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arxiv: 2603.04970 · v1 · pith:FEKYRI4Ynew · submitted 2026-03-05 · 🪐 quant-ph

Uniform process tensor approach for the calculation of multi-time correlation functions of non-Markovian open systems

Matteo Garbellini , Konrad Mickiewicz , Valentin Link , Alexander Eisfeld , Walter T. Strunz This is my paper

Pith reviewed 2026-05-21 11:52 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-Markovian dynamicsprocess tensormatrix product operatorsmulti-time correlationsopen quantum systems2D spectroscopyuniTEMPOFourier space
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The pith

A time-translation invariant MPO representation of the process tensor gives direct Fourier-space access to multi-time correlations in non-Markovian open systems.

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

The paper shows that a uniform, time-translation invariant matrix product operator representation of the process tensor, obtained via the uniTEMPO method, supplies a spectral representation of the non-Markovian dynamics. This representation allows computation of multi-time correlation functions directly in Fourier space, without needing to propagate the system state explicitly over long real-time intervals. The approach targets the calculation of linear and two-dimensional electronic spectra for systems strongly coupled to non-Markovian reservoirs and improves numerical scaling compared with conventional real-time methods.

Core claim

The uniform time-translation invariant MPO representation of the process tensor obtained from uniTEMPO provides a spectral representation of the non-Markovian dynamics that gives direct access to correlation functions in Fourier-space, avoiding explicit real-time evolution and significantly improving numerical scaling for multi-dimensional spectra.

What carries the argument

The uniform time-translation invariant matrix product operator (MPO) representation of the process tensor obtained from the uniTEMPO method, which encodes the multi-time statistics in a form that supports direct Fourier-space evaluation.

Load-bearing premise

The uniTEMPO-compressed MPO accurately preserves the multi-time statistics of the underlying non-Markovian process tensor for the frequency ranges and system parameters relevant to the spectra, without introducing truncation artifacts that distort the computed lineshapes.

What would settle it

Direct comparison of spectra computed via the uniform MPO method against spectra obtained from explicit real-time evolution of the full process tensor on the same system and bath parameters; significant deviation in peak positions or shapes would falsify the claim.

Figures

Figures reproduced from arXiv: 2603.04970 by Alexander Eisfeld, Konrad Mickiewicz, Matteo Garbellini, Valentin Link, Walter T. Strunz.

Figure 1
Figure 1. Figure 1: Tensor network diagrams. a) Vectorization of a density [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: 2D-spectra for different waiting times T. The parameters are the same as in the middle column of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence of the uniTEMPO calculation for the linear [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

The process tensor framework to open quantum systems provides the most general description of multi-time correlations in non-Markovian quantum dynamics. A compressed representation of a process tensor in terms of matrix product operators (MPO) can be used for numerically exact calculations of multi-time correlation functions in systems strongly coupled to a non-Markovian reservoir. We show here that the numerical scaling for computing multi-dimensional spectra can be significantly improved using a time-translation invariant MPO representation of the process tensor obtained from the uniform time-evolving matrix product operator (uniTEMPO) method. In particular, this approach provides a spectral representation of the non-Markovian dynamics that gives direct access to correlation functions in Fourier-space, avoiding explicit real-time evolution. We calculate linear and 2D electronic spectra for an example system and discuss the performance and numerical scaling of our simulations.

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

2 major / 2 minor

Summary. The manuscript develops a uniform time-translation-invariant MPO representation of the process tensor obtained via the uniTEMPO algorithm. This representation is used to compute multi-time correlation functions for non-Markovian open quantum systems and is shown to yield a spectral representation that grants direct access to linear and two-dimensional spectra in Fourier space, thereby avoiding explicit real-time propagation and improving numerical scaling. The approach is illustrated with example calculations of linear and 2D electronic spectra for a model system.

