical issues: the boundary vector vj is written as (1/sqrt(2), sqrt(theta_j/2))^T, which should likely be (1/sqrt(2), sqrt(theta_j/2))^T with the sign of theta_j carried separately, and the index structure of G_alpha beta is not fully defined.
  4. [Figs. 2-4] The y-axis label '1/N log (t)' is incomplete; it should indicate the rescaled Loschmidt echo, e.g., (1/N) log L(t), and the legends should spell out that chi is the TDVP bond dimension and chi_P the projection bond dimension in all panels, as is already done in some but not all figures.
  5. [Introduction] There are several grammatical and typographical issues in the introduction, e.g., 'recently it was proposed a novel technique' and inconsistent citation formatting; a careful proofread would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: overlap estimators derive from definitions and the numerical claims are benchmarked against standard TDVP, not fitted or self-referential inputs.

full rationale

The manuscript's derivation of the two overlap estimators is self-contained. Eq. (1) is an exact importance-sampling identity over the stabilizer-state Born distribution, and the stated variance property follows from the normalization of both states rather than from any fitted parameter. Method 2 uses the projector identity |s><s| = prod_j P_j, which is the definition of a stabilizer-state projector, and the MPO representation in Eq. (4) is a controlled-Pauli tensor of bond dimension 2; no target Loschmidt amplitude is fed back into the construction. The reduction of the Loschmidt amplitude to <s_m|phi_m> is the definition of the Clifford-dressed TDVP state and the dressed stabilizer state s_m, so the relation is not assumed in order to prove itself. The benchmark comparisons in Figs. 2-4 use plain TDVP, including a larger-bond-dimension reference, as an external baseline; no parameter is fitted to Loschmidt-echo data and then renamed as a prediction. The reliance on the Clifford-search heuristic of Ref. [19] by the same group is proper attribution of a numerical tool rather than a circular import of the paper's central claim, because the new contribution is the stabilizer-MPS overlap evaluation, which is derived explicitly from Pauli and MPS definitions. Even though the heuristic lacks a rigorous error bound, that absence is an accuracy or correctness limitation, not circularity. No self-definitional step, fitted-input-called-prediction step, or author-imported uniqueness step was found.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

Free parameters are standard numerical settings (dt, chi, chi_P, sample count) that affect the quantitative results but are not fitted to make a specific prediction come true. The axioms are standard stabilizer and tensor-network facts plus two heuristic domain assumptions inherited from the parent C-TDVP algorithm. No new physical entities are introduced.

free parameters (4)
  • time step dt = 0.1 (0.05 for 2D)
    Discretization step for TDVP evolution, chosen by hand; the comparison of methods is at a fixed dt.
  • bond dimension chi = 32 (1D benchmarks), 64 (2D)
    Truncation bond dimension for the TDVP update; the central comparison fixes chi between methods.
  • projection bond dimension chi_P = 32, 64 in figures
    Maximum bond dimension during stabilizer-group projection; the paper shows accuracy is retained with chi_P smaller than evolution chi.
  • number of Monte Carlo samples = 10^4
    Number of samples for the stochastic estimator (method 1) in Fig. 2; error bars are not shown.
assumptions (4)
  • standard math Stabilizer states are classically simulable via the Gottesman-Knill theorem.
    Invoked in the Framework section to justify the tableaux update of the initial state under Clifford gates.
  • standard math Any stabilizer state projector can be written as the product of single-generator projectors with bond-dimension-2 MPO representations.
    Used in Eqs. (3)-(5) and Fig. 1 for Method 2; this follows from the stabilizer formalism.
  • domain assumption The Clifford disentangling circuit found by the algorithm of ref [19] reduces the entanglement of the evolved state without biasing the quantities of interest.
    The entire C-TDVP accuracy relies on this heuristic; no error bound is given in this paper.
  • domain assumption TDVP truncated evolution on the dressed MPS remains a faithful approximation of the true time evolution for the times studied.
    The benchmarks compare against higher-bond-dimension TDVP rather than exact results, so the validity of the approximation is assumed.

