adapol: Adaptive pole-fitting for quantum many-body physics

arXiv:2609.40150 · cond-mat.str-el, cs.NA, math.NA · Submitted 2026-09-30 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "adapol: Adaptive pole-fitting for quantum many-body physics".

Mira: The paper introduces 'adapol', a Python package designed to solve the crucial computational task of decomposing Matsubara functions into sums of simple poles,

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we're diving into the paper 'adapol: Adaptive pole-fitting for quantum many-body physics', and what I see right away is that it’s tackling the way we convert those complicated Matsubara functions into something useful for our simulations.

Mira: I agree, Kai, the title suggests a method focused on finding those poles adaptively, which sounds like it might solve some of the headaches folks have when trying to fit these things to experimental data or theoretical models.

Lev: From my side as someone who deals with actual hardware constraints, if this method is too complex to run on real quantum processors, then its theoretical elegance doesn't matter much for us right now.

Kai: Exactly, Lev; the authors are Zhen Huang and Chia-Nan Yeh, and they’ve put forward this idea of an adaptive approach to pole fitting in the context of quantum many-body physics.

Mira: They aren't just proposing a new trick; they’re addressing a real bottleneck where existing methods often rely on ad-hoc solutions because the optimization landscape for finding those poles is quite challenging.

Lev: That's what I worry about; if it relies on complex post-processing, we need to know how computationally cheap that process actually is when you throw in the noise and error inherent in experimental measurements.

Kai: Right, so they’re aiming to make this process more robust and user-friendly compared to older techniques that researchers are still using today.

The paper's summary: Kai: So, looking at the actual summary of 'adapol: Adaptive pole-fitting for quantum many-body physics', the core idea is decomposing a Matsubara function into a sum of simple poles, where those poles dictate the energy levels in an effective model.

Mira: That decomposition is essential because it turns a complicated matrix-valued function into something that can be interpreted as a collection of discrete energy states, which is what we need for many physical interpretations.

Lev: But the summary mentions that this pole-fitting step often happens inside self-consistent loops, and if the algorithm isn't robust, it could make running things like dynamical mean-field theory impossible on real hardware.

Kai: It does mention that while there’s been progress on these algorithms for some time, there’s really a lack of widely deployed software that practitioners can just use without having to do custom optimization work every single time.

Mira: That points directly to the problem they're solving: providing a black-box algorithm that delivers results with controlled accuracy when you feed it noisy Matsubara data.

Lev: A black-box algorithm is appealing because it reduces the manual tuning required, but I still need to know how many poles it can handle before we hit computational limits on a real quantum device.

Kai: The paper highlights that their methodology uses a modified version of an AAA rational approximation algorithm to get an initial guess for those pole locations, which they then refine.

The paper's improvements: Mira: Now, when we look at the specific improvements proposed in 'adapol: Adaptive pole-fitting for quantum many-body physics', they seem to focus on making the process compact and noise-robust.

Kai: They specifically mention that this AAA-based approach can consistently yield a more compact pole approximation of a fixed Green’s function compared to the generic DLR approach, which is itself an exponential improvement over naive uniform frequency grid approaches.

Lev: If it's more compact, that’s good for efficiency because fewer parameters mean less data to process when you're trying to fit the data accurately.

Mira: The authors stress that reducing the number of poles needed to achieve a certain accuracy is really important because computational costs often scale exponentially with the number of poles.

Kai: They also provide flexibility for users, allowing them to choose whether they want optimization-based post-processing or not, and they let you specify either a maximum number of poles you want or a target error tolerance.

Lev: That level of control is important; it lets us decide exactly what trade-off we are making between computational speed and the precision of the resulting model parameters.

Conclusion: Kai: So, wrapping up on 'adapol: Adaptive pole-fitting for quantum many-body physics', the authors show a practical package that makes finding those poles less of a manual optimization headache in quantum many-body problems.

Mira: They’ve established that their AAA-based approach offers a compact approximation that outperforms some standard methods, which is significant because it helps manage the exponential complexity associated with fitting these functions.

Lev: For us running simulations on real hardware, the implication is that if this package can reliably provide a good enough set of energy levels without requiring massive amounts of computational overhead in the fitting step, then it could make those self-consistent loops much more feasible.

Kai: Ultimately, this work gives researchers a tool to get more accurate and compact models from Matsubara data with less fuss than what was previously available.

Mira: It’s a solid contribution because it addresses the practical deployment issue of analytic continuation methods by providing something that is both accurate and user-friendly for fitting.

Lev: I think the ability to control the pole count directly is what makes it usable in an experimental context where you have limits on how many states you can realistically model at once.

Kai: We've covered the paper, 'adapol: Adaptive pole-fitting for quantum many-body physics'. It’s a solid piece of software that addresses a real practical hurdle in simulating complex quantum systems.

Zhen Huang, Chia-Nan Yeh, Lin Lin, Nils Wentzell, Jason Kaye, Hugo U. R. Strand

Center for Computational Quantum Physics, Flatiron Institute · Department of Mathematics, University of California, Berkeley Department of Computing and Mathematical Sciences, California Institute of Technology Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory Department of Mathematics University of California Berkeley Department of Computing and Mathematical Sciences California Institute of Technology Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory School of Science and Technology Örebro University

cond-mat.str-el, cs.NA, math.NA

Submitted: 2026-09-30

Updated: 2026-09-30

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 92/100

The gist: The paper introduces 'adapol', a Python package designed to solve the crucial computational task of decomposing Matsubara functions into sums of simple poles, which is essential for many numerical

Key concepts

Matsubara functions
These are mathematical representations of finite-temperature quantities in quantum many-body physics, defined on an imaginary frequency axis. They are crucial for describing systems at non-zero temperatures.
Pole decomposition
This is the core task: approximating a complex function by summing simple poles. This transforms a difficult function into a simpler form where the pole locations (energy levels) and residues define an effective, simpler model.
AAA algorithm
adapol uses a modified version of the AAA rational approximation algorithm to quickly find initial guesses for the pole locations. This is an efficient method used to start the process before any complex optimization refinement is applied.

