Pith. sign in

REVIEW 2 major objections 5 minor 3 cited by

Quantum Utility-Scale Error Mitigation for Quantum Quench Dynamics in Heisenberg Spin Chains

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read A single noise correction from a forward-backward test circuit keeps 104-qubit spin-chain simulations accurate and beats zero-noise extrapolation.

desk verdict Useful SM validation at N=20 and a plausible scaling study, but the large-scale ZNE comparison reuses the N=20 fit and the p-transfer assumption is only checked at small size. read the letter →

arxiv 2506.20125 v1 pith:AENC7SIS submitted 2025-06-25 quant-ph

Seokwon Choi , Talal Ahmed Chowdhury , Kwangmin Yu This is my paper · ORCID
classification quant-ph
keywords quantumerrormitigationself-mitigationzero-noiseextrapolationTrotterizationHeisenbergXXZspinchainquenchdynamicsstaggeredmagnetizationentanglemententropy
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

The paper tries to establish that a normalization-based error-mitigation scheme, called self-mitigation, keeps time evolution on today's noisy superconducting hardware accurate even when the simulation is Trotterized into thousands of two-qubit gates and run on 104 qubits. The scheme runs a companion test circuit that performs half of the Trotter steps forward and half backward, so its ideal output is the known initial state; the decay of the expectation value in that test circuit gives a depolarizing noise factor $p$, and the target circuit is corrected through $\langle O\rangle = \langle O\rangle/(1-p)$. For quench dynamics of the Heisenberg XXZ spin chain, the paper reports mean absolute errors in staggered magnetization around 0.026 at 104 qubits with self-mitigation versus around 0.049 with zero-noise extrapolation, and errors that stay at 20-qubit levels rather than growing. A sympathetic reader would care because it points to an extrapolation-free way to get utility-scale accuracy for time-dependent many-body problems before fault-tolerant quantum computers arrive.

What carries the argument

The load-bearing object is the self-mitigation test circuit: the same optimized second-order Trotter circuit as the target, but with the first half of the steps executed with $+dt$ and the second half with $-dt$. Because forward evolution followed by the exact reverse evolution returns the initial state, the ideal expectation value of any observable in the test circuit is known in advance; the ratio between the measured noisy value and the known ideal value yields the depolarizing noise factor $p$, and the target circuit's noisy expectation value is divided by $1-p$. The paper chooses the time sequence $(dt, dt, \dots, -dt, -dt)$ rather than alternating signs so that only the middle layer of $U_j(\vec 0)$ identity gates drops out, keeping the test circuit structurally near the target. The work this machinery does is to replace extrapolation in noise strength with a single multiplicative correction; the whole argument rests on the test circuit and target circuit suffering the same noise factor.

What would settle it

At a system size where the ideal output of the target circuit is known at early times, for example the 20-qubit case with exact simulation, extract the depolarizing noise factor of the test circuit and of the target circuit separately, either by randomized benchmarking on the same gate layers or by comparing corrected early-time values, and check whether they agree within the reported error bars; a disagreement larger than the error bars would mean the $1/(1-p)$ correction is systematically biasing the staggered-magnetization results.

Watch

Extended reading notes

Core claim

On its own terms, the discovery is that a depolarizing-noise correction factor measured from a structurally similar test circuit stays valid as the system grows. For the XXZ model with open and periodic boundary conditions, the authors stack readout twirling, dynamical decoupling, and Pauli twirling, then add self-mitigation: a test circuit with the time sequence $(dt, dt, \dots, -dt, -dt)$ whose ideal final state is the initial Néel state. The measured expectation value in that test circuit gives $p$, and the target circuit is corrected by $1/(1-p)$. Across $N=20$, $N=84$ (periodic), and $N=104$ (open) qubits, the corrected staggered-magnetization curves track the classical tensor-network benchmark closely; mean absolute errors are 0.02760 (20, open), 0.02832 (84, periodic), and 0.02556 (104, open), versus 0.04106, 0.05044, and 0.04917 for zero-noise extrapolation. The claim is that self-mitigation's circuit-aware normalization captures the dominant noise without extrapolation, while ZNE's fitted curve, derived at 20 qubits, degrades at larger scale.

Load-bearing premise

The load-bearing premise is that the noise measured in the forward-backward test circuit is the same as the noise in the actual forward simulation, even though the test circuit leaves out a whole layer of identity gates; the paper states this as an assumption and checks it only at 20 qubits.