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Pith. "Pith review of Clifford-Dressed Variational Principles for Precise Loschmidt Echoes." pith.science (2026). https://pith.science/paper/XFTQUPKH

@misc{pith2026250201872,
  author       = {Pith},
  title        = {Pith review of: Clifford-Dressed Variational Principles for Precise Loschmidt Echoes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XFTQUPKH}},
  note         = {Machine review of arXiv:2502.01872}
}
read the original abstract

We extend the recently introduced Clifford dressed Time-Dependent Variational Principle (TDVP) to efficiently compute many-body wavefunction amplitudes in the computational basis. This advancement enhances the study of Loschmidt echoes, which generally require accurate calculations of the overlap between the evolved state and the initial wavefunction. By incorporating Clifford disentangling gates during TDVP evolution, our method effectively controls entanglement growth while keeping the computation of these amplitudes accessible. Specifically, it reduces the problem to evaluating the overlap between a Matrix Product State (MPS) and a stabilizer state, a task that remains computationally feasible within the proposed framework. To demonstrate the effectiveness of this approach, we first benchmark it on the one-dimensional transverse-field Ising model. We then apply it to more challenging scenarios, including a non-integrable next-to-nearest-neighbor Ising chain and a two-dimensional Ising model. Our results highlight the versatility and efficiency of the Clifford-augmented MPS, showcasing its capability to go beyond the evaluation of simple expectation values. This makes it a powerful tool for exploring various aspects of many-body quantum dynamics.

Figures

Figures reproduced from arXiv: 2502.01872 by the authors.

Figure 1
Figure 1. FIG. 1. Tensor network representation of the projection of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Rescaled Loschmidt echo for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Rescaled Loschmidt echo for a 2D system of size [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Rescaled Loschmidt echo for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Maximum entanglement entropy for a 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Reference graph

Works this paper leans on

57 extracted references · 26 canonical work pages

  1. [24]

    Liu and B

    Z. Liu and B. K. Clark, Classical simulability of clifford+t circuits with clifford-augmented matrix product states (2024), arXiv:2412.17209 [quant-ph]

    arXiv 2024

  2. [19]

    A. F. Mello, A. Santini, G. Lami, J. De Nardis, and M. Collura, Clifford dressed time-dependent variational principle, arXiv preprint arXiv:2407.01692 (2024)

    arXiv 2024

  3. [1]

    Alba and P

    V. Alba and P. Calabrese, Entanglement dynamics after quantum quenches in generic integrable systems, SciPost Phys. 4, 017 (2018)

    2018

  4. [2]

    Alba and P

    V. Alba and P. Calabrese, Entanglement and thermody- namics after a quantum quench in integrable systems, Proceedings of the National Academy of Sciences 114, 7947 (2016)

    work page 2016

  5. [3]

    Calabrese and J

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

    work page 2005

  6. [4]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entan- glement in many-body systems, Rev. Mod. Phys. 80, 517 (2008)

    2008

  7. [5]

    R. P. Feynman, Simulating physics with computers, Inter- national Journal of Theoretical Physics 21, 467 (1982). 6

    work page 1982

  8. [6]

    A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature 607, 667 (2022)

    2022

Show all 57 references
  1. [7]

    X. Xu, S. Benjamin, J. Sun, X. Yuan, and P. Zhang, A her- culean task: Classical simulation of quantum computers (2023), arXiv:2302.08880 [quant-ph]

    2023 arXiv

  2. [8]

    Pashayan, S

    H. Pashayan, S. D. Bartlett, and D. Gross, From estima- tion of quantum probabilities to simulation of quantum circuits, Quantum 4, 223 (2020)

    2020

  3. [9]

    Paeckel, T

    S. Paeckel, T. K¨ ohler, A. Swoboda, S. R. Manmana, U. Schollw¨ ock, and C. Hubig, Time-evolution methods for matrix-product states, Annals of Physics 411, 167998 (2019)

    2019

  4. [10]

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

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

    2011

  5. [11]

    Vidal, Efficient Simulation of One-Dimensional Quan- tum Many-Body Systems, Phys

    G. Vidal, Efficient Simulation of One-Dimensional Quan- tum Many-Body Systems, Phys. Rev. Lett. 93, 040502 (2004)

    2004

  6. [12]

    Collura, G

    M. Collura, G. Lami, N. Ranabhat, and A. Santini,Tensor Network Techniques for Quantum Computation (SISSA Medialab S.r.l., 2024)

    2024

  7. [13]

    Haegeman, C

    J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimiza- tion with matrix product states, Phys. Rev. B 94, 165116 (2016)

    2016

  8. [14]

    Haegeman, J

    J. Haegeman, J. I. Cirac, T. J. Osborne, I. Piˇ zorn, H. Ver- schelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett. 107, 070601 (2011)

    2011

  9. [15]

    Biamonte, Lectures on Quantum Tensor Networks (2020), arXiv:1912.10049 [quant-ph]

    J. Biamonte, Lectures on Quantum Tensor Networks (2020), arXiv:1912.10049 [quant-ph]

    2020 arXiv

  10. [16]

