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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 · Sebastiano Pilati |
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| Preprint Link: | scipost_202509_00030v2 (pdf) |
| Date submitted: | Jan. 26, 2026, 3 p.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
Author comments upon resubmission
We thank the Referees for their comments and overall positive assessments. Following their remarks, and in particular the objection by Referee #2, in the revised manuscript we provided relevant clarifications and additional analyses. In particular: i) we included a fermionic testbed considering a small quantum chemistry problem, ii) we expanded the analysis on (simulated) quantum hardware including a comparison with popular variational quantum algorithms, and iii) we investigated the role of hardware noise. We hope that the revised manuscript will be considered suitable for publication in SciPost Physics. Hereafter, we report the list of changes implemented in the revised manuscript.
List of changes
-) Extended the analysis on (simulated) quantum hardware, including a direct comparison between adaptive-basis SQD, standard SQD, and the variational quantum eigensolver (VQE).
-) Included a systematic study of the impact of hardware noise (modeled via depolarizing noise), showing improved robustness of the adaptive-basis approach compared to common variational quantum algorithms.
-) Clarified the role and scope of comparisons with existing classical methods (e.g., exact solutions in 1D, quantum Monte Carlo for sign-problem-free models).
-) Expanded the discussion of computational resources, including training costs and scalability, with comparisons between SND/AB-SND and NQS-based sampling.
-) Improved the presentation and discussion of challenging regimes (e.g., frustration in the 2D spin glass).
-) Added remarks in the conclusion and outlook on alternative basis choices, possible extensions to excited states, and entanglement-related limitations.
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My initial comment that "Overall, this is a novel idea that is not performing well." has not been changed in the revised manuscript.
Furthermore, we wish to clarify that the computational approaches we introduce, namely, neural sampling for SBD and the adaptive basis approach, are not intended to replace existing numerical methods, but rather to be integrated within classical and, chiefly, quantum algorithms for ground-state energy estimation. This point is better remarked in the revised manuscript.
Even in that case, at least some comparison with other classical methods such as CISD should be made. In the current version of the manuscript, I do not see any advantage of the proposed method except compared with "HF" which is much less costly compared with SND, or with "SBD", which is not widely used classically. In particular, the current result for LiH shows that the proposed method performs better than HF and worse than exact, which is trivial.
Also, my previous comment of
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. has not been addressed as far as I see. This question could be at least partially answered if the authors try both SND and AB-SND on LiH to see if AB can make any difference, because a standard LiH Hamiltonian is typically generated after operating the Hartree-Fock calculation. If AB is effective even after the Hartree-Fock calculation, it is more convincing that the AB is an effective method.
leading to a suppression of the effects of hardware errors. If this part is referring to the result of Fig. 6, the comparison is not properly addressing the issue of hardware noise. AB-SND requires deeper circuits for basis rotation, and the true question here is if the error due to the deeper circuits can be compensated by the better algorithmic accuracy due to the basis rotation. The comparison is not taking the former effect into account, so it is trivial that AB-SND, which is doing something extra compared to SND, performs better.
Recommendation
Reject