Editorial extensions

If this is right

  • Correcting by a single measured noise factor instead of extrapolating from amplified circuits keeps the mean absolute error in staggered magnetization near 0.026 at 104 qubits, essentially unchanged from the 20-qubit value.
  • Zero-noise extrapolation's error grows by about 20 percent from 20 to 104 qubits when its fitting curve is fixed at the small-system size, while self-mitigation does not require such a curve.
  • Self-mitigation adds exactly one test circuit per observable, whereas the zero-noise extrapolation schedule used here adds two noise-amplified circuits, so the scheme is more resource-efficient in both circuit count and shot overhead.
  • Combining the same mitigation stack with randomized measurement protocols and circuit parallelization yields Rényi entanglement-entropy estimates within a few percent of classical simulation, extending the mitigation benefit beyond simple observables.

Reading between the lines

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

  • Inference: the same forward-backward construction should transfer to other time-symmetric protocols, such as Loschmidt echoes, echo-type diagnostics, or out-of-time-order correlator measurements, where the ideal output is known; the transfer would still require the test/target noise-match assumption to hold.
  • Inference: because self-mitigation applies one global depolarizing factor, spatially non-uniform or correlated noise should reveal itself as a late-time systematic drift in the corrected curves; comparing self-mitigated results against independent benchmarks across circuit depths would quantify this sensitivity.
  • Inference: the method's applicability is limited to cases where the initial-state expectation value is known, so a practical protocol would use self-mitigation for Trotterized quench observables and fall back to zero-noise extrapolation for general circuits; the paper itself notes ZNE's broader applicability but not this hybrid.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a self-mitigation (SM) error mitigation method for Trotterized time evolution, extending the method of Ref. [12] to optimized second-order Trotterization, and applies it to quench dynamics of the Heisenberg XXZ spin chain on IBM quantum hardware. On N=20 systems, SM applied on top of TREX+DD+PT is compared with ZNE and the baseline, showing lower mean absolute errors in staggered magnetization. The method is then deployed at N=104 (OBC) and N=84 (PBC) with 3,246 and 2,646 CNOT gates, respectively, where the paper reports that SM maintains errors near the N=20 values while ZNE errors increase. The paper also demonstrates a randomized-measurement protocol for Rényi entropy on an IBM Heron processor.

Significance. If the central claim held, the paper would provide a valuable, resource-efficient QEM technique that avoids ZNE's extrapolation overhead and remains stable at utility-scale qubit counts. Strengths include the clean N=20 benchmark against exact simulation, the transparent reporting of 10 repetitions and 100,000 shots per circuit, and the validation of the MPS-TDVP reference at N=20. However, the large-scale comparison is compromised by an unfair ZNE baseline, and the SM noise-factor-transfer assumption is only validated at small size, so the headline claim of 'clearly outperforming both ZNE and the baseline' across all tested sizes is not yet established.

major comments (2)
  1. [§4.2, Figs. 11–13, Table A3] The claim that SM outperforms ZNE at N=104/84 is not supported by the reported data because the ZNE comparison is deliberately mismatched. The manuscript states 'For the ZNE, we apply the same extrapolation fitting curve used in the N=20 cases,' yet §3.4 argues that ZNE's assumption of a consistent noise model 'often breaks down when the number of qubits changes.' Reusing the N=20 curve at N=84/104 therefore tests ZNE under a known-failing condition, and the reported 43.9–48.0% error reductions are an artifact of this handicap. A fair comparison requires fitting the ZNE curve to measurements at each system size, or at least reporting both a re-fit ZNE and the transferred-curve ZNE. Without this, the headline conclusion that SM 'clearly outperforming both ZNE and the baseline' across all tested sizes is not established.
  2. [§3.5, Figs. 2–3] The self-mitigation correction factor 1/(1−p) rests on the assumption that the noise factor p measured on the test circuit equals that of the target circuit. The test circuit with the (dt, dt, −dt, −dt) sequence still has a vanishing center layer (the U_j(0) layer), so it is not gate-identical to the target; the paper acknowledges this only by assertion ('we assume the discrepancy ... is negligible'). The equality is checked empirically only at N=20. Because a size-dependent bias in p would propagate multiplicatively into every reported SM value, the N=20 check is not sufficient to justify the large-scale results. Please provide a quantitative test of the size dependence of p (e.g., compare p extracted at N=20 and N=104, or show that the reported errors are insensitive to a ±20% perturbation of p).
minor comments (5)
  1. [§4, hardware identification] The names 'ibm yonsei' and 'ibm marrakesh' should be formatted consistently (e.g., IBM Yonsei, IBM Marrakesh) and the processor generation should be stated uniformly for each device used.
  2. [Table A1 caption] The caption 'Averaged staggered magnetization of ten repetition' should read 'ten repetitions'; similar grammar issues appear in a few other places.
  3. [Figure 16] The axis labels in Figure 16 appear corrupted by unicode escape sequences (e.g., '/uni00000013'); the figure should be regenerated with correct font encoding so that 'Rényi Entropy, S(2)' is readable.
  4. [§5.2] The typo 'entanglment' should be corrected, and the number of shots per random-unitary circuit in the RM protocol should be stated explicitly.
  5. [Eq. (3)] The product notation in Eq. (3) is ambiguous: the second product is written as ∏_{n=M}^1, which is nonstandard; please clarify the ordering of factors.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the SM noise factor is measured from a separate test circuit, the target results are benchmarked against external exact/MPS simulations, and the self-citations are not load-bearing.