    Silvi, F

    P. Silvi, F. Tschirsich, M. Gerster, J. J¨ unemann, D. Jaschke, M. Rizzi, and S. Montangero, The Tensor Networks Anthology: Simulation techniques for many- body quantum lattice systems, SciPost Physics Lecture Notes 10.21468/scipostphyslectnotes.8 (2019)

    2019 doi

  11. [17]

    A.-K. Wu, M. T. Fishman, J. H. Pixley, and E. M. Stoudenmire, Disentangling interacting systems with fermionic gaussian circuits: Application to the single impurity anderson model (2022), arXiv:2212.09798 [cond- mat.str-el]

    2022 arXiv

  12. [18]

    X. Qian, J. Huang, and M. Qin, Clifford circuits augmented time-dependent variational principle, arXiv preprint arXiv:2407.03202 (2024)

    2024 arXiv

  13. [20]

    Dehaene and B

    J. Dehaene and B. De Moor, Clifford group, stabilizer states, and linear and quadratic operations over GF(2), Physical Review A 68, 10.1103/physreva.68.042318 (2003)

    2003 doi

  14. [21]

    Gottesman, Theory of fault-tolerant quantum compu- tation, Phys

    D. Gottesman, Theory of fault-tolerant quantum compu- tation, Phys. Rev. A 57, 127 (1998)

    1998

  15. [22]

    Gottesman, The Heisenberg Representation of Quantum Computers, arXiv preprint arXiv:9807006 10.48550/ARXIV.QUANT-PH/9807006 (1998)

    D. Gottesman, The Heisenberg Representation of Quantum Computers, arXiv preprint arXiv:9807006 10.48550/ARXIV.QUANT-PH/9807006 (1998)

    1998 doi

  16. [23]

    Gottesman, Stabilizer Codes and Quantum Error Cor- rection (1997)

    D. Gottesman, Stabilizer Codes and Quantum Error Cor- rection (1997)

    1997

  17. [25]

    Macr ` ı, A

    T. Macr ` ı, A. Smerzi, and L. Pezz` e, Loschmidt echo for quantum metrology, Phys. Rev. A 94, 010102 (2016)

    2016

  18. [26]

    ˇZunkoviˇ c, A

    B. ˇZunkoviˇ c, A. Silva, and M. Fabrizio, Dynam- ical phase transitions and loschmidt echo in the infinite-range xy model, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374, 20150160 (2016), https://royalsocietypublishing.o...

    2016

  19. [27]

    M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field ising model, Phys. Rev. Lett. 110, 135704 (2013)

    2013

  20. [28]

    Jurcevic, H

    P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos, Direct observation of dynamical quantum phase transitions in an interacting many-body system, Phys. Rev. Lett. 119, 080501 (2017)

    2017

  21. [29]

    ˇZunkoviˇ c, M

    B. ˇZunkoviˇ c, M. Heyl, M. Knap, and A. Silva, Dynamical quantum phase transitions in spin chains with long-range interactions: Merging different concepts of nonequilibrium criticality, Phys. Rev. Lett. 120, 130601 (2018)

    2018

  22. [30]

    Peotta, F

    S. Peotta, F. Brange, A. Deger, T. Ojanen, and C. Flindt, Determination of dynamical quantum phase transitions in strongly correlated many-body systems using loschmidt cumulants, Phys. Rev. X 11, 041018 (2021)

    2021

  23. [31]

    Heyl, Dynamical quantum phase transitions: a review, Reports on Progress in Physics 81, 054001 (2018)

    M. Heyl, Dynamical quantum phase transitions: a review, Reports on Progress in Physics 81, 054001 (2018)

    2018

  24. [32]

    Karch, S

    S. Karch, S. Bandyopadhyay, Z.-H. Sun, A. Impertro, S. Huh, I. P. Rodr ´ ıguez, J. F. Wienand, W. Ketterle, M. Heyl, A. Polkovnikov, I. Bloch, and M. Aidelsburger, Probing quantum many-body dynamics using subsys- tem loschmidt echos (2025), arXiv:2501.16995 [cond- mat.quant-gas]

    2025 arXiv

  25. [33]

    Mazzola, G

    L. Mazzola, G. De Chiara, and M. Paternostro, Measuring the characteristic function of the work distribution, Phys. Rev. Lett. 110, 230602 (2013)

    2013

  26. [34]

    Dorner, S

    R. Dorner, S. R. Clark, L. Heaney, R. Fazio, J. Goold, and V. Vedral, Extracting quantum work statistics and fluctuation theorems by single-qubit interferometry, Phys. Rev. Lett. 110, 230601 (2013)

