insight - Quantum Computing - # Hamiltonian Structure Learning

Efficient Quantum Algorithm for Learning Local Hamiltonians from Real-Time Evolution

Q: How can the techniques developed in this work be extended to learn more general quantum dynamics, such as Lindbladian evolution or open quantum systems

In the context of learning more general quantum dynamics, such as Lindbladian evolution or open quantum systems, the techniques developed in this work can be extended by adapting the algorithms to handle the specific characteristics of these systems. For Lindbladian evolution, which describes the time evolution of open quantum systems under the influence of noise and decoherence, the key challenge lies in dealing with the non-unitary nature of the dynamics. One approach could be to modify the estimation improvement subroutine to account for the Lindblad operators and their corresponding coefficients. By incorporating the structure of Lindbladian evolution into the algorithm, it may be possible to estimate the parameters of the Lindblad operators and the corresponding Hamiltonian terms simultaneously. Additionally, for open quantum systems, where the dynamics are influenced by interactions with an external environment, the algorithms can be adapted to incorporate the effects of environmental noise. This may involve introducing additional terms in the Hamiltonian to model the interaction with the environment and developing techniques to estimate these terms from experimental data. Overall, by customizing the algorithms to capture the specific features of Lindbladian evolution and open quantum systems, it is possible to extend the techniques developed in this work to learn more general quantum dynamics beyond simple Hamiltonians.

Q: What are the limitations of the assumptions made in this work, such as the bounded B1 norm of the Hamiltonian, and how can they be relaxed further

The assumptions made in this work, such as the bounded B1 norm of the Hamiltonian, serve as simplifying constraints that enable the development of efficient algorithms for structure learning from real-time evolution. However, these assumptions also impose limitations on the applicability of the algorithms to more general quantum systems. One limitation of the bounded B1 norm assumption is that it restricts the class of Hamiltonians that can be effectively learned using the proposed techniques. To relax this assumption, one possible approach is to consider more general classes of Hamiltonians with unbounded norms or non-local interactions. This would require adapting the algorithm to handle a wider range of interaction strengths and structures, potentially by introducing regularization techniques or alternative parameterizations. Another limitation is the assumption of a fixed locality for the Hamiltonian terms. Relaxing this assumption would involve developing algorithms that can handle varying degrees of locality, allowing for interactions that extend over larger distances or involve more qubits. This would require novel approaches to structure learning that can capture the complexity of non-local interactions while maintaining efficiency and accuracy. By addressing these limitations and relaxing the assumptions made in the work, it may be possible to extend the applicability of the algorithms to a broader class of quantum systems with more diverse dynamics and interactions.

Q: Can the ideas of efficient structure learning and Goldreich-Levin-like subroutines be applied to other inverse problems in quantum computing, such as learning quantum circuits or quantum channels

The ideas of efficient structure learning and Goldreich-Levin-like subroutines developed in this work can indeed be applied to other inverse problems in quantum computing, such as learning quantum circuits or quantum channels. For learning quantum circuits, the algorithms can be adapted to estimate the parameters of the gates and operations in the circuit from experimental data. By treating the circuit as a parameterized quantum evolution, similar techniques to those used for Hamiltonian learning can be employed to efficiently learn the structure and parameters of the circuit. Similarly, for learning quantum channels, which describe the evolution of quantum states under the influence of noise and decoherence, the algorithms can be modified to estimate the parameters of the channel operations. By formulating the problem as a structure learning task for the quantum channel, it is possible to apply the same principles of efficient estimation and parameter recovery to learn the dynamics of the channel. Overall, the concepts of efficient structure learning and Goldreich-Levin-like subroutines can be generalized to a variety of inverse problems in quantum computing, providing a versatile framework for characterizing and learning the dynamics of quantum systems beyond Hamiltonians.

Conceitos Básicos

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.

Resumo

The key highlights and insights of the content are:

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.
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.
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.
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.
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.

