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arxiv: 2410.10794 · v2 · submitted 2024-10-14 · 🪐 quant-ph · cond-mat.mes-hall· cond-mat.stat-mech· cond-mat.str-el

Robustness of near-thermal dynamics on digital quantum computers

Eli Chertkov , Yi-Hsiang Chen , Michael Lubasch , David Hayes , Michael Foss-Feig This is my paper

Pith reviewed 2026-05-23 18:37 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallcond-mat.stat-mechcond-mat.str-el
keywords Trotterizationquantum simulationthermal statesgate errorsquantum computingHamiltonian dynamicserror robustnessnear-term devices
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0 comments X

The pith

Trotterized circuits simulating near-thermal quantum evolution withstand gate errors and discretization errors better than expected.

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

The paper argues that Trotterized quantum circuits for time evolution near thermal equilibrium resist both quantum gate errors and Trotter discretization errors more effectively than is commonly assumed. This robustness stems from the preservation of thermal-like behavior under imperfections, combined with the fact that weakly entangling gates incur lower error when they produce less entanglement. A new theoretical tool, a statistical ensemble of random product states that approximates a thermal state, supports the analysis because the ensemble can be prepared efficiently with low noise. The linear decrease in gate error with the angle tau further amplifies the gains. These elements together point to concrete improvements in simulation accuracy on current hardware.

Core claim

Trotterized quantum circuits simulating the time-evolution of systems near thermal equilibrium are substantially more robust to both quantum gate errors and Trotter (discretization) errors than is widely assumed. Analytical arguments, numerical simulations, and experiments on trapped-ion hardware show that the error rate for gates of the form exp[-i(Z⊗Z)τ] decreases roughly linearly with τ. The statistical ensemble of random product states approximates a thermal state, can be prepared with low noise, and enables prediction and optimization of such experiments.

What carries the argument

Statistical ensemble of random product states that approximates a thermal state and can be efficiently prepared with low noise on quantum computers.

If this is right

  • Substantial improvements become possible in the achievable accuracy of Trotterized dynamics on near-term quantum computers.
  • The random product state ensemble can be used to predict, optimize, and design Hamiltonian simulation experiments on near-thermal quantum systems.
  • Weakly entangling gates can be implemented natively with error rates that decrease when they generate less entanglement.
  • Near-thermal dynamics maintain their properties despite imperfections in the simulation circuit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Simulations of longer evolution times could become practical on noisy hardware without full error correction.
  • The same error-tolerance mechanism may apply to other algorithms that operate on states close to thermal equilibrium.
  • Experiment design could systematically favor smaller-angle gates to exploit the observed error scaling.

Load-bearing premise

The statistical ensemble of random product states approximates a thermal state and can be efficiently prepared with low noise, while gate error rates decrease roughly linearly with the angle tau.

What would settle it

An experiment measuring error accumulation in Trotterized evolution of a near-thermal state that shows error rates comparable to those in non-thermal initial states, or a benchmark where gate error for exp[-i(Z⊗Z)τ] fails to decrease with smaller τ.

Figures

Figures reproduced from arXiv: 2410.10794 by David Hayes, Eli Chertkov, Michael Foss-Feig, Michael Lubasch, Yi-Hsiang Chen.

Figure 1
Figure 1. Figure 1: FIG. 1. Insertion of an error (here occurring on the gate [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Experimentally measured average gate infidelity for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The observable error versus time [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The observable error versus Trotter step [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Histogram of the log-scaled off-diagonals of [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The average [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The change in single-site observables [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: d, appears nearly conserved in time, with only mi￾nor temporal oscillations that appear to decay in ampli￾tude quickly with time. However, for individual samples from the RPE (marked as dashed lines), there are signif￾icant oscillations in the trace distance that are not being damped within the time scales of our simulations. Such large oscillations were also seen for the |0 · · · 0⟩ state. Note that the … view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20 [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Energy density versus time for the XY model with [PITH_FULL_IMAGE:figures/full_fig_p023_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The autocorrelation function versus MCMC sweep [PITH_FULL_IMAGE:figures/full_fig_p026_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. The average [PITH_FULL_IMAGE:figures/full_fig_p028_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: a shows the energy variance per site as a func￾tion of energy density for the RPE. The borders of the shaded region correspond to the energy densities of the mean-field ground state and anti-ground state (i.e., the lowest and highest energy product states for the Hamiltonian). For the ferromagnetic mixed-field Ising model in Eq. (14), the mean-field ground state is unique. Therefore, as the RPE energy app… view at source ↗
read the original abstract

