Hamiltonian Learning and Certification via Eigenphase Engineering
Abstract
We study Hamiltonian learning for k-local Hamiltonians at large step size, using only forward time evolution at integer multiples of a fixed time interval. For this task, our new technique, which we call eigenphase engineering, achieves Heisenberg-limited learning with a step size that is optimal up to logarithmic factors. For an unknown s-sparse Hamiltonian with fixed locality k and |H|op≤Λ, our algorithm recovers its Pauli coefficients to ℓ2 error ε in total evolution time Õ(s/ε), matching the best known scaling while allowing a near-maximal step size Θ̃(1/Λ). As an independent contribution, we strengthen the evolution-time lower bound to Ωk(s1−1/(2k)/ε), placing our algorithm within a factor Õ(s1/(2k)) of being optimal. Eigenphase engineering departs from approaches based on dynamical approximation or iteratively canceling the unknown Hamiltonian. It encodes energy expectation differences in the perturbative response of eigenphases of the actual controlled evolution, then extracts them through phase estimation and high-order extrapolation. This avoids the need for polynomially shorter evolution steps as the target precision improves. A version using only single-qubit operations retains the same step size with a √n overhead in ℓ2 learning time for n qubits. The method also enables tolerant Hamiltonian certification and estimation of a specified Pauli coefficient, the latter without locality or sparsity assumptions. These results establish eigenphase engineering as a versatile approach to extracting Hamiltonian information with infrequent control.
Related
- Hamiltonian locality testing and certification do not achieve the Heisenberg limit
- Robust Hamiltonian engineering with subensemble control
- Hamiltonian engineering via pulses: beyond group averaging
- Fermionic Hamiltonian engineering with local control
- Fast Hamiltonian engineering from cut polytope geometry
BibTeX
@misc{shin2026hamiltonian,
title = {{Hamiltonian Learning and Certification via Eigenphase Engineering}},
author = {Myeongjin Shin and Yu Tong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26596},
eprint = {2609.26596},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26596},
}
RIS
TY - UNPB
TI - Hamiltonian Learning and Certification via Eigenphase Engineering
AU - Myeongjin Shin
AU - Yu Tong
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26596
UR - https://arxiv.org/abs/2609.26596
AB - We study Hamiltonian learning for k-local Hamiltonians at large step size, using only forward time evolution at integer multiples of a fixed time interval. For this task, our new technique, which we call eigenphase engineering, achieves Heisenberg-limited learning with a step size that is optimal up to logarithmic factors. For an unknown s-sparse Hamiltonian with fixed locality k and |H|_op≤Λ, our algorithm recovers its Pauli coefficients to ℓ_2 error ε in total evolution time Õ(s/ε), matching the best known scaling while allowing a near-maximal step size Θ̃(1/Λ). As an independent contribution, we strengthen the evolution-time lower bound to Ω_k(s^1−1/(2k)/ε), placing our algorithm within a factor Õ(s^1/(2k)) of being optimal. Eigenphase engineering departs from approaches based on dynamical approximation or iteratively canceling the unknown Hamiltonian. It encodes energy expectation differences in the perturbative response of eigenphases of the actual controlled evolution, then extracts them through phase estimation and high-order extrapolation. This avoids the need for polynomially shorter evolution steps as the target precision improves. A version using only single-qubit operations retains the same step size with a √n overhead in ℓ_2 learning time for n qubits. The method also enables tolerant Hamiltonian certification and estimation of a specified Pauli coefficient, the latter without locality or sparsity assumptions. These results establish eigenphase engineering as a versatile approach to extracting Hamiltonian information with infrequent control.
ER -