full rationale

The paper's central derivation is not circular. In Section 3.5 the SM correction is defined by <O> = <O>/(1-p), where p is inferred by running a test circuit whose ideal output is the known Néel-state expectation value; the target circuit's noisy expectation is then measured separately on hardware and scaled by this p. No equation defines the target expectation in terms of p alone, and no fitted parameter is later renamed a prediction. The target dynamics are validated against external benchmarks: exact QuSpin diagonalization at N=20 and MPS-TDVP at N=84/104. The admitted assumption that test-circuit and target-circuit noise factors coincide is an explicit empirical ansatz, not a circular reduction. The ZNE comparison at large scale reuses the N=20 fitting curve ('For the ZNE, we apply the same extrapolation fitting curve used in the N=20 cases,' Section 4.2), and the paper itself states that this assumption 'often breaks down when the number of qubits changes'; this is a disclosed benchmark-fairness caveat for the SM-vs-ZNE comparison, not a derivation of the SM result from its own inputs. Self-citations (Refs. [14,33,44,45]) provide circuit-optimization and QMP implementation details but do not carry the central claim; the SM concept is attributed to external Ref. [12] and the depolarizing-noise framework to external Ref. [46]. Overall, no self-referential reduction is load-bearing, so the circularity score is low.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The ledger is dominated by measured parameters and domain assumptions about noise and classical benchmarks. The method introduces no free parameters fit to the target observable: p is measured from a test circuit with known ideal output, and the ZNE curve is fit to small-system data. The largest unquantified risks are the equal-noise-factor assumption and the MPS benchmark fidelity at large sizes.

free parameters (4)
  • noise factor p (self-mitigation) = per-circuit measurement, not stated numerically; corrected expectation = measured/(1-p)
    Estimated from the SM test circuit whose ideal output is the known initial state; the central correction parameter of the method. It is measured rather than fit to the target observable, but its value directly determines the reported results.
  • ZNE extrapolation curve coefficients = not stated numerically; fitted at N=20 with fold factors 1, 3, 5 and reused for N=84/104
    The paper states the same extrapolation fitting curve from the 20-qubit case is used at large scale (Section 4.2); these coefficients determine the ZNE benchmark values and the comparison outcome.
  • MPS-TDVP truncation parameters = chi_max = 1000, epsilon = 1e-12, dt = 0.1
    Chosen maximum bond dimension and truncation error define the classical reference at N=84/104; no convergence check vs chi is shown at those sizes (Section 4).
  • Trotter time step dt = 0.5 (dimensionless, J1=1)
    Chosen after comparing first- and second-order Trotterization at N=20 (figure 4); all hardware results use it, and the second-order Trotter error at dt=0.5 sets the accuracy floor at late times.
assumptions (5)
  • standard math Trotter-Suzuki decomposition with dt=0.5 approximates e^{-iHt} within acceptable error for the observables.
    Used in Eqs. (3)-(4) and validated only at N=20 in figure 4; the Trotter error scales as O(t dt^2), and at N=104 the Trotter error is not separately quantified.
  • domain assumption Device noise acts as a uniform depolarizing channel with a single factor p per circuit.
    The SM correction (Section 3.5, citing Ref. [46]) assumes a depolarizing channel; spatially or temporally correlated noise would break the single-parameter correction.
  • domain assumption Test-circuit and target-circuit noise factors are equal.
    Stated in Section 3.5: 'we assume the discrepancy between the noise factors of the test circuit and the target circuit is negligible.' This is the load-bearing premise of the SM correction.
  • domain assumption MPS-TDVP with chi_max=1000 and epsilon=1e-12 reproduces the exact dynamics at N=84/104 up to t=5.
    Used as the large-system benchmark (Section 4); only validated against exact at N=20 (figure 4).
  • standard math The RM protocol estimator X = (1/N_U) sum_a X_a estimates the purity Tr(rho_A^2).
    Standard result from Refs. [63-69] assumed in Section 5.1; the estimator variance with N_U=60 is not reported.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Utility-Scale Error Mitigation for Quantum Quench Dynamics in Heisenberg Spin Chains." pith.science (2026). https://pith.science/paper/AENC7SIS