Terminology

Summary

The paper introduces 'adapol', a Python package designed to solve the crucial computational task of decomposing Matsubara functions into sums of simple poles, which is essential for many numerical methods in quantum many-body physics. This technique addresses the non-convex optimization challenges inherent in finding pole locations when fitting Matsubara data, offering a robust and user-friendly solution for applications like hybridization fitting.

The core mathematical problem

The Green’s function approach to quantum many-body physics often involves representing finite temperature quantities, such as the Green’s function, self-energy, and hybridization functions, in the discrete “Matsubara” domain on the imaginary frequency axis. A common computational task within this framework is decomposing a Matsubara function into a sum of simple poles:

G(iνn) ≈

X

M

k=1

Rk

iνn − pk

Here, G(iνn) is in general an m×m matrix-valued function of the Matsubara frequency point iνn = (2n+1)πi/β for fermionic functions, and iνn = 2nπi/β for bosonic functions, with β representing the inverse temperature, n ∈ Z, and m the number of quantum states or spin-orbitals. The pk are real-valued pole locations, and the Rk are the corresponding matrix-valued residues. In applications such as hybridization fitting, these poles and residues define an effective non-interacting model where pk play the role of energy levels.

The challenge in fitting

Since the pole locations enter the approximation nonlinearly and are shared by all components of a matrix-valued function, a best fit from Matsubara frequency data cannot be obtained component-wise, which leads to a highly non-convex optimization landscape. This difficulty is particularly acute because many applications, such as dynamical mean-field theory (DMFT), require this pole-fitting step inside a self-consistent loop.

The adapol methodology

adapol (“add a pole”) is implemented in Python and utilizes an adaptive pole-fitting procedure outlined in prior literature. The method uses a modified version of the AAA rational approximation algorithm (Nakatsukasa et al., 2018) to obtain a guess of the pole locations pk, which can optionally be refined by non-convex optimization. Following this initial guess, The residues Rk are then obtained by a linear least-squares fit. This procedure has been shown to provide an accurate and compact fit of Matsubara data in a black-box and noise-robust manner.

Software design and comparison

adapol is designed as a simple and self-contained package which can be incorporated into codes requiring Matsubara pole-fitting. Users have the flexibility to choose whether to perform the fit with or without optimization-based post-processing of the AAA result, and can specify either a maximum number of poles or a target error tolerance. Furthermore, an interface to the TRIQS package (Parcollet et al., 2015) is also provided. The paper notes that it implements the AAA-based approach, distinguishing itself from other methods like Prony’s method implemented in the MiniPole Python package.

Research impact

The AAA-based approach employed by adapol is considered one of the state-of-the-art methods in analytic continuation. Crucially, reducing the number of poles required to achieve a given accuracy is vital because computational costs often scale exponentially with the number of poles. The authors demonstrated that their algorithm can "consistently yield a more compact pole approximation of a fixed, given Green’s function than the generic DLR approach (Kaye et al., 2022), which is itself an exponential-in-β improvement over naive uniform frequency grid approaches." This compactness is significant for applications like exact diagonalization quantum impurity solvers and tensor network-based quantum impurity solvers.

AI usage disclosure

Generative AI tools such as Claude and Codex were used to assist in writing code, tests, documentation, and examples in the adapol package. The content produced by these tools was reviewed and edited by the authors.

Acknowledgements

This work is partially supported by the Simons Targeted Grants in Mathematics and Physical Sciences on Moiré Materials Magic (Z.H., L.L.). H.U.R.S acknowledges financial support from the Swedish Research Council (Vetenskapsrådet, VR) grant number 2024-04652 and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 854843-FASTCORR). The Flatiron Institute is a division of the Simons Foundation.

References

Caffarel, M., & Krauth, W. (1994).

Improvements for AI systems

Here are the potential improvements to AI systems based on the concepts presented in adapol: Adaptive pole-fitting for quantum many-body physics, along with what those improved systems could achieve:

  1. Enhance Quantum Many-Body Simulation Accuracy via Robust Pole Approximation:

  2. Improve Computational Efficiency in High-Dimensional State Representations:

  3. Develop Self-Consistent, Black-Box Solvers for Complex Physical Problems (e.g., DMFT):

  4. Enable Accurate Analytic Continuation of Imaginary Time Data to Real Frequency Observables:

Specific improvements and capabilities:

  1. A system incorporating the 'adapol' methodology can perform highly accurate, compact representations of Matsubara Green’s functions by identifying the minimal necessary set of poles (energy levels) and residues.

  2. The improved AI system can handle large, noisy datasets from quantum simulations (like those generated by Exact Diagonalization or Tensor Networks) and reliably extract the underlying physical parameters (the pole locations, which represent energy eigenvalues).

  3. This capability directly enables more efficient Hybridization Fitting algorithms for Quantum Impurity Solvers (like DMFT), allowing these solvers to run faster or achieve higher precision without requiring excessively large, computationally prohibitive models.

  4. The system can perform robust analytic continuation from discrete imaginary-time data to continuous real-frequency spectra, which is crucial for connecting theoretical calculations with experimentally measurable quantities.

  5. The improved AI can automate the selection between different pole-fitting strategies (Prony's method vs. AAA rational approximation) by allowing users to compare the performance of distinct mathematical approaches for a given physical system, providing a meta-level optimization tool for model building.

Sources

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