    2013

  27. [35]

    F. M. Cucchietti, D. A. Dalvit, J. P. Paz, and W. H. Zurek, Decoherence and the loschmidt echo, Physical review letters 91, 210403 (2003)

    2003

  28. [36]

    Haikka, J

    P. Haikka, J. Goold, S. McEndoo, F. Plastina, and S. Man- iscalco, Non-markovianity, loschmidt echo, and criticality: A unified picture, Phys. Rev. A 85, 060101 (2012)

    2012

  29. [37]

    Piroli, B

    L. Piroli, B. Pozsgay, and E. Vernier, From the quan- tum transfer matrix to the quench action: the loschmidt echo in xxz heisenberg spin chains, Journal of Statistical Mechanics: Theory and Experiment 2017, 023106 (2017)

    2017

  30. [38]

    Gorin, T

    T. Gorin, T. Prosen, T. H. Seligman, and M. ˇZnidariˇ c, Dynamics of loschmidt echoes and fidelity decay, Physics Reports 435, 33 (2006)

    2006

  31. [39]

    B. Yan, L. Cincio, and W. H. Zurek, Information scram- bling and loschmidt echo, Phys. Rev. Lett. 124, 160603 (2020)

    2020

  32. [40]

    H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Decay of loschmidt echo enhanced by quantum criticality, Phys. Rev. Lett. 96, 140604 (2006)

    2006

  33. [41]

    H´ emery, K

    K. H´ emery, K. Ghanem, E. Crane, S. L. Campbell, J. M. Dreiling, C. Figgatt, C. Foltz, J. P. Gaebler, J. Johansen, M. Mills, S. A. Moses, J. M. Pino, A. Ransford, M. Rowe, P. Siegfried, R. P. Stutz, H. Dreyer, A. Schuckert, and R. Nigmatullin, Measuring the loschmidt amplitud...

    2024

  34. [42]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005)

    2005

  35. [43]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

    2010

  36. [44]

    A. Y. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and Quantum Computation , Graduate studies in mathematics (American Mathematical Society, USA, 2002)

    2002

  37. [45]

    G. Lami, T. Haug, and J. De Nardis, Quantum state de- signs with clifford enhanced matrix product states, arXiv preprint arXiv:2404.18751 (2024)

    2024 arXiv

  38. [46]

    X. Qian, J. Huang, and M. Qin, Augmenting density matrix renormalization group with clifford circuits, arXiv preprint arXiv:2405.09217 (2024)

    2024 arXiv

  39. [47]

    Masot-Llima and A

    S. Masot-Llima and A. Garcia-Saez, Stabilizer tensor networks: universal quantum simulator on a basis of stabilizer states (2024), arXiv:2403.08724 [quant-ph]

    2024 arXiv

  40. [48]

    A. F. Mello, A. Santini, and M. Collura, Hybrid stabilizer matrix product operator, Physical Review Letters 133, 150604 (2024)

    2024

  41. [49]

    Huang, X

    J. Huang, X. Qian, and M. Qin, Clifford circuits aug- mented matrix product states for fermion systems (2024), arXiv:2501.00413 [cond-mat.str-el]

    2024 arXiv

  42. [51]

    G. E. Fux, B. B´ eri, R. Fazio, and E. Tirrito, Disentan- gling unitary dynamics with classically simulable quantum circuits (2024), arXiv:2410.09001 [quant-ph]

    2024

  43. [52]

    S. R. White, Density-matrix algorithms for quantum renormalization groups, Physical review b 48, 10345 (1993)

    1993

  44. [53]

    X. Qian, J. Huang, and M. Qin, Augmenting finite temper- ature tensor network with clifford circuits, arXiv preprint arXiv:2410.15709 (2024)

    2024 arXiv

  45. [54]

    E. M. Stoudenmire and S. R. White, Minimally entangled typical thermal state algorithms, New Journal of Physics 12, 055026 (2010)

    2010

  46. [55]

    Aaronson and D

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

    2004

  47. [56]

    ManyQLowD

    for the stabilzer formalism operations and ITensor for the TN simulations [ 57]. We are particularly grateful to G. Lami, J. De Nardis, M. Dalmonte, E. Tirrito for collab- orations on topics connected with this work. This work was supported by the PNRR MUR project PE0000023- N...

    2022

  48. [57]

    Gidney, Stim: a fast stabilizer circuit simulator, Quan- tum 5, 497 (2021)

    C. Gidney, Stim: a fast stabilizer circuit simulator, Quan- tum 5, 497 (2021)

    2021

  49. [58]

    Fishman, S

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

    2022

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