In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that $\ell_2$ -regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results.In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that $\ell_2$-regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results.

arXiv:1904.11839Apr 26, 20190 citations

Quantum kernel estimation-based quantum support vector regression

Xiaojian Zhou, Jieyao Yu, Junfan Tan +1 more

Quantum Information ProcessingJan 23, 202427 citations

Quantum regularized least squares solver with parameter estimate

Changpeng Shao, Hua Xiang

Quantum Information ProcessingFeb 14, 20207 citations

Quantum algorithms for ridge regression

Chao‐Hua Yu, Fei Gao, Qiaoyan Wen

Ridge regression (RR), also called regularized linear regression, is an important machine learning technique which introduces a regularization hyperparameter to ordinary multiple linear regression for analyzing data suffering from multicollinearity. Here we provide an efficient quantum algorithm for RR. Specifically, by giving the technique of parallel Hamiltonian simulation that can simulate a number of Hermitian matrices in parallel, we develop a quantum version of $K$ -fold cross-validation approach that can efficiently estimate the predictive performance of RR. Our algorithm involves two phases: (1) we first use the quantum $K$ -fold cross-validation to efficiently determine an approximately optimal regularization hyperparameter for RR with which RR can achieve very good predictive performance, and (2) then generate a quantum state encoding the optimal fitting parameters of RR with such hyperparameter, which can be further utilized to predict new data. Our algorithm can handle nonsparse data matrices, and is exponentially faster than the classical algorithm for (low-rank) design matrices with relatively small elements and low condition numbers.Ridge regression (RR), also called regularized linear regression, is an important machine learning technique which introduces a regularization hyperparameter to ordinary multiple linear regression for analyzing data suffering from multicollinearity. Here we provide an efficient quantum algorithm for RR. Specifically, by giving the technique of parallel Hamiltonian simulation that can simulate a number of Hermitian matrices in parallel, we develop a quantum version of $K$-fold cross-validation approach that can efficiently estimate the predictive performance of RR. Our algorithm involves two phases: (1) we first use the quantum $K$-fold cross-validation to efficiently determine an approximately optimal regularization hyperparameter for RR with which RR can achieve very good predictive performance, and (2) then generate a quantum state encoding the optimal fitting parameters of RR with such hyperparameter, which can be further utilized to predict new data. Our algorithm can handle nonsparse data matrices, and is exponentially faster than the classical algorithm for (low-rank) design matrices with relatively small elements and low condition numbers.

Jul 29, 20176 citations

\ell_2

-regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results.

K

-fold cross-validation approach that can efficiently estimate the predictive performance of RR. Our algorithm involves two phases: (1) we first use the quantum

K

-fold cross-validation to efficiently determine an approximately optimal regularization hyperparameter for RR with which RR can achieve very good predictive performance, and (2) then generate a quantum state encoding the optimal fitting parameters of RR with such hyperparameter, which can be further utilized to predict new data. Our algorithm can handle nonsparse data matrices, and is exponentially faster than the classical algorithm for (low-rank) design matrices with relatively small elements and low condition numbers.