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arXiv:1905.01625 · IEEE Transactions on Control Systems Technology

Quantum Hamiltonian Identification With Classical Colored Measurement Noise

Lingyu Tan, Daoyi Dong, Dewei Li, Shibei Xue

Published May 8, 2020• 15 citations
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Abstract

In this brief, we present a Hamiltonian-identification method for a closed quantum system whose time-trace observables are measured with colored measurement noise. The dynamics of the quantum system are described by a Liouville equation that can be converted into a coherence vector representation. Since the measurement process is disturbed by classical colored noise, we introduce an augmented system model to describe the total dynamics, where the classical colored noise is parameterized. Based on the augmented system model as well as the measurement data, we can find a realization of the quantum system with unknown parameters by employing an eigenstate-realization algorithm. The unknown parameters can be identified using a transfer-function-based technique. An example of a two-qubit system with colored measurement noise is shown to verify the effectiveness of our method.

DOI
10.1109/tcst.2020.2991611

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Precise identification of parameters governing quantum processes is a critical task for quantum information and communication technologies. In this Letter, we consider a setting where system evolution is determined by a parametrized Hamiltonian, and the task is to estimate these parameters from temporal records of a restricted set of system observables (time traces). Based on the notion of system realization from linear systems theory, we develop a constructive algorithm that provides estimates of the unknown parameters directly from these time traces. We illustrate the algorithm and its robustness to measurement noise by applying it to a one-dimensional spin chain model with variable couplings.
In this paper, we focus on detection of quantum colored noise and present a novel detection method which employs a quantum harmonic oscillator as a noise probe. With respect to an unknown spectrum of the quantum colored noise, quantum spectral decomposition theorem is applied to obtain a linear system model for the internal modes of the quantum colored noise, where the spectrum is parameterized. Then we establish a linear augmented model for the whole system which consists of a probe of a quantum oscillator and the quantum linear system model of the quantum colored noise. Due to the equivalence between the augmented model and the actual system, we estimate unknown parameters of the linear system model for the noise by solving an optimization problem. The effectiveness of our method is verified by an example of identification of quantum Lorentzian noise.
In this paper, we present a gradient algorithm for identifying unknown parameters in an open quantum system from the measurements of time traces of local observables. The open system dynamics is described by a general Markovian master equation based on which the Hamiltonian identification problem can be formulated as minimizing the distance between the real time traces of the observables and those predicted by the master equation. The unknown parameters can then be learned with a gradient descent algorithm from the measurement data. We verify the effectiveness of our algorithm in a circuit QED system described by a Jaynes-Cummings model whose Hamiltonian identification has been rarely considered. We also show that our gradient algorithm can learn the spectrum of a non-Markovian environment based on an augmented system model.
The dynamics of an unknown closed quantum system can be characterized by a Hamiltonian. We discuss the identification of this Hamiltonian from quantum process tomography data at several time points for systems with a time-independent Hamiltonian. Quantum process tomography (QPT) yields a set of unitary operators and allows us to convert the identification problem into an estimation problem for the eigenfrequencies of the system. Spectral decomposition of Hermitian and unitary operators shows that process tomography needs to be performed at least at two different time points and the identifiability of the Hamiltonian depends on the choice of these points. Sufficient conditions on the timing to ensure complete identifiability of the Hamiltonian are presented under the assumption that unitary matrices obtained from QPT are exact without numerical errors.
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