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A quantum central path algorithm for linear optimization

Brandon Augustino, Jiaqi Leng, Giacomo Nannicini, Tamás Terlaky, Xiaodi Wu

TL;DR

A novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path by performing a single simulation working directly with the nonlinear complementarity equations.

Abstract

We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a single simulation working directly with the nonlinear complementarity equations. This approach yields an algorithm for solving linear optimization problems involving $m$ constraints and $n$ variables to $\varepsilon$-optimality using $\mathcal{O} \left( \sqrt{m + n} \frac{R_{1}}{\varepsilon}\right)$ queries to an oracle that evaluates a potential function, where $R_{1}$ is an $\ell_{1}$-norm upper bound on the size of the optimal solution. In the standard gate model (i.e., without access to quantum RAM) our algorithm can obtain highly-precise solutions to LO problems using at most $$\mathcal{O} \left( \sqrt{m + n} \textsf{nnz} (A) \frac{R_1}{\varepsilon}\right)$$ elementary gates, where $\textsf{nnz} (A)$ is the total number of non-zero elements found in the constraint matrix.

A quantum central path algorithm for linear optimization

TL;DR

A novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path by performing a single simulation working directly with the nonlinear complementarity equations.

Abstract

We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a single simulation working directly with the nonlinear complementarity equations. This approach yields an algorithm for solving linear optimization problems involving constraints and variables to -optimality using queries to an oracle that evaluates a potential function, where is an -norm upper bound on the size of the optimal solution. In the standard gate model (i.e., without access to quantum RAM) our algorithm can obtain highly-precise solutions to LO problems using at most elementary gates, where is the total number of non-zero elements found in the constraint matrix.
Paper Structure (22 sections, 17 theorems, 127 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 22 sections, 17 theorems, 127 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Technical summary and roadmap
  5. Preliminaries
  6. Interior Point Methods for Linear Optimization
  7. Primal and dual LO problems
  8. The self-dual embedding model
  9. The central path and its neighborhood
  10. Quantum adiabatic theorem for unbounded Hamiltonians
  11. Quantum algorithms for Schrödinger equations
  12. A quantum algorithm that traces the central path
  13. Quantum representation of the central path
  14. Quantum simulation of the central path
  15. Quantum central path algorithm
  16. ...and 7 more sections

Key Result

Theorem 1

Let ${\cal H}(\mu(t))$ be a time-dependent quantum Hamiltonian with potential function $f(\mu(t))$. Suppose that for every value of $\mu(t)$, the ground state of ${\cal H} (\mu(t))$ encodes a probability distribution centered on the central path of e:LP-e:LP-D. For $\varepsilon > 0$ there is a quant and The algorithm can be implemented using queries to an evaluation oracle for $f$. If the proble

Figures (1)

  • Figure 1: Visualization of the quantum central path algorithm. The dotted lines define the boundary of a neighborhood of the central path. The dashed circles indicate the progression of the wave packet from time $t= 0$ to $t= T$. The wave packet begins to concentrate on a small ball centered at $z(\mu(T))$, near the optimal solution to the linear optimization problem $z_*$.

Theorems & Definitions (32)

  • Theorem 1: Main result, informal
  • Definition 1: Definition I.4 in roos2005interior
  • Theorem 2: Theorem I.6 in roos2005interior
  • Theorem 3: Exponential estimate
  • proof
  • Theorem 4: Improved version of Theorem 8 in childs2022quantumSim
  • Definition 2: Central-path Hamiltonian
  • Lemma 1
  • proof
  • Proposition 1
  • ...and 22 more