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IEEE Conference on Decision and Control (CDC)

Hamiltonian Identification of Quantum Networks from Sequential Boolean Measurements

Biqiang Mu, Hongsheng Qi, Ian R. Petersen, Guodong Shi

Published Dec 14, 2020• 1 citation
Publisher

Abstract

Hamiltonian identification plays a key role in learning the dynamics of closed quantum systems. A novel strategy for Hamiltonian identification is developed in the paper by applying repeated projective measurements with a fixed period. The corresponding measurement outcome sequence forms a time-homogeneous Markov chain, the transition matrix of which is a quadratic function of the Hamiltonian. As a result, the identification method is achieved by two steps: 1) learning the transition matrix of the observed Markov chains; 2) identifying the Hamiltonian using the function relating it to the transition matrix, which is transformed to a manifold optimization problem. A simulation example is provided to show the efficacy of the proposed method.

DOI
10.1109/cdc42340.2020.9304353

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Hamiltonian identification plays a key role in learning the dynamics of closed quantum systems. A novel strategy for Hamiltonian identification is developed in the paper by applying repeated projective measurements with a fixed period. The corresponding measurement outcome sequence forms a time-homogeneous Markov chain, the transition matrix of which is a quadratic function of the Hamiltonian. As a result, the identification method is achieved by two steps: 1) learning the transition matrix of the observed Markov chains; 2) identifying the Hamiltonian using the function relating it to the transition matrix, which is transformed to a manifold optimization problem. A simulation example is provided to show the efficacy of the proposed method.
An approximate quantum Hamiltonian identification algorithm is presented with the assumption that the system initial state and observation matrix can be set appropriately. We sample the system with a fixed period and using the sampled data we estimate the Hamiltonian based on a Taylor expansion of the matrix exponential function. We prove the estimation error is linear in the variance of the additive Gaussian noise. We also propose a heuristic formula to find the order of magnitude of the optimal sampling period. Two numerical examples are presented to validate the theoretical results.
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.
By considering the identification problem of unknown but fixed Hamiltonian H = S 0 σ 0 + Σ i = x, y, z S i σ i where σ i (i = x, y, z) are pauli matrices and σ 0 = I, we explore the feasibility and limitation of empirically determining the Hamiltonian parameters for quantum systems under experimental conditions imposed by projective measurements and initialization procedures. It may be unsurprising to physicists that one can not obtain the knowledge of So no matter what kind of projective measurements and initialization are permitted, but the observation draws our attention to the importance of the parameter identifiability under different experimental condition. It has also been revealed that one can obtain the knowledge of |S z | and S x 2 + S y 2 at most when only the projective measurement {|0⟩⟨0|, |1⟩⟨1|} is permitted to perform on and initialize the qubit. Further more, we demonstrated that it is feasible to distinguish |S x |, |S y |, and |S Z | even without any a priori information about Hamiltonian if at least two kinds of projective measurement or initialization procedures are permitted. It should be emphasized that both projective measurements and initialization procedures play an important role in quantum system identification.
Identifying parameters in the system Hamiltonian is a vitally important task in the development of quantum technology. This paper investigates the problem of Hamiltonian identification for closed quantum systems and develops a new algorithm to achieve the task of identifying the Hamiltonian. Under the framework of standard quantum process tomography and without prior assumptions, we first convert the problem of Hamiltonian identification into an optimization problem and then design an iterative algorithm to numerically solve this problem. Numerical results for a two-qubit system are presented to show the effectiveness of the proposed algorithm through analyzing the mean squared error.
In this paper, we study dynamical quantum networks which evolve according to Schrödinger equations but subject to sequential local or global quantum measurements. A network of qubits forms a composite quantum system whose state undergoes unitary evolution in between periodic measurements, leading to hybrid quantum dynamics with random jumps at discrete time instances along a continuous orbit. The measurements either act on the entire network of qubits, or only a subset of qubits. First of all, we reveal that this type of hybrid quantum dynamics induces probabilistic Boolean recursions representing the measurement outcomes. With global measurements, it is shown that such resulting Boolean recursions define Markov chains whose state-transitions are fully determined by the network Hamiltonian and the measurement observables. Particularly, we establish an explicit and algebraic representation of the underlying recursive random mapping driving such induced Markov chains. Next, with local measurements, the resulting probabilistic Boolean dynamics is shown to be no longer Markovian. The state transition probability at any given time becomes dependent on the entire history of the sample path, for which we establish a recursive way of computing such non-Markovian probability transitions. Finally, we adopt the classical bilinear control model for the continuous Schrödinger evolution, and show how the measurements affect the controllability of the quantum networks.
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.