@misc{pith2026250620125,
  author       = {Pith},
  title        = {Pith review of: Quantum Utility-Scale Error Mitigation for Quantum Quench Dynamics in Heisenberg Spin Chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AENC7SIS}},
  note         = {Machine review of arXiv:2506.20125}
}
read the original abstract

We propose a quantum error mitigation method termed self-mitigation, which is comparable with zero-noise extrapolation, to achieve quantum utility on near-term, noisy quantum computers. We investigate the effectiveness of several quantum error mitigation strategies, including self-mitigation, by simulating quantum quench dynamics for Heisenberg spin chains with system sizes up to 104 qubits using IBM quantum processors. In particular, we discuss the limitations of zero-noise extrapolation and the advantages offered by self-mitigation at a large scale. The self-mitigation method shows stable accuracy with the large systems of 104 qubits with more than 3,000 CNOT gates. Also, we combine the discussed quantum error mitigation methods with practical entanglement entropy measuring methods, and it shows a good agreement with the theoretical estimation. Our study illustrates the usefulness of near-term noisy quantum hardware in examining the quantum quench dynamics of many-body systems at large scales, and lays the groundwork for surpassing classical simulations with quantum methods prior to the development of fault-tolerant quantum computers.

Figures

Figures reproduced from arXiv: 2506.20125 by the authors.

