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Quantum Alternating Direction Method of Multipliers for Semidefinite Programming

Hantao Nie, Dong An, Zaiwen Wen

TL;DR

The paper develops QADMM, a quantum-accelerated ADMM framework for semidefinite programming that tolerates quantum-induced inexactness. By replacing costly eigenvalue decompositions with quantum spectral transformations (via QSVT) and leveraging block-encodings, LCU, and tomography, it achieves ε-optimal SDP solutions with favorable scaling in dimension compared to classical ADMM and quantum IPMs. The authors prove ergodic convergence for the inexact framework and provide detailed complexity bounds, highlighting potential quantum advantages in large-scale SDP settings while noting the QRAM data-access requirement as a key limitation. This work advances quantum optimization by blending first-order methods with quantum linear algebra to tackle convex conic problems at scale.

Abstract

Semidefinite programming (SDP) is a fundamental convex optimization problem with wide-ranging applications. However, solving large-scale instances remains computationally challenging due to the high cost of solving linear systems and performing eigenvalue decompositions. In this paper, we present a quantum alternating direction method of multipliers (QADMM) for SDPs, building on recent advances in quantum computing. An inexact ADMM framework is developed, which tolerates errors in the iterates arising from block-encoding approximation and quantum measurement. Within this robust scheme, we design a polynomial proximal operator to address the semidefinite conic constraints and apply the quantum singular value transformation to accelerate the most costly projection updates. We prove that the scheme converges to an $ε$-optimal solution of the SDP problem under the strong duality assumption. A detailed complexity analysis shows that the QADMM algorithm achieves favorable scaling with respect to dimension compared to the classical ADMM algorithm and quantum interior point methods, highlighting its potential for solving large-scale SDPs.

Quantum Alternating Direction Method of Multipliers for Semidefinite Programming

TL;DR

The paper develops QADMM, a quantum-accelerated ADMM framework for semidefinite programming that tolerates quantum-induced inexactness. By replacing costly eigenvalue decompositions with quantum spectral transformations (via QSVT) and leveraging block-encodings, LCU, and tomography, it achieves ε-optimal SDP solutions with favorable scaling in dimension compared to classical ADMM and quantum IPMs. The authors prove ergodic convergence for the inexact framework and provide detailed complexity bounds, highlighting potential quantum advantages in large-scale SDP settings while noting the QRAM data-access requirement as a key limitation. This work advances quantum optimization by blending first-order methods with quantum linear algebra to tackle convex conic problems at scale.

Abstract

Semidefinite programming (SDP) is a fundamental convex optimization problem with wide-ranging applications. However, solving large-scale instances remains computationally challenging due to the high cost of solving linear systems and performing eigenvalue decompositions. In this paper, we present a quantum alternating direction method of multipliers (QADMM) for SDPs, building on recent advances in quantum computing. An inexact ADMM framework is developed, which tolerates errors in the iterates arising from block-encoding approximation and quantum measurement. Within this robust scheme, we design a polynomial proximal operator to address the semidefinite conic constraints and apply the quantum singular value transformation to accelerate the most costly projection updates. We prove that the scheme converges to an -optimal solution of the SDP problem under the strong duality assumption. A detailed complexity analysis shows that the QADMM algorithm achieves favorable scaling with respect to dimension compared to the classical ADMM algorithm and quantum interior point methods, highlighting its potential for solving large-scale SDPs.

Paper Structure

This paper contains 31 sections, 20 theorems, 95 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Notations
  4. Preliminaries
  5. Optimality conditions
  6. Block-encodings and quantum random access memory
  7. Tomography
  8. Eigenvalue transformation
  9. Quatum linear solver
  10. Quantum alternating direction method of multipliers for SDP
  11. Inexact ADMM framework for SDP
  12. QADMM for SDP
  13. Polynomial proximal operator
  14. Complexity analysis
  15. Iteration number of QADMM for SDP
  16. ...and 16 more sections

Key Result

Proposition 2.1

(Optimality conditions) $(X^*, y^*, S^*)$ is a primal-dual optimal solution of the primal-dual pair of SDPs eq: primal SDP and eq: dual SDP if and only if the following conditions hold:

Theorems & Definitions (21)

  • Proposition 2.1
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1
  • Corollary 4.1
  • ...and 11 more