[Paper Review] Quantum Spectral Clustering through a Biased Phase Estimation Algorithm

Ammar Daskin|arXiv (Cornell University)|Mar 16, 2017

Quantum Computing Algorithms and Architecture21 references4 citations

TL;DR

This paper proposes a quantum algorithm for spectral clustering using a biased phase estimation approach to accelerate amplitude amplification. By representing the data matrix as a sum of Householder matrices and applying phase estimation with a modified unitary, the method achieves a quantum complexity of O(2^m L N), offering potential exponential speedup over classical spectral clustering when the matrix is sparse or decomposable into simple unitaries.

ABSTRACT

In this brief paper, we go through the theoretical steps of the spectral clustering on quantum computers by employing the phase estimation and the amplitude amplification algorithms. We discuss circuit designs for each step and show how to obtain the clustering solution from the output state. In addition, we introduce a biased version of the phase estimation algorithm which significantly speeds up the amplitude amplification process. The complexity of the whole process is analyzed: it is shown that when the circuit representation of a data matrix of order $N$ is produced through an ancilla based circuit in which the matrix is written as a sum of $L$ number of Householder matrices; the computational complexity is bounded by $O(2^mLN)$ number of quantum gates. Here, $m$ represents the number of qubits (e.g., 6) involved in the phase register of the phase estimation algorithm.

Motivation & Objective

• To develop a quantum algorithm for spectral clustering using phase estimation and amplitude amplification.
• To reduce the number of iterations in amplitude amplification through a biased phase estimation algorithm.
• To analyze the quantum circuit complexity of spectral clustering and compare it with classical counterparts.
• To demonstrate how eigenvectors of the data or Laplacian matrix can be extracted efficiently on a quantum computer for clustering.

Proposed method

• Uses phase estimation on a unitary e^H constructed from Householder matrices derived from data vectors.
• Represents the matrix H = XX^T as a sum of L Householder matrices, each implementable with O(N) gates.
• Applies a biased phase estimation algorithm that replaces the quantum Fourier transform with a tailored operator to reduce iterations in amplitude amplification.
• Employs amplitude amplification to amplify the amplitude of the desired eigenvector state corresponding to the largest eigenvalue.
• Uses a control register of m qubits to estimate eigenvalues with precision related to 2^{-m}.
• Measures the output state using a Pauli-based or Y-operator-based initial state to extract clustering labels.

Experimental results

Research questions

• RQ1 Can spectral clustering be efficiently implemented on a quantum computer using phase estimation and amplitude amplification?
• RQ2 How does the use of a biased phase estimation operator reduce the number of iterations in amplitude amplification compared to standard QPE?
• RQ3 What is the quantum circuit complexity of spectral clustering when the data matrix is decomposed into Householder matrices?
• RQ4 Under what conditions does the quantum approach outperform classical spectral clustering in terms of computational complexity?
• RQ5 How can the clustering solution be extracted from the quantum state output of the phase estimation process?

Key findings

• The quantum algorithm achieves a computational complexity of O(2^m L N) for spectral clustering, where m is the number of qubits in the phase register and L is the number of Householder matrices.
• The biased phase estimation algorithm significantly reduces the number of iterations required in amplitude amplification, improving efficiency.
• When the matrix H can be decomposed into simple unitaries requiring O(log N) gates each, the complexity reduces to O(2^m L log N).
• The method provides exponential speedup over classical spectral clustering when the matrix is sparse or has a low-rank decomposition into simple unitaries.
• The clustering solution is extracted via measurement of the output state using an initial superposition state and a Y-operator (sum of X-gates), requiring only one trial.
• The approach maintains a favorable complexity scaling even when multiple trials are needed to find the optimal clustering, as the number of trials c is shared with classical methods.