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arXiv:1808.09607

Nonlinear regression based on a hybrid quantum computer

Dan-Bo Zhang, Shi-Liang Zhu, Z. D. Wang

Published Aug 29, 2018• 4 citations
arXivHTMLPDF

Abstract

Incorporating nonlinearity into quantum machine learning is essential for learning a complicated input-output mapping. We here propose quantum algorithms for nonlinear regression, where nonlinearity is introduced with feature maps when loading classical data into quantum states. Our implementation is based on a hybrid quantum computer, exploiting both discrete and continuous variables, for their capacity to encode novel features and efficiency of processing information. We propose encoding schemes that can realize well-known polynomial and Gaussian kernel ridge regressions, with exponentially speed-up regarding to the number of samples.

DOI
10.48550/arxiv.1808.09607

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Incorporating nonlinearity into quantum machine learning is essential for learning a complicated input-output mapping. We here propose quantum algorithms for nonlinear regression, where nonlinearity is introduced with feature maps when loading classical data into quantum states. Our implementation is based on a hybrid quantum computer, exploiting both discrete and continuous variables, for their capacity to encode novel features and efficiency of processing information. We propose encoding schemes that can realize well-known polynomial and Gaussian kernel ridge regressions, with exponentially speed-up regarding to the number of samples.
In order to exploit quantum advantages, quantum algorithms are indispensable for operating machine learning with quantum computers. We here propose an intriguing hybrid approach of quantum information processing for quantum linear regression, which utilizes both discrete and continuous quantum variables, in contrast to existing wisdoms based solely upon discrete qubits. In our framework, data information is encoded in a qubit system, while information processing is tackled using auxiliary continuous qumodes via qubit-qumode interactions. Moreover, it is also elaborated that finite squeezing is quite helpful for efficiently running the quantum algorithms in a realistic setup. Comparing with an all-qubit approach, the present hybrid approach is more efficient and feasible for implementing quantum algorithms, still retaining exponential quantum speed-up.
Hybrid variational quantum algorithms are promising for solving practical problems, such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. However, variational quantum algorithms (derived from randomized hardware-efficient ansatz or adaptive ansatz) become a black box, not trustworthy for model interpretation, and not to mention for application deployment in informing critical decisions. In this paper, we construct the first interpretable quantum regression algorithm, in which the quantum state exactly encodes the classical data table and the variational parameters correspond directly to the regression coefficients, which are real numbers by construction, providing a high degree of model interpretability and minimal cost to optimize due to the right expressiveness. We also exploit the encoded data structure to reduce the gate complexity of computing the regression map. To reduce circuit depth in nonlinear regression, our algorithm can be extended by directly constructing nonlinear features via classical preprocessing, such as independent encoded column vectors. By design, the model performance is determined by the cost function measurement results C\mathcal{C}C synchronous to the mean squared errors (MSE) for the regression models. We derived the read-out errors induced by one-hot encoding and compact encoding; the required physical qubit resources are exponentially compressed for the compact encoding to be favorable for noisy quantum devices. We also derive the cost function dependent sample complexity $ \in \mathcal{O}\left(σ^{2}(\mathcal{C}) \ln (1/α)/ε^{2}\right)$ under the error budget εεε and confidence tolerance ααα.
A basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely, to efficiently perform computations in an intractably large Hilbert space. In this Letter we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyze the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. The kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualize the working principle with two-dimensional minibenchmark datasets.
Kernel methods are of current interest in quantum machine learning due to similarities with quantum computing in how they process information in high-dimensional feature (Hilbert) spaces. Kernels are believed to offer particular advantages when they cannot be computed classically, so a kernel matrix with indisputably nonclassical elements is desirable provided it can be generated efficiently in a particular physical machine. Kerr nonlinearities, known to be a route to universal continuous variable (CV) quantum computation, may be able to play this role for quantum machine learning. We propose a quantum hardware kernel implementation scheme based on superconducting quantum circuits. The scheme does not use qubits or quantum circuits but rather exploits the analogue features of Kerr coupled modes. Our approach is more akin to the growing number of analog machine learning schemes based on sampling quantum probabilities directly in an engineered device by stochastic quantum control.
Hybrid variational quantum algorithms (VQAs) are promising for solving practical problems such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. However, with typical random ansatz or quantum alternating operator ansatz, derived variational quantum algorithms become a black box that cannot be trusted for model interpretation, not to mention deploying as applications in informing critical decisions: the results of these variational parameters are just rotational angles for the quantum gates and have nothing to do with interpretable values that a model can provide directly. In this paper, we construct the first interpretable quantum regression algorithm, in which the quantum state exactly encodes the classical data table and the variational parameters correspond directly to the regression coefficients, which are real numbers by construction, providing a high degree of model interpretability and minimal cost to optimize due to the right expressiveness. We also take advantage of the encoded data structure to reduce the time complexity of computing the regression map. To shorten the circuit depth for nonlinear regression, our algorithm can be extended by building nonlinear features by classical preprocessing as the independent encoded column vectors. Even though the realization of compressed encoding in superconducting qubits has been achieved by the less noisy compressed encoding recently by the authors, we envision potential quantum utilities with multi-qubit gates implemented in neutral cold atoms and ions.
With the significant advancement in quantum computation during the past couple of decades, the exploration of machine-learning subroutines using quantum strategies has become increasingly popular. Gaussian process regression is a widely used technique in supervised classical machine learning. Here we introduce an algorithm for Gaussian process regression using continuous-variable quantum systems that can be realized with technology based on photonic quantum computers under certain assumptions regarding distribution of data and availability of efficient quantum access. Our algorithm shows that by using a continuous-variable quantum computer a dramatic speedup in computing Gaussian process regression can be achieved, i.e., the possibility of exponentially reducing the time to compute. Furthermore, our results also include a continuous-variable quantum-assisted singular value decomposition method of nonsparse low rank matrices and forms an important subroutine in our Gaussian process regression algorithm.