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Chinese Control Conference (CCC)

A Brief Review of Linear Regression Estimation in Quantum Tomography

Yanran Cheng, Zhihui Lou

Published Jul 1, 2020• 1 citation
Publisher

Abstract

Quantum tomography is one of the essential tasks in quantum information technology. Much effort has been devoted to increasing the precision and efficiency of quantum tomography. In this paper, we only focus on a specific tomography method, i.e., the linear regression estimation (LRE). We give a brief review on how to employ LRE for quantum state tomography, adaptive quantum state tomography and quantum Hamiltonian identification.

DOI
10.23919/ccc50068.2020.9188692

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Quantum tomography is one of the essential tasks in quantum information technology. Much effort has been devoted to increasing the precision and efficiency of quantum tomography. In this paper, we only focus on a specific tomography method, i.e., the linear regression estimation (LRE). We give a brief review on how to employ LRE for quantum state tomography, adaptive quantum state tomography and quantum Hamiltonian identification.
A simple yet efficient state reconstruction algorithm of linear regression estimation (LRE) is presented for quantum state tomography. In this method, quantum state reconstruction is converted into a parameter estimation problem of a linear regression model and the least-squares method is employed to estimate the unknown parameters. An asymptotic mean squared error (MSE) upper bound for all possible states to be estimated is given analytically, which depends explicitly upon the involved measurement bases. This analytical MSE upper bound can guide one to choose optimal measurement sets. The computational complexity of LRE is O(d(4)) where d is the dimension of the quantum state. Numerical examples show that LRE is much faster than maximum-likelihood estimation for quantum state tomography.
Quantum state tomography is one of the critical tasks in the development of quantum technology. This paper further investigates the adaptive linear regression estimation (ALRE) method that was developed recently for state reconstruction of qubit systems. In this method, adaptive measurements are used to improve the performance of state reconstruction where current measurement basis is adaptively optimized based on previous knowledge through minimizing an index related to the mean squared error (MSE). The numerical results on single-qubit and two-qubit systems demonstrate the effectiveness of this proposed adaptive estimation method over static one for quantum state reconstruction.
This paper proposes a recursively adaptive tomography protocol to improve the precision of quantum process estimation for finite dimensional systems. The problem of quantum process tomography is firstly formulated as a parameter estimation problem which can then be solved by the linear regression estimation method. An adaptive algorithm is proposed for the selection of subsequent input states given the previous estimation results. Numerical results show that the proposed adaptive process tomography protocol can achieve an improved level of estimation performance.
Abstract Adaptive techniques have great potential for wide application in enhancing the precision of quantum parameter estimation. We present an adaptive quantum state tomography protocol for finite dimensional quantum systems and experimentally implement the adaptive tomography protocol on two-qubit systems. In this adaptive quantum state tomography protocol, an adaptive measurement strategy and a recursive linear regression estimation algorithm are performed. Numerical results show that our adaptive quantum state tomography protocol can outperform tomography protocols using mutually unbiased bases and the two-stage mutually unbiased bases adaptive strategy, even with the simplest product measurements. When nonlocal measurements are available, our adaptive quantum state tomography can beat the Gill–Massar bound for a wide range of quantum states with a modest number of copies. We use only the simplest product measurements to implement two-qubit tomography experiments. In the experiments, we use error-compensation techniques to tackle systematic error due to misalignments and imperfection of wave plates, and achieve about a 100-fold reduction of the systematic error. The experimental results demonstrate that the improvement of adaptive quantum state tomography over nonadaptive tomography is significant for states with a high level of purity. Our results also show that this adaptive tomography method is particularly effective for the reconstruction of maximally entangled states, which are important resources in quantum information.
Quantum state tomography is a core task in quantum system identification. Real experimental conditions often deviate from nominal designs, introducing errors in both the measurement devices and the Hamiltonian governing the system's dynamics. In this paper, we investigate the robustness of quantum state tomography against such perturbations in both static and dynamic settings using linear regression estimation. We derive explicit bounds that quantify how bounded errors in the measurement devices and the Hamiltonian affect the mean squared error (MSE) upper bound in each scenario. Numerical simulations for qubit systems illustrate how these bounds scale with resources.
Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we design the optimal probe states for quantum detector tomography via linear regression estimation. In particular, two optimal criteria are adopted where one is related to upper bound of the mean squared error and the other concerns how robust the estimation result is with respect to measurement noise. We prove the optimal value based on the two criteria and provide two illustrative examples of optimal probe states.