Figure 1
Figure 1. The optimized second-order Trotterization of the Hamiltonian for the XXZ spin chain model with open boundary conditions. The index j represents the first qubit index where Uj is placed. For a periodic boundary condition, the odd layers have the two-qubit gates, Uj ( ⃗θ), between qn−1 and q0. where the antiferromagnetic nearest-neighbor coupling J1 > 0 and the exchange￾anisotropy parameter ∆ controls the parameter sp… view at source ↗
Figure 2
Figure 2. The test circuit diagram of (dt, −dt, dt, −dt) time sequence. This test circuit has (dt, −dt, dt, −dt) time sequence for the optimized second-order Trotterization (figure 1). q0 : Uj ( θ⃗ 2 ) Uj ( ⃗θ) Uj (⃗0) Uj (−⃗θ) Uj (− θ⃗ 2 ) q1 : Uj ( ⃗θ) Uj ( ⃗θ) Uj (−⃗θ) Uj (−⃗θ) q2 : Uj ( θ⃗ 2 ) Uj ( ⃗θ) Uj (⃗0) Uj (−⃗θ) Uj (− θ⃗ 2 ) q3 : Uj ( ⃗θ) Uj ( ⃗θ) Uj (−⃗θ) Uj (−⃗θ) q4 : Uj ( θ⃗ 2 ) Uj ( ⃗θ) Uj (⃗0) Uj (−⃗θ) Uj (− θ… view at source ↗
Figure 3
Figure 3. The test circuit diagram of (dt, dt, −dt, −dt) time sequence. This test circuit has (dt, dt, −dt, −dt) time sequence for the optimized second-order Trotterization (figure 1). implementation having (dt, dt, −dt, −dt) time sequence. In the (dt, dt, −dt, −dt) time sequence, only the center layer has ⃗0 parameter and the former (the left) has the positive parameters (⃗θ and θ⃗ 2 ) and the latter (the right) has the nega… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Qiskit simulation of the observable measurement after time evolution of the XXZ model with open and periodic boundary conditions. The Qiskit implementations of the first-order Trotterization and the second-order Trotterization are compared with the direct computation b…
Figure 5
Figure 5. Figure 5: TREX, DD, and PT Comparison with OBC and N = 20. ‘ The comparison between TREX+DD and TREX+PT in figure 7 shows that the contribution by DD is slight, even though the performance of DD is practically validated in several experiments [56, 57, 58, 59]. We conjecture that…
Figure 6
Figure 6. Figure 6: TREX, DD, and PT Comparison with PBC and N = 20. insertion [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: This bar chart presents the error rates of various Quantum Error Mitigation (QEM) strategies under Open Boundary Conditions (OBC, blue) and Periodic Boundary Conditions (PBC, red) with N = 20. 4.2. Comparison of ZNE and SM This section extends the QEM comparison to the…
Figure 8
Figure 8. Figure 8: Comparison of ZNE and SM with OBC and N = 20. (a) Staggered Magnetization (b) Absolute Error [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Comparison of ZNE and SM with PBC and N = 20 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Comparison of the mean absolute errors for the ZNE and the SM with OBC and PBC when N = 20. The TREX+DD+PT result is presented again to compare with the ZNE and the SM. The corresponding average absolute error rates of these results are summarized in figure 10 and tab…
Figure 11
Figure 11. Figure 11: Comparison of ZNE and SM with OBC and N = 104. (a) Staggered Magnetization (b) Absolute Error [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Comparison of ZNE and SM with PBC and N = 84. We extend the system size to N = 104 for OBC and N = 84 for PBC. The PBC system is limited to 84 qubits due to the requirement of connecting the first and last qubits. For the ZNE, we apply the same extrapolation fitting c…
Figure 13
Figure 13. Figure 13: Comparison of the mean absolute errors of the TREX+DD+PT, the ZNE, and the SM with OBC (N = 104) and PBC (N = 84). are presented in figure 13 and table A3. The average absolute error value with ZNE is 0.04917 and 0.05044 on OBC and PBC, respectively, whereas the SM ha…
Figure 14
Figure 14. Figure 14: The randomized measurement (RM) protocol, where the Trotter steps prepare the quantum state. Then randomized measurements on a subsystem A (system size L; the upper-subsystem as an example) are performed by applying NU product of local random unitaries Uˆ a = ⊗L i=1U …
Figure 15
Figure 15. Figure 15: The Quantum Multi-Programming (QMP) implementation of the randomized measurement (RM) protocol in figure 14. result includes both Trotterization and RM protocol errors. Finally, the ibm marrakesh results are obtained from an IBM Quantum device, ibm marrakesh, which fe…
Figure 16
Figure 16. Figure 16: R´enyi entropy of order 2 after time evolution of the Hamiltonian. 6. Conclusion and outlook In this study, we compare a range of QEM methods, examining their characteristics as well as the advantages and limitations of each. Building on this comparative analysis, we …

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Noninvertible Kramers-Wannier duality symmetries for the discrete-time quantum Ising chain

    quant-ph 2025-11 conditional novelty 6.0 of 10

    The trotterized critical Ising chain is integrable and carries two non-invertible Kramers-Wannier symmetry operators that act as half space-time translations.

  2. Universal Quantum Error Mitigation via Random Inverse Depolarizing Approximation

    quant-ph 2025-08 conditional novelty 6.0 of 10

    RIDA estimates a circuit's global depolarization probability from a random half-gate identity circuit and uses it to amplify noisy expectation values.

  3. Quantum Utility in Simulating the Real-time Dynamics of the Fermi-Hubbard Model using Superconducting Quantum Computers

    quant-ph 2025-09 conditional novelty 5.0 of 10

    A 104-qubit IBM quantum computer simulates the 1D Fermi-Hubbard model's staggered-magnetization dynamics with constant-depth Trotter circuits, matching MPS-TDVP up to time 4 but not at later times.

Reference graph

Works this paper leans on

73 extracted references · 32 canonical work pages · cited by 3 Pith papers

  1. [12]

    Rahman S A, Lewis R, Mendicelli E and Powell S 2022 Physical Review D106 074502 (Preprint2205.09247)

    arXiv 2022

  2. [46]

    Urbanek M, Nachman B, Pascuzzi V R, He A, Bauer C W and de Jong W A 2021 Physical review letters127270502

    work page 2021

  3. [1]

    2023 Nature614676–681

    work page 2023

  4. [2]

    2024 Nature

    Acharya R, Abanin D A, Aghababaie-Beni L, Aleiner I, Andersen T I, Ansmann M, Arute F, Arya K, Asfaw A, Astrakhantsev N et al. 2024 Nature

    work page 2024

  5. [3]

