Основные понятия
We present a new quantum algorithm that can efficiently learn the structure and parameters of an unknown local Hamiltonian from its real-time evolution, without any prior knowledge about the interaction terms. The algorithm achieves Heisenberg-limited scaling in the total evolution time and constant time resolution.
Аннотация
The key highlights and insights of the content are:
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The authors initiate the study of Hamiltonian structure learning, where the goal is to recover an unknown local Hamiltonian H = ∑m
a=1 λaEa without prior knowledge of the interaction terms Ea.
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They present a new algorithm that solves the challenging structure learning problem, while also resolving other open questions in Hamiltonian learning. The algorithm has the following appealing properties:
- It does not need to know the Hamiltonian terms in advance.
- It works beyond the short-range setting, extending to Hamiltonians where the sum of terms interacting with a qubit has bounded norm.
- It achieves Heisenberg-limited scaling in the total evolution time and constant time resolution.
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The algorithm works by recursively improving an initial estimate of the Hamiltonian, using a novel Trotter approximation that allows for constant-time resolution. It also employs a Goldreich-Levin-like subroutine to efficiently identify the large Hamiltonian coefficients without knowing the interaction terms.
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As applications, the authors show that their algorithm can learn Hamiltonians exhibiting power-law decay up to accuracy ε with total evolution time beating the standard limit of 1/ε2.
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The authors demonstrate that their techniques can achieve fixed-parameter tractable classical running time in the locality of the Hamiltonian, in contrast to prior algorithms that scale linearly with the locality.
Статистика
The content does not provide any specific numerical data or metrics to support the key claims. It focuses on describing the algorithmic techniques and theoretical guarantees.
Цитаты
"We initiate the study of Hamiltonian structure learning from real-time evolution: given the ability to apply e−iHt for an unknown local Hamiltonian H = ∑m
a=1 λaEa on n qubits, the goal is to recover H."
"To our knowledge, no prior algorithm with Heisenberg-limited scaling existed with even one of these properties."
"Curiously, as first proved by Huang, Tong, Fang, and Su [HTFS23], to estimate H to ε error, one can achieve the "Heisenberg-limited" scaling ttotal = 1/ε, better than the "standard limit" of 1/ε2 which one might expect."