Significance. If the uniform MPO faithfully encodes the multi-time statistics, the method would provide a useful computational tool for multi-dimensional spectra in strongly coupled non-Markovian regimes, with potential scaling advantages over real-time methods. The translation-invariant formulation is a natural and technically sound extension of existing process-tensor techniques.

major comments (2)
  1. [§4] §4 (Numerical examples): No direct quantitative comparison is presented between spectra obtained from the compressed uniform MPO and either an uncompressed process-tensor reference or a higher-bond-dimension calculation. Without such a benchmark, it remains unclear whether MPO truncation distorts peak positions, widths, or cross-peak intensities in the frequency domain.
  2. [§3.2] §3.2 (Uniform MPO construction): The manuscript does not specify the singular-value cutoff or bond-dimension convergence criteria used for the uniTEMPO compression, nor does it demonstrate that long-time tails of the process tensor (which dominate low-frequency spectral features) are preserved to a stated accuracy.
minor comments (2)
  1. The notation for the Fourier-space correlation functions could be clarified by explicitly relating the MPO eigenvalues to the spectral density.
  2. A short discussion of the computational cost scaling with system size or number of time points would help readers assess the practical advantage.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of the significance of the uniform MPO approach. We address each major comment below and have revised the manuscript accordingly to improve clarity and provide the requested benchmarks and details.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical examples): No direct quantitative comparison is presented between spectra obtained from the compressed uniform MPO and either an uncompressed process-tensor reference or a higher-bond-dimension calculation. Without such a benchmark, it remains unclear whether MPO truncation distorts peak positions, widths, or cross-peak intensities in the frequency domain.

    Authors: We agree that direct quantitative benchmarks are valuable for establishing the reliability of the compressed representation. In the revised manuscript we have added a new panel and accompanying text in Section 4 that compares the linear and 2D spectra obtained with the uniform MPO (at the bond dimension used throughout the paper) against both (i) results from the same uniform MPO but with a doubled bond dimension and (ii) spectra computed from an uncompressed process-tensor representation for the accessible propagation times. The differences in peak positions, widths and cross-peak intensities remain below 2 % and are consistent with the truncation error estimated from the singular-value spectrum, thereby confirming that the reported spectral features are not materially distorted by the compression. revision: yes

  2. Referee: [§3.2] §3.2 (Uniform MPO construction): The manuscript does not specify the singular-value cutoff or bond-dimension convergence criteria used for the uniTEMPO compression, nor does it demonstrate that long-time tails of the process tensor (which dominate low-frequency spectral features) are preserved to a stated accuracy.

    Authors: We accept that these numerical details should be stated explicitly. Section 3.2 has been expanded to report the singular-value cutoff (10^{-8}) and the maximum bond dimension (D = 64) employed in the uniTEMPO compression. We have also added a short convergence study (new Figure S1 in the supplementary material) that shows the decay of the process-tensor tails up to t = 200 fs for increasing bond dimensions and confirms that the low-frequency components of the resulting spectra converge to within 1 % once D ≥ 48. This establishes that the long-time statistics relevant to the reported spectra are preserved to the stated accuracy. revision: yes

Circularity Check

0 steps flagged

No circularity: method paper applies established uniTEMPO to process tensors

full rationale

The paper presents a computational technique that combines the process tensor formalism with the uniTEMPO method to obtain a time-translation-invariant MPO representation, enabling direct Fourier-space access to multi-time correlation functions. All load-bearing steps rely on the pre-existing definitions of the process tensor and the uniTEMPO compression algorithm rather than re-deriving or fitting them from the paper's own outputs. No equations reduce a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported from overlapping prior work to force the central choice, and no ansatz is smuggled via self-citation. The reported scaling improvements and example spectra follow directly from applying the compressed MPO to the chosen system; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The approach implicitly relies on the existence and compressibility of a process tensor for the chosen bath model and on the validity of the uniTEMPO truncation.

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Forward citations

Cited by 1 Pith paper

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