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arXiv:1303.3771 · IEEE Transactions on Automatic Control

System Identification for Passive Linear Quantum Systems

Mădălin Guţǎ, Naoki Yamamoto

Published Jun 22, 2015• 61 citations
arXivHTMLPDF

Abstract

System identification is a key enabling component for the implementation of quantum technologies, including quantum control. In this paper, we consider the class of passive linear input-output systems, and investigate several basic questions: 1) which parameters can be identified? 2) Given sufficient input-output data, how do we reconstruct the system parameters? 3) How can we optimize the estimation precision by preparing appropriate input states and performing measurements on the output? We show that minimal systems can be identified up to a unitary transformation on the modes, and systems satisfying a Hamiltonian connectivity condition called “infecting” are completely identifiable. We propose a frequency domain design based on a Fisher information criterion, for optimizing the estimation precision for coherent input state. As a consequence of the unitarity of the transfer function, we show that the Heisenberg limit with respect to the input energy can be achieved using non-classical input states.

DOI
10.1109/tac.2015.2448491

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The purpose of this paper is to investigate system identification for single-input--single-output general (active or passive) quantum linear systems. For a given input we address the following questions: (1) Which parameters can be identified by measuring the output? (2) How can we construct a system realization from sufficient input-output data? We show that for time-dependent inputs, the systems which cannot be distinguished are related by symplectic transformations acting on the space of system modes. This complements a previous result of Gu\ifmmode \mbox{\c{t}}\else \c{t}\fi{}\ifmmode \u{a}\else \u{a}\fi{} and Yamamoto [IEEE Trans. Autom. Control 61, 921 (2016)] for passive linear systems. In the regime of stationary quantum noise input, the output is completely determined by the power spectrum. We define the notion of global minimality for a given power spectrum, and characterize globally minimal systems as those with a fully mixed stationary state. We show that in the case of systems with a cascade realization, the power spectrum completely fixes the transfer function, so the system can be identified up to a symplectic transformation. We give a method for constructing a globally minimal subsystem direct from the power spectrum. Restricting to passive systems the analysis simplifies so that identifiability may be completely understood from the eigenvalues of a particular system matrix.
The system identification problem is to estimate dynamical parameters from the output data, obtained by performing measurements on the output fields. We investigate system identification for quantum linear systems. Our main objectives are to address the following general problems: (1) Which parameters can be identified by measuring the output? (2) How can we construct a system realisation from sufficient input-output data? (3) How well can we estimate the parameters governing the dynamics? We investigate these problems in two contrasting approaches; using time-dependent inputs or time-stationary (quantum noise) inputs.
The aim of quantum system identification is to estimate the ingredients inside a black box, in which some quantum-mechanical unitary process takes place, by just looking at its input-output behavior. Here we establish a basic and general framework for quantum system identification, that allows us to classify how much knowledge about the quantum system is attainable, in principle, from a given experimental setup. We show that controllable closed quantum systems can be estimated up to unitary conjugation. Prior knowledge on some elements of the black box helps the system identification. We present an example in which a Bell measurement is more efficient to identify the system. When the topology of the system is known, the framework enables us to establish a general criterion for the estimability of the coupling constants in its Hamiltonian.
In this paper, we develop a system identification algorithm to identify a model for unknown linear quantum systems driven by time-varying coherent states, based on empirical single-shot continuous homodyne measurement data of the system’s output. The proposed algorithm identifies a model that satisfies the physical realizability conditions for linear quantum systems, challenging constraints not encountered in classical (non-quantum) linear system identification. Numerical examples on a multiple-input multiple-output optical cavity model are presented to illustrate an application of the identification algorithm.
Identifying an unknown unitary quantum process is a fundamental task in the development of quantum technology. In this paper, we propose an efficient identification algorithm for estimating unitary processes. In the method, input pure states are used and a fast quantum state tomography algorithm is developed to reconstruct the output states. Then the information of the output states is used to estimate the unitary process. The identification algorithm has computational complexity O(d3) for a d-dimensional system. Numerical results show that the proposed identification algorithm is much more efficient than the maximum likelihood estimation method and works well for input mixed states with high purity. An analytical upper bound for the identification error is also provided, and numerical simulations and experimental results on quantum optical systems verify the theoretical results.
The identification of time-varying parameters (e.g., directly inaccessible in situ signals in vacuum and low-temperature environments) is prevalent for characterizing the dynamics of quantum processes. Under certain circumstances, they can be identified from time-resolved measurements via Ramsey interferometry experiments, but only with specially designed probe systems can the parameters be explicitly read out, and a rigorous identifiability analysis is lacking, i.e., whether the measurement data are sufficient for unambiguous identification. In this paper we formulate this problem as the invertibility of the input-output mapping associated with the quantum system for which an algebraic identifiability criterion is derived based on the system's relative degree. The invertibility analysis also leads to an inversion-based algorithm for numerically identifying the parameters, which is computationally much more efficient than nonlinear regression methods. The effectiveness of the criterion are demonstrated by numerical examples.