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Quantum Recommendation Systems

Iordanis Kerenidis, Anupam Prakash

TL;DR

The paper addresses scalable personalization under a low-rank user–item model by introducing a quantum approach that samples from a low-rank approximation rather than reconstructing the full preference matrix. It develops a data-structure-ready framework and a quantum projection procedure that, together with quantum singular-value estimation, enables sampling from the top-k subspace with polylogarithmic dependence on the matrix dimensions and polynomial dependence on the rank k. The authors prove that this approach yields high-quality recommendations for most users and establish rigorous error and runtime guarantees, avoiding reliance on matrix sparsity or conditioning. This work demonstrates a practical real-world quantum machine-learning application, enabling efficient online recommendations for very large-scale systems.

Abstract

A recommendation system uses the past purchases or ratings of $n$ products by a group of $m$ users, in order to provide personalized recommendations to individual users. The information is modeled as an $m \times n$ preference matrix which is assumed to have a good rank-$k$ approximation, for a small constant $k$. In this work, we present a quantum algorithm for recommendation systems that has running time $O(\text{poly}(k)\text{polylog}(mn))$. All known classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for a real world application.

Quantum Recommendation Systems

TL;DR

The paper addresses scalable personalization under a low-rank user–item model by introducing a quantum approach that samples from a low-rank approximation rather than reconstructing the full preference matrix. It develops a data-structure-ready framework and a quantum projection procedure that, together with quantum singular-value estimation, enables sampling from the top-k subspace with polylogarithmic dependence on the matrix dimensions and polynomial dependence on the rank k. The authors prove that this approach yields high-quality recommendations for most users and establish rigorous error and runtime guarantees, avoiding reliance on matrix sparsity or conditioning. This work demonstrates a practical real-world quantum machine-learning application, enabling efficient online recommendations for very large-scale systems.

Abstract

A recommendation system uses the past purchases or ratings of products by a group of users, in order to provide personalized recommendations to individual users. The information is modeled as an preference matrix which is assumed to have a good rank- approximation, for a small constant . In this work, we present a quantum algorithm for recommendation systems that has running time . All known classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for a real world application.

Paper Structure

This paper contains 20 sections, 13 theorems, 35 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Comparisons with related work.
  4. Preliminaries
  5. Linear algebra
  6. Quantum information
  7. A model for recommendation systems
  8. The preference matrix
  9. Sampling an approximation of the preference matrix
  10. Matrix Sampling
  11. Quantum projections in polylogarithmic time
  12. The data structure
  13. Quantum Singular Value Estimation
  14. Analysis
  15. Quantum projection with threshold
  16. ...and 5 more sections

Key Result

Theorem 2.2

Phase estimation K95: Let $U$ be a unitary operator, with eigenvectors $\mathinner{|{v_j}\rangle}$ and eigenvalues $e^{ \iota \theta_{j}}$ for $\theta_{j} \in [-\pi, \pi]$, i.e. we have $U\mathinner{|{v_{j}}\rangle} = e^{ \iota \theta_{j}} \mathinner{|{v_{j}}\rangle}$ for $j \in [n]$. For a precisio

Figures (1)

  • Figure 1: Vector state preparation illustrated for $4$-dimensional state $\mathinner{|{\phi}\rangle}$.

Theorems & Definitions (22)

  • Definition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • ...and 12 more