Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Scaling Quantum Interior Point Method for Linear Optimization

Mohammadhossein Mohammadisiahroudi, Zeguan Wu, Pouya Sampourmahani, Jun-Kai You, Tamás Terlaky

TL;DR

This work addresses scaling challenges in solving large-scale linear optimization problems with interior point methods by introducing an almost-exact quantum interior point method (QIPM). The method constructs and solves the Newton system on a quantum computer while performing updates on a classical processor and executing matrix-vector products on quantum hardware, augmented by iterative refinement to achieve exponentially small errors. It achieves optimal dimension scaling of $\mathcal{O}(n^2)$ with a quantum-specific complexity footprint that surpasses prior classical and quantum IPMs, albeit relying on QRAM. The approach is extended with an iterative refinement framework (IR-AE-QIPM) to stabilize precision across iterations, and the authors discuss QRAM-free variants and future primal-dual extensions for broader optimization settings.

Abstract

The emergence of huge-scale, data-intensive linear optimization (LO) problems in applications such as machine learning has driven the need for more computationally efficient interior point methods (IPMs). While conventional IPMs are polynomial-time algorithms with rapid convergence, their per-iteration cost can be prohibitively high for dense large-scale LO problems. Quantum linear system solvers have shown potential in accelerating the solution of linear systems arising in IPMs. In this work, we introduce a novel almost-exact quantum IPM, where the Newton system is constructed and solved on a quantum computer, while solution updates occur on a classical machine. Additionally, all matrix-vector products are performed on the quantum hardware. This hybrid quantum-classical framework achieves an optimal worst-case scaling of $\mathcal{O}(n^2)$ for fully dense LO problems. To ensure high precision, despite the limited accuracy of quantum operations, we incorporate iterative refinement techniques both within and outside the proposed IPM iterations. The proposed algorithm has a quantum complexity of $\mathcal{O}(n^{1.5} κ_A \log(\frac{1}ε))$ queries to QRAM and $\mathcal{O}(n^2 \log(\frac{1}ε))$ classical arithmetic operations. Our method outperforms the worst-case complexity of prior classical and quantum IPMs, offering a significant improvement in scalability and computational efficiency.

Optimal Scaling Quantum Interior Point Method for Linear Optimization

TL;DR

This work addresses scaling challenges in solving large-scale linear optimization problems with interior point methods by introducing an almost-exact quantum interior point method (QIPM). The method constructs and solves the Newton system on a quantum computer while performing updates on a classical processor and executing matrix-vector products on quantum hardware, augmented by iterative refinement to achieve exponentially small errors. It achieves optimal dimension scaling of with a quantum-specific complexity footprint that surpasses prior classical and quantum IPMs, albeit relying on QRAM. The approach is extended with an iterative refinement framework (IR-AE-QIPM) to stabilize precision across iterations, and the authors discuss QRAM-free variants and future primal-dual extensions for broader optimization settings.

Abstract

The emergence of huge-scale, data-intensive linear optimization (LO) problems in applications such as machine learning has driven the need for more computationally efficient interior point methods (IPMs). While conventional IPMs are polynomial-time algorithms with rapid convergence, their per-iteration cost can be prohibitively high for dense large-scale LO problems. Quantum linear system solvers have shown potential in accelerating the solution of linear systems arising in IPMs. In this work, we introduce a novel almost-exact quantum IPM, where the Newton system is constructed and solved on a quantum computer, while solution updates occur on a classical machine. Additionally, all matrix-vector products are performed on the quantum hardware. This hybrid quantum-classical framework achieves an optimal worst-case scaling of for fully dense LO problems. To ensure high precision, despite the limited accuracy of quantum operations, we incorporate iterative refinement techniques both within and outside the proposed IPM iterations. The proposed algorithm has a quantum complexity of queries to QRAM and classical arithmetic operations. Our method outperforms the worst-case complexity of prior classical and quantum IPMs, offering a significant improvement in scalability and computational efficiency.

Paper Structure

This paper contains 8 sections, 7 theorems, 25 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Almost-exact Quantum interior point method
  4. Proof of Theorem \ref{['theo: IPM']}
  5. Quantum Subroutine
  6. Proposed IR-AE-QIPM
  7. Total Complexity
  8. Conclusion

Key Result

Lemma 1

Let $(x^*,y^*,s^*)\in \mathcal{PD}^*$ be a basic solution. If $x_i^*>0$, then we have $x_i^*\geq 2^{-L}$. If $s_i^*>0$, then we have $s_i^*\geq 2^{-L}$.

Figures (1)

  • Figure 1: The effect of IR on the condition number of linear systems arising in QIPM for degenerate LO mohammadisiahroudi2023inexact.

Theorems & Definitions (8)

  • Lemma 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Theorem 6: Lemma 13 of wu2024quantum
  • Theorem 7
  • proof