[Paper Review] Nonlinear regression based on a hybrid quantum computer

Dan-Bo Zhang, Shi-Liang Zhu+1 more|arXiv (Cornell University)|Aug 29, 2018

Quantum Computing Algorithms and Architecture3 references4 citations

TL;DR

This paper proposes a hybrid quantum algorithm for nonlinear regression using both discrete qubits and continuous-variable qumodes to encode classical data via feature maps. By leveraging quantum random access memory (qRAM), the method achieves O(log M) runtime scaling with the number of training samples M, enabling exponential speedup over classical methods for polynomial and Gaussian kernel ridge regressions.

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.

Motivation & Objective

• To address the challenge of incorporating nonlinearity into quantum machine learning while preserving quantum speedup.
• To develop efficient quantum algorithms for nonlinear regression using hybrid quantum systems combining qubits and qumodes.
• To achieve exponential speedup in training sample size M for kernel ridge regression by exploiting quantum state encoding and qRAM.
• To explore novel feature maps based on quantum evolution for predicting quantum physical properties, such as ground state energies.
• To demonstrate that quantum feature maps can reproduce standard kernel ridge regressions (polynomial and Gaussian) with quantum advantage.

Proposed method

• Uses a hybrid quantum architecture combining discrete qubits and continuous-variable qumodes to encode classical data into quantum states.
• Employs feature maps that encode input vectors into entangled superpositions of qubit and qumode states, enabling nonlinear transformations.
• Applies quantum random access memory (qRAM) to prepare the encoded data state |ψ_A⟩ in O(log M) time, independent of data dimension N.
• Utilizes the HHL-like algorithm framework to solve the linear system arising from kernel ridge regression, with quantum state preparation and measurement.
• Constructs the kernel matrix via inner products of encoded quantum states, enabling efficient computation of polynomial and Gaussian kernels.
• Proposes a quantum evolution-based feature map using time-evolution under a parameterized Hamiltonian H′(a) to encode atomic configurations into quantum states for physical property prediction.

Experimental results

Research questions

• RQ1 Can nonlinear regression in quantum machine learning be achieved with provable quantum speedup using hybrid quantum systems?
• RQ2 How can feature maps be encoded in a hybrid quantum system of qubits and qumodes to realize standard kernel ridge regressions?
• RQ3 What is the runtime scaling of quantum kernel ridge regression when using qRAM for state preparation?
• RQ4 Can quantum evolution-based feature maps improve the prediction of quantum physical properties like ground state energies?
• RQ5 Is it possible to achieve exponential speedup in the number of training samples M while maintaining accuracy in nonlinear regression tasks?

Key findings

• The proposed algorithm achieves O(log M) runtime scaling for both polynomial and Gaussian kernel ridge regression, independent of the input dimension N.
• For polynomial kernel ridge regression, the runtime is O(log M), matching the speedup of linear regression and providing exponential speedup over classical O(M) methods.
• For Gaussian kernel ridge regression, the runtime scales as O(log M) when using qRAM to access hybrid quantum states of qubits and qumodes.
• The use of qRAM enables efficient preparation of the data encoding state |ψ_A⟩ in O(log M) time, crucial for achieving the exponential speedup.
• The method reproduces well-known kernel ridge regressions through quantum feature maps, demonstrating feasibility and efficiency on hybrid quantum hardware.
• Quantum evolution-based feature maps, such as |ϕ_a⟩ = e^{iH′(a)t₀}|0⟩, are proposed as a promising approach for predicting quantum physical properties from classical inputs like atomic configurations.