Understanding the impact of gate errors on quantum circuits is crucial to determining the potential applications of quantum computers, especially in the absence of large-scale error-corrected hardware. We put forward analytical arguments, corroborated by extensive numerical and experimental evidence, that Trotterized quantum circuits simulating the time-evolution of systems near thermal equilibrium are substantially more robust to both quantum gate errors and Trotter (discretization) errors than is widely assumed. In Quantinuum's trapped-ion computers, the weakly entangling gates that appear in Trotterized circuits can be implemented natively, and their error rate is smaller when they generate less entanglement; from benchmarking, we know that the error for a gate $\exp[-i (Z\otimes Z) \tau]$ decreases roughly linearly with $\tau$, up to a small offset at $\tau = 0$. We provide extensive evidence that this scaling, together with the robustness of near-thermal dynamics to both gate and discretization errors, facilitates substantial improvements in the achievable accuracy of Trotterized dynamics on near-term quantum computers. We make heavy use of a new theoretical tool -- a statistical ensemble of random product states that approximates a thermal state, which can be efficiently prepared with low noise on quantum computers. We outline how the random product state ensemble can be used to predict, optimize, and design Hamiltonian simulation experiments on near-thermal quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that Trotterized circuits simulating near-thermal Hamiltonian dynamics are substantially more robust to both gate errors and discretization (Trotter) errors than is widely assumed. This robustness is supported by analytical arguments, extensive numerical simulations, and experimental results on Quantinuum trapped-ion hardware; the key enabling observation is that the error rate of native weakly entangling gates exp[-i(Z⊗Z)τ] decreases roughly linearly with τ (up to a small offset), combined with the use of a new statistical ensemble of random product states that approximates thermal states and can be prepared with low noise.

Significance. If the central claims hold, the work would be significant for near-term quantum simulation: it indicates that practical accuracy gains are achievable for thermal systems without error correction by exploiting the observed error scaling and the random-product-state ensemble as a predictive and optimization tool. The ensemble itself is presented as an independent theoretical device that can be used to design experiments.

minor comments (3)
  1. [Abstract] Abstract: the statement that the gate error 'decreases roughly linearly with τ, up to a small offset at τ=0' is central to the robustness argument; a brief quantitative indication of the fitted slope, offset magnitude, and range of τ over which the linear regime holds would strengthen the claim without requiring additional data.
  2. The manuscript introduces the random product state ensemble as a new tool that 'approximates a thermal state' and 'can be efficiently prepared with low noise.' A short paragraph clarifying the precise sense in which the ensemble approximates thermal expectation values (e.g., in trace distance, in local observables, or in the thermodynamic limit) would help readers assess its scope.
  3. [Abstract] The abstract asserts that the robustness 'facilitates substantial improvements in the achievable accuracy.' A concrete example—e.g., a factor by which the effective error is reduced for a fixed total evolution time—would make the practical impact easier to evaluate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. We appreciate the recognition that the central claims, if they hold, would be significant for near-term quantum simulation of thermal systems.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external evidence

full rationale

The paper advances analytical arguments for robustness of near-thermal Trotterized dynamics, supported by extensive numerical simulations and experimental benchmarking on Quantinuum hardware (including observed linear gate-error scaling with τ). The random-product-state ensemble is introduced as a new independent tool for approximating thermal states and is not derived from or fitted to the target predictions. No load-bearing steps reduce by construction to self-citations, fitted inputs renamed as predictions, or ansatzes smuggled via prior work. The central claims remain externally falsifiable and do not collapse to internal definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Central claim rests on the domain assumption that the random product state ensemble approximates thermal states and on the observed linear error scaling from hardware benchmarking; no free parameters or invented entities beyond the ensemble itself are stated in the abstract.

axioms (2)
  • domain assumption The error rate for exp[-i (Z⊗Z) τ] decreases roughly linearly with τ up to a small offset at τ=0
    Stated as known from benchmarking on Quantinuum trapped-ion computers
  • domain assumption Statistical ensemble of random product states approximates a thermal state
    Introduced as new theoretical tool in the abstract
invented entities (1)
  • statistical ensemble of random product states no independent evidence
    purpose: Approximates thermal state and can be prepared efficiently with low noise
    New tool introduced to enable the robustness study; independent evidence outside the paper not stated in abstract

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Reference graph

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    Change the spin to ⃗ σ′ j = Ej/||⃗hj|| ˆnj,∥ + 1 − E2 j /||⃗hj||2 1/2 ˆnj,⊥. Since rotating the spin j about the local field ⃗hj does not change the energy of the state, the new state produced at step 3 must have the same energy as the original state. Since the new state proposed (the cho- sen ˆnj,∥ vector) does not depend on the current state, T (σ → σ′)...

