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arXiv:1903.06569 · New Journal of Physics

Determining system Hamiltonian from eigenstate measurements without correlation functions

Shi‐Yao Hou, Ningping Cao, Sirui Lu, Yi Shen, Yiu‐Tung Poon, Bei Zeng

Published Jul 30, 2020• 17 citations
arXivHTMLPDF

Abstract

Abstract Local Hamiltonians arise naturally in physical systems. Despite their seemingly ‘simple’ local structures, exotic features such as non-local correlations and topological orders exhibit in eigenstates of these systems. Previous studies for recovering local Hamiltonians from measurements on an eigenstate ψ require information of nonlocal correlation functions. In this work, we argue that local measurements on ψ is enough to recover the Hamiltonian in most of the cases. Specially, we develop an algorithm to demonstrate the observation. Our algorithm is tested numerically for randomly generated local Hamiltonians of different system sizes and returns promising reconstructions with desired accuracy. Additionally, for random generated Hamiltonians (not necessarily local), our algorithm also provides precise estimations.

DOI
10.1088/1367-2630/abaacf

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Abstract Local Hamiltonians arise naturally in physical systems. Despite their seemingly ‘simple’ local structures, exotic features such as non-local correlations and topological orders exhibit in eigenstates of these systems. Previous studies for recovering local Hamiltonians from measurements on an eigenstate ψ require information of nonlocal correlation functions. In this work, we argue that local measurements on ψ is enough to recover the Hamiltonian in most of the cases. Specially, we develop an algorithm to demonstrate the observation. Our algorithm is tested numerically for randomly generated local Hamiltonians of different system sizes and returns promising reconstructions with desired accuracy. Additionally, for random generated Hamiltonians (not necessarily local), our algorithm also provides precise estimations.
We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the `` k -correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.
Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.
Reconstructing a system Hamiltonian through measurements on its eigenstates is an important inverse problem in quantum physics. Recently, it was shown that generic many-body local Hamiltonians can be recovered by local measurements without knowing the values of the correlation functions. In this work, we discuss this problem in more depth for different systems and apply supervised learning method via neural networks to solve it. For low-lying eigenstates, the inverse problem is well-posed, neural networks turn out to be efficient and scalable even with a shallow network and a small data set. For middle-lying eigenstates, the problem is ill-posed, we present a modified method based on transfer learning accordingly. Neural networks can also efficiently generate appropriate initial points for numerical optimization based on the BFGS method.
Abstract Recent works have shown that generic local Hamiltonians can be efficiently inferred from local measurements performed on their eigenstates or thermal states. Realistic quantum systems are often affected by dissipation and decoherence due to coupling to an external environment. This raises the question whether the steady states of such open quantum systems contain sufficient information allowing for full and efficient reconstruction of the system’s dynamics. We find that such a reconstruction is possible for generic local Markovian dynamics. We propose a recovery method that uses only local measurements; for systems with finite-range interactions, the method recovers the Lindbladian acting on each spatial domain using only observables within that domain. We numerically study the accuracy of the reconstruction as a function of the number of measurements, type of open-system dynamics and system size. Interestingly, we show that couplings to external environments can in fact facilitate the reconstruction of Hamiltonians composed of commuting terms.
A quantum state for being an eigenstate of some local Hamiltonian should be constrained by zero energy variance and consequently, the constraint is rather strong that a single eigenstate may uniquely determine the Hamiltonian. For non-Hermitian systems, it is natural to expect that determining the Hamiltonian requires a pair of both left and right eigenstates. Here, we observe that it can be sufficient to determine a non-Hermitian Hamiltonian from a single right or left eigenstate. Our approach is based on the quantum covariance matrix, where the solution of Hamiltonian corresponds to the complex null vector. Our scheme favors non-Hermitian Hamiltonian learning on experimental quantum systems, as only the right eigenstates there can be accessed. Furthermore, we use numerical simulations to examine the effects of measurement errors and show the stability of our scheme.
With the fast development of quantum technology, the sizes of both digital and analog quantum systems increase drastically. In order to have better control and understanding of the quantum hardware, an important task is to characterize the interaction, i.e., to learn the Hamiltonian, which determines both static and dynamic properties of the system. Conventional Hamiltonian learning methods either require costly process tomography or adopt impractical assumptions, such as prior information on the Hamiltonian structure and the ground or thermal states of the system. In this work, we present a robust and efficient Hamiltonian learning method that circumvents these limitations based only on mild assumptions. The proposed method can efficiently learn any Hamiltonian that is sparse on the Pauli basis using only short-time dynamics and local operations without any information on the Hamiltonian or preparing any eigenstates or thermal states. The method has a scalable complexity and a vanishing failure probability regarding the qubit number. Meanwhile, it performs robustly given the presence of state preparation and measurement errors and resiliently against a certain amount of circuit and shot noise. We numerically test the scaling and the estimation accuracy of the method for transverse field Ising Hamiltonian with random interaction strengths and molecular Hamiltonians, both with varying sizes and manually added noise. All these results verify the robustness and efficacy of the method, paving the way for a systematic understanding of the dynamics of large quantum systems.