    2025 Nature638927–934

    Putterman H, Noh K, Hann C T, MacCabe G S, Aghaeimeibodi S, Patel R N, Lee M, Jones W M, Moradinejad H, Rodriguez R et al. 2025 Nature638927–934

    work page 2025

  6. [4]

    Endo S, Benjamin S C and Li Y 2018 Physical Review X8031027 (Preprint 1712.09271)

    arXiv 2018

  7. [5]

    Temme K, Bravyi S and Gambetta J M 2017 Physcial Review Letters119180509 (Preprint1612.02058)

    arXiv 2017

  8. [6]

    Li Y and Benjamin S C 2016 arXiv e-prints arXiv:1611.09301 (Preprint1611. 09301)

    arXiv 2016

Show all 73 references
  1. [7]

    Kandala A, Temme K, C´ orcoles A D, Mezzacapo A, Chow J M and Gambetta J M 2019 Nature567491–495 (Preprint1805.04492)

    2019 arXiv

  2. [8]

    van den Berg E, Minev Z K, Kandala A and Temme K 2023 Nature Physics19 1116–1121 (Preprint2201.09866)

    2023 arXiv

  3. [9]

    Yu H, Zhao Y and Wei T C 2023 Physical Review Research5013183 (Preprint 2207.09994)

    2023 arXiv

  4. [10]

    Kim Y, Wood C J, Yoder T J, Merkel S T, Gambetta J M, Temme K and Kandala A 2023 Nature Phys.19752–759 (Preprint2108.09197)

    2023 arXiv

  5. [11]

    Kim Y, Eddins A, Anand S, Wei K X, van den Berg E, Rosenblatt S, Nayfeh H, Wu Y, Zaletel M, Temme K and Kandala A 2023 Nature618500–505

    2023

  6. [13]

    Giurgica-Tiron T, Hindy Y, LaRose R, Mari A and Zeng W J 2020 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 306– 316 (Preprint2005.10921)

    2020 arXiv

  7. [14]

    Chowdhury T A, Yu K, Shamim M A, Kabir M and Sufian R S 2024 Physical Review Research6033107

    2024

  8. [15]

    Orbach R 1958 Physical Review112(2) 309–316 URLhttps://link.aps.org/ doi/10.1103/PhysRev.112.309

    1958 doi

  9. [16]

    Lieb E, Schultz T and Mattis D 1961 Annals of Physics16407–466 ISSN 0003-4916 URLhttps://www.sciencedirect.com/science/article/pii/ 0003491661901154

    1961

  10. [17]

    Des Cloizeaux J and Gaudin M 1966 Journal of Mathematical Physics71384–1400 ISSN 0022-2488 URLhttps://doi.org/10.1063/1.1705048

    1966 doi

  11. [18]

    Yang C N and Yang C P 1966 Physical Review150(1) 321–327 URLhttps: //link.aps.org/doi/10.1103/PhysRev.150.321 REFERENCES27

    1966 doi

  12. [19]

    Alcaraz F C, Barber M N and Batchelor M T 1987 Phys. Rev. Lett.58(8) 771–774 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.58.771

    1987 doi

  13. [20]

    de Vega H and Woynarovich F 1985 Nuclear Physics B251439–456 ISSN 0550-3213 URLhttps://www.sciencedirect.com/science/article/pii/ 0550321385902718

    1985

  14. [21]

    Luther A and Peschel I 1975 Phys. Rev. B12(9) 3908–3917 URLhttps://link. aps.org/doi/10.1103/PhysRevB.12.3908

    1975 doi

  15. [22]

    Lukyanov S 1998 Nuclear Physics B522533–549 ISSN 0550-3213 URLhttps: //www.sciencedirect.com/science/article/pii/S0550321398002491

    1998

  16. [23]

    Calabrese P and Cardy J 2005 Journal of Statistical Mechanics: Theory and Experiment200504010 (Preprintcond-mat/0503393)

    2005

  17. [24]

    DeChiara G, Montangero S, Calabrese P and Fazio R 2006 Journal of Statistical Mechanics: Theory and Experiment200603001 (Preprintcond-mat/0512586)

    2006

  18. [25]

    Fagotti M, Collura M, Essler F H L and Calabrese P 2014 Physical Review B89 125101 (Preprint1311.5216)

    2014 arXiv

  19. [26]

    Bonnes L, Essler F H L and L¨ auchli A M 2014Physical Review Letters113187203 (Preprint1404.4062)

    arXiv

  20. [27]