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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.

Citações

"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."

Principais Insights Extraídos De

Structure learning of Hamiltonians from real-time evolution

by Ainesh Baksh... às arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00082.pdf

Structure learning of Hamiltonians from real-time evolution

Perguntas Mais Profundas

How can the techniques developed in this work be extended to learn more general quantum dynamics, such as Lindbladian evolution or open quantum systems

In the context of learning more general quantum dynamics, such as Lindbladian evolution or open quantum systems, the techniques developed in this work can be extended by adapting the algorithms to handle the specific characteristics of these systems.
For Lindbladian evolution, which describes the time evolution of open quantum systems under the influence of noise and decoherence, the key challenge lies in dealing with the non-unitary nature of the dynamics. One approach could be to modify the estimation improvement subroutine to account for the Lindblad operators and their corresponding coefficients. By incorporating the structure of Lindbladian evolution into the algorithm, it may be possible to estimate the parameters of the Lindblad operators and the corresponding Hamiltonian terms simultaneously.
Additionally, for open quantum systems, where the dynamics are influenced by interactions with an external environment, the algorithms can be adapted to incorporate the effects of environmental noise. This may involve introducing additional terms in the Hamiltonian to model the interaction with the environment and developing techniques to estimate these terms from experimental data.
Overall, by customizing the algorithms to capture the specific features of Lindbladian evolution and open quantum systems, it is possible to extend the techniques developed in this work to learn more general quantum dynamics beyond simple Hamiltonians.

What are the limitations of the assumptions made in this work, such as the bounded B1 norm of the Hamiltonian, and how can they be relaxed further

The assumptions made in this work, such as the bounded B1 norm of the Hamiltonian, serve as simplifying constraints that enable the development of efficient algorithms for structure learning from real-time evolution. However, these assumptions also impose limitations on the applicability of the algorithms to more general quantum systems.
One limitation of the bounded B1 norm assumption is that it restricts the class of Hamiltonians that can be effectively learned using the proposed techniques. To relax this assumption, one possible approach is to consider more general classes of Hamiltonians with unbounded norms or non-local interactions. This would require adapting the algorithm to handle a wider range of interaction strengths and structures, potentially by introducing regularization techniques or alternative parameterizations.
Another limitation is the assumption of a fixed locality for the Hamiltonian terms. Relaxing this assumption would involve developing algorithms that can handle varying degrees of locality, allowing for interactions that extend over larger distances or involve more qubits. This would require novel approaches to structure learning that can capture the complexity of non-local interactions while maintaining efficiency and accuracy.
By addressing these limitations and relaxing the assumptions made in the work, it may be possible to extend the applicability of the algorithms to a broader class of quantum systems with more diverse dynamics and interactions.

Can the ideas of efficient structure learning and Goldreich-Levin-like subroutines be applied to other inverse problems in quantum computing, such as learning quantum circuits or quantum channels

The ideas of efficient structure learning and Goldreich-Levin-like subroutines developed in this work can indeed be applied to other inverse problems in quantum computing, such as learning quantum circuits or quantum channels.
For learning quantum circuits, the algorithms can be adapted to estimate the parameters of the gates and operations in the circuit from experimental data. By treating the circuit as a parameterized quantum evolution, similar techniques to those used for Hamiltonian learning can be employed to efficiently learn the structure and parameters of the circuit.
Similarly, for learning quantum channels, which describe the evolution of quantum states under the influence of noise and decoherence, the algorithms can be modified to estimate the parameters of the channel operations. By formulating the problem as a structure learning task for the quantum channel, it is possible to apply the same principles of efficient estimation and parameter recovery to learn the dynamics of the channel.
Overall, the concepts of efficient structure learning and Goldreich-Levin-like subroutines can be generalized to a variety of inverse problems in quantum computing, providing a versatile framework for characterizing and learning the dynamics of quantum systems beyond Hamiltonians.