SciPost Submission Page
Adaptive-basis sample-based neural diagonalization for quantum many-body systems
by Simone Cantori, Luca Brodoloni, Edoardo Recchi, Emanuele Costa, Bruno Juliá-Díaz, Sebastiano Pilati
Submission summary
| Authors (as registered SciPost users): | Simone Cantori |
| Submission information | |
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| Preprint Link: | scipost_202509_00030v1 (pdf) |
| Code repository: | https://github.com/simonecantori/Sample-based-Neural-Diagonalization |
| Date submitted: | Sept. 16, 2025, 10:11 a.m. |
| Submitted by: | Simone Cantori |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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Abstract
Estimating ground-state energies of quantum many-body systems is challenging due to the exponential growth of Hilbert space. Sample-based diagonalization (SBD) addresses this by projecting the Hamiltonian onto a subspace of selected basis configurations but works only for concentrated ground-state wave functions. We propose two neural network-enhanced SBD methods: sample-based neural diagonalization (SND) and adaptive-basis SND (AB-SND). Both leverage autoregressive neural networks for efficient sampling; AB-SND also optimizes a basis transformation to concentrate the wave function. We explore classically tractable single- and two-spin rotations, and more expressive unitaries implementable on quantum computers. On quantum Ising models, SND performs well for concentrated states, while AB-SND consistently outperforms SND and standard SBD in less concentrated regimes.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1- New numerical method to diagonalize quantum Hamiltonians.
2- Interesting link with near-term quantum algorithms
Weaknesses
1- Lack of resource (CPU-time, training steps, ...) comparison with state-of-the-art methods.
Report
For all the examples shown, SND performs similarly or worse than SBD and suffers from the same shortcomings when the ground state is not concentrated. Because it is not made clear what resources are necessary to train the NQS ansatz and the autoregressive neural network respectively, it is difficult to assess the impact of these results.
The adaptive-basis version shows more promise in tackling a wider range of parameters. Once again, a clearer discussion on the additional resources needed to train the basis rotation on top of the autoregressive neural network would have helped evaluate the technique's scalability.
The link with sample-based quantum diagonalisation is also interesting at a time when this algorithm is of great interest in the quantum computing community.
Given the computational focus of the work, I think greater care should be taken to present a fair comparison between different state-of-the-art techniques. For example, what about adding the adaptive-basis idea to the NQS ansatz ? What is the computational cost of all these techniques ? Can AB-SND succeed where Monte-Carlo methods fail (sign-problem) ?
In conclusion, I think the manuscript presents a new quantum/quantum-inspired technique to estimate the ground state of correlated Hamiltonians. The potential extension to sample-based quantum diagonalization highlights a novel link between different areas of research. I would recommend publication after minor revisions.
Requested changes
1- A more thorough discussion of the computational resources required by the different techniques should appear in the main text. Although details for the training of the autoregressive neural network are presented in Appendix G, a comparison to the NQS ansatz (and maybe the Monte-Carlo reference) is not present.
2- More context and discussion of the validity of the methods beyond spin systems, e.g. for exhibiting sign problems in Monte-Carlo techniques, would strengthen the manuscript.
Recommendation
Ask for minor revision
Strengths
1- Novelity of the proposed framework
Weaknesses
1- Lack of performance compared to other methods 2- Lack of sufficient comparison with other methods
Report
The work presents a novel framework for ground-state energy estimation, sample-based neural diagonalization (SND) and adaptive-basis SND (AB-SND).
Overall, this is a novel idea that is not performing well.
- The authors compare SND and AB-SND against "Jordan-Wigner transoformations" for the 1D model, and some quantum Monte-Carlo simlations for the 2D models for benchmarking. (The Jordan-Wigner transformation is not a method for estimating the ground-state energy, so I assume some other methods are also involved.) I do not see any advantage for (AB-)SND over those standard methods. The authors should demonstrate or at least make a convincing argument that the proposed method can be advantageous over existing methods.
- There are some standard techniques such as the Hartree-Fock calculation to obtain a set of basis rotations that gives a basis where the exact ground state has concentrated configurations. This type of comparison is also lacking in this manuscript.
Recommendation
Reject
Strengths
(2) Thorough analysis and benchmarking.
(3) Potential applicability on quantum hardware.
Weaknesses
Report
The paper is well written, clearly structured, and includes a thorough analysis and benchmarking of the presented algorithms. It meets the journal’s standards, opens a new pathway within an existing research direction, and provides a novel link between different areas of research. Therefore, I recommend publication.
Given the computational focus of the paper and the availability of a public code repository, the manuscript could also be a good fit for SciPost Physics Codebases. I leave this decision to the editor and authors.
Requested changes
A few points should be addressed before publication:
(i) The models used for testing (in particular the Ising models) are relatively simple, as acknowledged by the authors. It would be valuable to apply the method to a more challenging problem and see the comparison between an NQS-based calculation and the SND ansatz with comparable computational cost, e.g., in a frustrated two-dimensional model.
(ii) A straightforward modification would be to employ momentum eigenstates as basis states, which would allow the computation of excited states. Did the authors consider this?
(iii) Do the authors have an understanding of the scaling of eps(S) in Fig. 3 for the AB-SND? (I am impressed by how well S=10 works for AB-SND.)
(iv) For large system sizes, one might expect the truncated Hamiltonian to split into disconnected sectors. Does the neural network implicitly select connected basis states to ensure a lower variational energy?
(v) I assume that entangled ground states (e.g., with entanglement entropy scaling with linear system size L in two dimensions) will not be well captured by this method. Do the authors agree with this assessment?
(vi) The appendix shows that the relative error is maximal at h_c in the limit S to infinity. Do the authors understand why, for finite S, the maximum does not occur exactly at h_c?
(vii) How do the authors assess the challenges associated with noise and circuit depth on a real quantum machine?
Minor Comments:
(i) The y-axis in Fig. 3 should be presented on a logarithmic scale for better readability and comparison to Fig. 2.
(ii) Figures 4 and 5 currently include substantial white space and could be reformatted to make better use of the available area.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)