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Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any inverse temperature β, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of a n-qubit D-dimensional Hamiltonian to an additive error ϵ with sample complexity $$\tilde{O}\left(\frac{e^{\mathrm{poly}(\beta)}}{\beta^2\epsilon^2}\right)\log(n)$$. The protocol uses parallelizable local quantum measurements that act within bounded regions of the lattice and near-linear-time classical post-processing. Thus, our complexity is near optimal with respect to n, ϵ and is polynomially tight with respect to β. We also give a learning algorithm for Hamiltonians with bounded interaction degree with sample and time complexities of similar scaling on n but worse on β, ϵ. At the heart of our algorithm is the interplay between locality, the Kubo-Martin-Schwinger condition, and the operator Fourier transform at arbitrary temperatures.
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Parameterized Inverse Eigenvalue Problem for Quantum Sensing
A system of tunnel-coupled quantum dots is considered in the presence of an applied electric field. Given the measurements of differences between ground state and excited state energy levels as the electric field is varied, we seek to recover the quantum Hamiltonians that describe this system. We formulate this as a parameterized inverse eigenvalue problem and develop algebraic and computational methods for solving for parameters to represent these Hamiltonians. The results demonstrate that this approach is highly precise even when there is error present within the measurements. This theory could aid in the design of high resolution tunable quantum sensors.
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- PAR ID:
- 10505319
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- 2023 IEEE 9th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
- ISBN:
- 979-8-3503-4452-3
- Page Range / eLocation ID:
- 351 to 355
- Format(s):
- Medium: X
- Location:
- Herradura, Costa Rica
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/CAMSAP58249.2023.10403442