    Essler F H L and Fagotti M 2016 Journal of Statistical Mechanics: Theory and Experiment6064002 (Preprint1603.06452)

    2016 arXiv

  21. [28]

    Trotter H F 1959 Proceedings of the American Mathematical Society10545–551 ISSN 00029939, 10886826 URLhttp://www.jstor.org/stable/2033649

    1959

  22. [29]

    Suzuki M 1976 Communications in Mathematical Physics51183–190

    1976

  23. [30]

    Suzuki M 1977 Communications in Mathematical Physics57193–200

    1977

  24. [31]

    Vanicat M, Zadnik L and Prosen T 2018 Phys. Rev. Lett.121030606

    2018

  25. [32]

    Smith A, Kim M S, Pollmann F and Knolle J 2019npj Quantum Inf.5106 (Preprint 1906.06343)

    1906 arXiv

  26. [33]

    Zhang K, Yu K, Hao K and Korepin V 2024 Advanced Quantum Technologies 2300345

    2024

  27. [34]

    Kivlichan I D, Gidney C, Berry D W, Wiebe N, McClean J, Sun W, Jiang Z, Rubin N, Fowler A, Aspuru-Guzik A, Neven H and Babbush R 2020 Quantum4 296 (Preprint1902.10673)

    2020 arXiv

  28. [35]

    Lee J, Berry D W, Gidney C, Huggins W J, McClean J R, Wiebe N and Babbush R 2021 PRX Quantum2030305 (Preprint2011.03494)

    2021 arXiv

  29. [36]

    Charles C, Gustafson E J, Hardt E, Herren F, Hogan N, Lamm H, Starecheski S, Van de Water R S and Wagman M L SimulatingZ 2 lattice gauge theory on a quantum computer (Preprint2305.02361)

    arXiv

  30. [37]

    van den Berg E, Minev Z K and Temme K 2022 Phys. Rev. A105(3) 032620 URL https://link.aps.org/doi/10.1103/PhysRevA.105.032620 REFERENCES28

    2022 doi

  31. [38]

    Viola L, Knill E and Lloyd S 1999 Phys. Rev. Lett.822417–2421 (Preprint quant-ph/9809071)

    1999 arXiv

  32. [39]

    Ezzell N, Pokharel B, Tewala L, Quiroz G and Lidar D A 2023 Phys. Rev. Applied 20064027 (Preprint2207.03670)

    2023 arXiv

  33. [40]

    Quantum Eng.31–10 (Preprint 2204.01471)

    Niu S and Todri-Sanial A 2022 IEEE Trans. Quantum Eng.31–10 (Preprint 2204.01471)

    2022 arXiv

  34. [41]

    Bennett C H, Brassard G, Popescu S, Schumacher B, Smolin J A and Wootters W K 1996 Physcial Review Letters76722–725 (Preprintquant-ph/9511027)

    1996

  35. [42]

    Wallman J J and Emerson J 2016 Physcial Review A94052325 (Preprint 1512.01098)

    2016 arXiv

  36. [43]

    Cai Z and Benjamin S C 2019 Scientific Reports9ISSN 2045-2322 (Preprint 1807.04973) URLhttp://dx.doi.org/10.1038/s41598-019-46722-7

    2019 arXiv

  37. [44]

    Chowdhury T A, Yu K, Asaduzzaman M and Sufian R S 2024 arXiv preprint arXiv:2412.15180

    2024

  38. [45]

    Chowdhury T A, Yu K and Sufian R S 2025 arXiv preprint arXiv:2503.18580

    2025

  39. [47]

    Weinberg P and Bukov M 2017 SciPost Phys.2003 URLhttps://scipost.org/ 10.21468/SciPostPhys.2.1.003

    2017 doi

  40. [48]

    Paeckel S, K¨ ohler T, Swoboda A, Manmana S R, Schollw¨ ock U and Hubig C 2019 Annals Phys.411167998 (Preprint1901.05824)

    2019 arXiv

  41. [49]

    Cirac J I, Perez-Garcia D, Schuch N and Verstraete F 2021 Rev. Mod. Phys.93 045003 (Preprint2011.12127)

    2021 arXiv

  42. [50]

    Haegeman J, Cirac J I, Osborne T J, Piˇ zorn I, Verschelde H and Verstraete F 2011 Physcial Review Letters107070601 (Preprint1103.0936)

    2011 arXiv

  43. [51]

