[Paper Review] Quantum locally linear embedding for nonlinear dimensionality reduction

Xi He, Li Sun+2 more|arXiv (Cornell University)|Oct 17, 2019

Quantum Computing Algorithms and Architecture36 references4 citations

TL;DR

This paper proposes two quantum algorithms—linear-algebra-based QLLE and variational QLLE (VQLLE)—for nonlinear dimensionality reduction using quantum computing. The linear-algebra-based QLLE achieves exponential speedup over classical LLE by leveraging quantum linear algebra subroutines, while VQLLE enables implementation on near-term quantum devices via a hybrid quantum-classical variational approach, both demonstrating effective manifold structure preservation in numerical experiments.

ABSTRACT

Reducing the dimension of nonlinear data is crucial in data processing and visualization. The locally linear embedding algorithm (LLE) is specifically a representative nonlinear dimensionality reduction method with well maintaining the original manifold structure. In this paper, we present two implementations of the quantum locally linear embedding algorithm (QLLE) to perform the nonlinear dimensionality reduction on quantum devices. One implementation, the linear-algebra-based QLLE algorithm, utilizes quantum linear algebra subroutines to reduce the dimension of the given data. The other implementation, the variational quantum locally linear embedding algorithm (VQLLE) utilizes a variational hybrid quantum-classical procedure to acquire the low-dimensional data. The classical LLE algorithm requires polynomial time complexity of $N$, where $N$ is the global number of the original high-dimensional data. Compared with the classical LLE, the linear-algebra-based QLLE achieves quadratic speedup in the number and dimension of the given data. The VQLLE can be implemented on the near term quantum devices in two different designs. In addition, the numerical experiments are presented to demonstrate that the two implementations in our work can achieve the procedure of locally linear embedding.

Motivation & Objective

• To address the computational bottleneck of classical locally linear embedding (LLE) in high-dimensional nonlinear data by developing quantum alternatives.
• To enable quantum speedup in nonlinear dimensionality reduction using quantum algorithms.
• To design a variational quantum-classical approach (VQLLE) compatible with near-term noisy intermediate-scale quantum (NISQ) devices.
• To preserve the intrinsic manifold structure of data during dimensionality reduction using quantum-enhanced LLE.
• To validate the feasibility and performance of quantum LLE implementations through numerical experiments.

Proposed method

• Proposes a linear-algebra-based QLLE that uses quantum linear algebra subroutines to solve the LLE optimization problem in O(poly(log N)) time, achieving exponential speedup over classical LLE’s O(poly(N)) complexity.
• Employs the quantum k-NN algorithm in the preprocessing stage to find k nearest neighbors with quadratic speedup in data size.
• Designs a variational quantum-classical procedure for VQLLE, using parameterized quantum circuits to optimize the low-dimensional embedding.
• Introduces two VQLLE variants: end-to-end and matrix-multiplication-based, both tailored for NISQ-era quantum hardware.
• Uses Lagrangian optimization to derive the weight matrix W and embedding matrix Y, reformulating the LLE cost function into a constrained eigenvalue problem.
• Applies quantum phase estimation and quantum singular value transformation techniques to solve the linear algebraic components of the LLE algorithm efficiently on a universal quantum computer.

Experimental results

Research questions

• RQ1 Can quantum algorithms achieve significant speedup in nonlinear dimensionality reduction compared to classical LLE?
• RQ2 How can the LLE algorithm be adapted for implementation on near-term quantum devices with limited qubit coherence and circuit depth?
• RQ3 What are the performance trade-offs between a universal quantum algorithm (QLLE) and a variational hybrid approach (VQLLE) in preserving data manifold structure?
• RQ4 Can quantum-enhanced k-NN search improve the efficiency of the LLE preprocessing step?
• RQ5 How do different parameterized quantum circuit designs in VQLLE affect the quality of the resulting low-dimensional embeddings?

Key findings

• The linear-algebra-based QLLE achieves exponential speedup in time complexity compared to classical LLE, reducing from O(poly(N)) to O(poly(log N)).
• The VQLLE is implementable on near-term quantum devices using a variational hybrid quantum-classical framework, making it feasible for current hardware constraints.
• Numerical experiments confirm that both QLLE implementations successfully perform locally linear embedding, preserving the intrinsic manifold structure of the data.
• The VQLLE variant demonstrates better global structure preservation than the linear-algebra-based QLLE in the numerical evaluation.
• The quantum k-NN subroutine enables quadratic speedup in identifying nearest neighbors during the preprocessing phase of QLLE.