[Paper Review] Quantum algorithms for supervised and unsupervised machine learning

Seth Lloyd, Masoud Mohseni+1 more|arXiv (Cornell University)|Jul 1, 2013

Quantum Computing Algorithms and ArchitectureComputer Science5 references563 citations

TL;DR

This paper proposes quantum algorithms for supervised and unsupervised machine learning that achieve exponential speedups over classical methods by leveraging quantum random access memory (qRAM) and quantum linear algebra techniques. It demonstrates that cluster assignment and k-means clustering can be performed in time O(log(MN)), compared to classical poly(MN) time, enabling efficient processing of high-dimensional 'big quantum data' with enhanced privacy.

ABSTRACT

Machine-learning tasks frequently involve problems of manipulating and classifying large numbers of vectors in high-dimensional spaces. Classical algorithms for solving such problems typically take time polynomial in the number of vectors and the dimension of the space. Quantum computers are good at manipulating high-dimensional vectors in large tensor product spaces. This paper provides supervised and unsupervised quantum machine learning algorithms for cluster assignment and cluster finding. Quantum machine learning can take time logarithmic in both the number of vectors and their dimension, an exponential speed-up over classical algorithms.

Motivation & Objective

• To develop quantum algorithms that provide exponential speedups for machine learning tasks involving high-dimensional vectors and large datasets.
• To address the computational bottleneck of classical algorithms that scale polynomially with the number of vectors and dimensionality.
• To enable efficient supervised cluster assignment and unsupervised k-means clustering using quantum computation.
• To explore privacy advantages in quantum machine learning where only O(log(MN)) qubits are needed to query a data base of size O(MN).

Proposed method

• Utilizes quantum random access memory (qRAM) to load classical data into quantum states in O(log N) time, enabling superposition over high-dimensional vectors.
• Employs quantum algorithms such as the quantum Fourier transform and quantum linear systems algorithm to perform post-processing on quantum states in poly(log N) time.
• Applies the quantum phase estimation algorithm to compute inner products and distances between quantum states in O(log N) time, achieving exponential speedup over classical sampling.
• Develops a quantum version of Lloyd’s k-means algorithm using the quantum adiabatic algorithm to find cluster assignments in O(k log k MN) time.
• Generalizes the distance evaluation to nonlinear metrics using multiple copies of quantum states and Hermitian operators, enabling higher-order polynomial distance measures.
• Uses quantum counting to estimate expectation values of operators with precision ε in O(ε⁻¹ q log N) steps, where q is the number of state copies.

Experimental results

Research questions

• RQ1 Can quantum algorithms achieve exponential speedup over classical algorithms for vector classification and clustering in high-dimensional spaces?
• RQ2 How can quantum random access memory (qRAM) be used to efficiently load and process classical data in quantum form for machine learning?
• RQ3 What is the quantum complexity of computing distances and inner products between large, high-dimensional vectors compared to classical methods?
• RQ4 Can the quantum adiabatic algorithm be used to approximately solve the NP-hard k-means clustering problem with a quantum speedup?
• RQ5 To what extent does quantum machine learning enhance data privacy by limiting the number of queries to the original data base?

Key findings

• Cluster assignment for M vectors in N-dimensional space can be performed in O(log(MN)) time on a quantum computer, representing an exponential speedup over classical algorithms that require poly(MN) time.
• The quantum k-means algorithm, based on the quantum adiabatic algorithm, runs in O(k log k MN) time, offering exponential speedup for unsupervised clustering tasks.
• Quantum algorithms can estimate inner products and distances between high-dimensional vectors in O(log N) time, while classical methods require O(poly(N)) time.
• The use of qRAM allows the preparation of quantum states encoding classical data in O(log N) steps, enabling efficient access to exponentially large vector spaces.
• Quantum machine learning provides enhanced privacy: only O(log(MN)) qubits are needed to query a data base of size O(MN), limiting exposure to an exponentially small fraction of the data.
• Even for the entire information content of the observable universe (~2³⁰⁰ bits), a small quantum computer could represent and analyze the data in a few hundred operations.