    Haegeman J, Lubich C, Oseledets I, Vandereycken B and Verstraete F 2016Physical Review B94ISSN 2469-9969 URLhttp://dx.doi.org/10.1103/PhysRevB.94. 165116

    doi

  44. [52]

    Codebases 4 URLhttps://scipost.org/10.21468/SciPostPhysCodeb.4

    Fishman M, White S R and Stoudenmire E M 2022 SciPost Phys. Codebases 4 URLhttps://scipost.org/10.21468/SciPostPhysCodeb.4

    2022 doi

  45. [53]

    Codebases 4–r0.3 URLhttps://scipost.org/10.21468/SciPostPhysCodeb.4-r0.3

    Fishman M, White S R and Stoudenmire E M 2022 SciPost Phys. Codebases 4–r0.3 URLhttps://scipost.org/10.21468/SciPostPhysCodeb.4-r0.3

    2022 doi

  46. [54]

    https://github.com/ITensor/ITensorTDVP.jl.git URLhttps://github.com/ ITensor/ITensorTDVP.jl.git

  47. [55]

    Childs A M, Su Y, Tran M C, Wiebe N and Zhu S 2021 Physical Review X11 011020

    2021

  48. [56]

    Coote P, Dimov R, Maity S, Hartnett G S, Biercuk M J and Baum Y 2025 PRX Quantum6010332

    2025

  49. [57]

    Tong C, Zhang H and Pokharel B 2024 arXiv preprint arXiv:2403.02294 REFERENCES29

    2024 arXiv

  50. [58]

    Ezzell N, Pokharel B, Tewala L, Quiroz G and Lidar D A 2023 Physical Review Applied20064027

    2023

  51. [59]

    Liu G Q, Po H C, Du J, Liu R B and Pan X Y 2013 Nature communications4 2254

    2013

  52. [60]

    Buhrman H, Cleve R, Watrous J and de Wolf R 2001 Phys. Rev. Lett.87167902 (Preprintquant-ph/0102001)

    2001

  53. [61]

    Horodecki P and Ekert A 2002 Physcial Review Letters89127902 (Preprint quant-ph/0111064)

    2002 arXiv

  54. [62]

    Ekert A K, Alves C M, Oi D K L, Horodecki M, Horodecki P and Kwek L C 2002 Phys. Rev. Lett.88(21) 217901 (Preprintquant-ph/0203016) URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.88.217901

    2002

  55. [63]

    Brydges T, Elben A, Jurcevic P, Vermersch B, Maier C, Lanyon B P, Zoller P, Blatt R and Roos C F 2019 Science364260–263 (Preprint1806.05747)

    2019 arXiv

  56. [64]

    van Enk S J and Beenakker C W J 2012 Phys. Rev. Lett.108(11) 110503 (Preprint1112.1027) URLhttps://link.aps.org/doi/10.1103/PhysRevLett. 108.110503

    2012 arXiv

  57. [65]

    Elben A, Vermersch B, Dalmonte M, Cirac J I and Zoller P 2018 Physcial Review Letters120050406 (Preprint1709.05060)

    2018 arXiv

  58. [66]

    Vermersch B, Elben A, Dalmonte M, Cirac J I and Zoller P 2018 Physcial Review A97023604 (Preprint1801.00999)

    2018 arXiv

  59. [67]

    Elben A, Vermersch B, Roos C F and Zoller P 2019 Physcial Review A99052323 (Preprint1812.02624)

    2019 arXiv

  60. [68]

    Rath A, van Bijnen R, Elben A, Zoller P and Vermersch B 2021 Physcial Review Letters127200503 (Preprint2102.13524)

    2021 arXiv

  61. [69]

    Elben A, Flammia S T, Huang H Y, Kueng R, Preskill J, Vermersch B and Zoller P 2023 Nature Reviews Physics59–24 (Preprint2203.11374)

    2023 arXiv

  62. [70]

    Park G, Zhang K, Yu K and Korepin V 2023 Quantum Information Processing22 54 (Preprint2207.14464)

    2023 arXiv

  63. [71]

    SPIE Int

    Rao P, Choi S and Yu K 2024 Proc. SPIE Int. Soc. Opt. Eng.12911129110E

    2024

  64. [72]

    Baker J S, Park G, Yu K, Ghukasyan A, Goktas O and Radha S K 2024 Quantum Machine Intelligence618 (Preprint2305.05881)

    2024 arXiv

  65. [73]

    Huang H Y, Kueng R and Preskill J 2020 Nature Physics161050–1057

    2020

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.