Scaling Theory for Learning Low-Energy Quantum Subspaces
Abstract
Phase diagrams and energy surfaces require low-energy states across a family of Hamiltonians, but solving each parameter point independently is prohibitively expensive. An important question is not how a single labeled eigenstate behaves under parameter variation, but how much information about the low-energy sector is contained in a small set of reference wavefunctions. We show that the physically relevant object is the isolated low-energy subspace itself: it remains well defined through degeneracies and level crossings, and it can be learned efficiently from nearby sampled states. Working in this subspace, we prove that a sampling pattern cancelling the first q nonconstant response orders yields an energy-density error bounded by d2(q+1) for every retained level, where d is the parameter-space sampling distance. For a gapped local ground state on n sites, locality sharpens this to d2(nd2)q, identifying nd2 as the natural scaling variable governing the effectiveness of higher-order information from sampled wavefunctions. In a local analytic regime this gives a sampling cost K=𝒪([log(1/ε)]D) at fixed parameter dimension D. Numerical results on Heisenberg and transverse-field Ising chains confirm the distance scaling law and the nd2 collapse. Beyond characterizing learnability, the scaling behavior further provides a finite-size probe of phase transitions near criticality.
Related
- Linear-Scaling Quantum Transport from Machine-Learning Density Functional Theory Hamiltonians
- Edge energy fluctuation: A unified theory of friction scaling laws in twisted layered material interfaces
- Finite-size scaling theory of Rényi thermal entropy for second-order, first-order, and weakly first-order quantum phase transitions
- Thermal screening and critical scaling of quantum energy teleportation in a harmonic chain
- Flow of local quantum fluids: Conservation laws and vertex corrections from many-body linear-response theory with local self-energy
BibTeX
@misc{duan2026scaling,
title = {{Scaling Theory for Learning Low-Energy Quantum Subspaces}},
author = {Guijing Duan and Chen Mo and Di Luo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33777},
eprint = {2609.33777},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33777},
}
RIS
TY - UNPB
TI - Scaling Theory for Learning Low-Energy Quantum Subspaces
AU - Guijing Duan
AU - Chen Mo
AU - Di Luo
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33777
UR - https://arxiv.org/abs/2609.33777
AB - Phase diagrams and energy surfaces require low-energy states across a family of Hamiltonians, but solving each parameter point independently is prohibitively expensive. An important question is not how a single labeled eigenstate behaves under parameter variation, but how much information about the low-energy sector is contained in a small set of reference wavefunctions. We show that the physically relevant object is the isolated low-energy subspace itself: it remains well defined through degeneracies and level crossings, and it can be learned efficiently from nearby sampled states. Working in this subspace, we prove that a sampling pattern cancelling the first q nonconstant response orders yields an energy-density error bounded by d^2(q+1) for every retained level, where d is the parameter-space sampling distance. For a gapped local ground state on n sites, locality sharpens this to d^2(nd^2)^q, identifying nd^2 as the natural scaling variable governing the effectiveness of higher-order information from sampled wavefunctions. In a local analytic regime this gives a sampling cost K=𝒪([log(1/ε)]^D) at fixed parameter dimension D. Numerical results on Heisenberg and transverse-field Ising chains confirm the distance scaling law and the nd^2 collapse. Beyond characterizing learnability, the scaling behavior further provides a finite-size probe of phase transitions near criticality.
ER -