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    Importantly, this update allows energy to dis- tribute non-locally in space and thereby helps avoid the slow diffusive spreading caused by the local move

    Two-site move To reduce the autocorrelation time of the Markov chain, we consider a more complicated two-site move that produces larger changes to the product state in a single proposal. Importantly, this update allows energy to dis- tribute non-locally in space and thereby helps avoid the slow diffusive spreading caused by the local move. The two-site mo...

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    The local energies of the spins are Ej1 , Ej2

    Choose uniformly at random two spins j1, j2 ∈ [1, N] for an N-site system such that j1 and j2 are not neighboring spins according to the Hamiltonian 27 (i.e., Jj1,j2 = 0). The local energies of the spins are Ej1 , Ej2

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    Compute the unit vectors ˆ nj1/2,∥ ≡ ⃗hj1/2 /||⃗hj1/2 || parallel to the effective fields and choose random unit vectors ˆnj1/2,⊥ that are orthogonal to ˆnj1/2,∥

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    Set the new energies for the two spins to E′ j1 = Ej1 + ∆E and E′ j2 = Ej2 − ∆E

    Pick an energy change ∆ E uniformly at random in the interval ∆E ≥ min(−||⃗hj1 || − Ej1 , ||⃗hj2 || − Ej2) ∆E ≤ max(||⃗hj1 || − Ej1 , −||⃗hj2 || − Ej2). Set the new energies for the two spins to E′ j1 = Ej1 + ∆E and E′ j2 = Ej2 − ∆E

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    Change the spins to ⃗ σ′ j1/2 = E′ j1/2 /||⃗hj1/2 || ˆnj1/2,∥ + 1 − E′2 j1/2 /||⃗hj1/2 ||2 1/2 ˆnj1/2,⊥. In addition to rotating each spin about its local field, in the two-site move we also redistribute energy ∆ E be- tween the two spins by changing how much each spin points along its local field. Since the range of possible ∆E is the same for the propos...

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    The m-site move has the steps:

    m-site move Finally, we generalize to an m-site move with m ≥ 2 that can produce even more non-local changes to the state in a single proposal. The m-site move has the steps:

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    , jm ∈ [1, N] for an N-site system such that none of the j1,

    Choose uniformly at random m spins j1, j2, . . . , jm ∈ [1, N] for an N-site system such that none of the j1, . . . , jm spins are neighboring according to the Hamiltonian (i.e., Jja,jb = 0 ∀a ̸= b) (This is not possible if m is too large. For example, for a 1D chain m ≤ N/3 must hold for this to always be possible.). The local energies of the spins are E...

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    Compute the unit vectors ˆ nja,∥ ≡ ⃗hja /||⃗hja || par- allel to the effective fields and choose random unit vectors ˆnja,⊥ that are orthogonal to ˆnja,∥, for a = 1, . . . , m

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    Choose a uniformly random m-dimensional unit vector ˆr that is orthogonal to the all-ones vec- tor (1 , . . . ,1). Define the energy change vector as ∆Eja = (∆E)ˆra

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    Choose ∆ E uni- formly at random in this range ∆ Emin ≤ ∆E ≤ ∆Emax

    Determine the minimum ∆ Emin and maximum ∆Emax allowed energy changes along the ˆ r direc- tion (∆ Emin can be negative). Choose ∆ E uni- formly at random in this range ∆ Emin ≤ ∆E ≤ ∆Emax. Set E′ ja = Eja + ∆Eja

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    energy difference space

    Change the spins to ⃗ σ′ ja = E′ ja /||⃗hja || ˆnja,∥ + 1 − E′2 ja /||⃗hja ||2 1/2 ˆnja,⊥. In step 3, the unit vector ˆ r specifies the direction in “energy difference space” to move. The vector is con- strained so that P a ˆra = 0 which ensures that the to- tal energy change in the proposal is zero: P ja ∆Eja = (∆E)P a ˆra = 0. Step 4 can be done efficie...

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  72. [72]

    The algorithm is quite similar to the one described in the previous section, but with a few minor changes

    m-site move with energy window We also develop a variant of the MCMC algorithm that allows for sampling states in an energy window [E−ε, E+ ε] with target energy E. The algorithm is quite similar to the one described in the previous section, but with a few minor changes. During this sampling, the current energy Ecurr is logged. Step 3 is modified, so that...

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  73. [73]

    Validation of algorithm To validate that our MCMC sampling algorithms are sampling the intended distribution, we compare them against the rejection sampling approach described in Sec. VI. Figure 23 shows a comparison of the MCMC sampling algorithms with and without an energy win- dow against rejection sampling. The target energy cor- responds to the |0 · ...

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