[Paper Review] Quantum Interior Point Methods for Semidefinite Optimization

Brandon Augustino, Giacomo Nannicini+2 more|arXiv (Cornell University)|Dec 11, 2021

Advanced Optimization Algorithms Research74 references4 citations

TL;DR

This paper presents two quantum interior point methods (QIPMs) for semidefinite optimization (SDO), leveraging quantum linear system algorithms. The first method computes inexact search directions without guaranteeing feasibility; the second uses a nullspace representation to ensure feasibility even with inexact directions, enabling hybrid quantum-classical implementation. Both schemes achieve polynomial convergence to optimal solutions under standard assumptions, with the second offering a quantum speedup in problem dimension $ n $, though with worse dependence on other parameters.

ABSTRACT

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem $n$, but has worse dependence on other numerical parameters.

Motivation & Objective

• To develop quantum algorithms that accelerate semidefinite optimization (SDO), a fundamental problem in convex optimization with broad applications.
• To extend classical interior point methods (IPMs) to the quantum domain using recent advances in quantum linear system solvers.
• To design a novel quantum scheme that ensures feasibility of iterates even with inexact search directions, enabling practical hybrid quantum-classical implementation.
• To analyze the theoretical complexity of the proposed quantum IPMs and compare their performance to classical counterparts.
• To establish polynomial convergence of both quantum schemes under standard interior point method assumptions.

Proposed method

• The first scheme, Inexact-Feasible QIPM (IF-QIPM), computes inexact search directions using quantum linear system algorithms, similar to classical IPMs, but without enforcing feasibility during iterations.
• The second scheme, Inexact-Infeasible QIPM (II-QIPM), employs a nullspace representation of the Newton system to project inexact directions into the feasible space, ensuring iterates remain feasible despite approximation errors.
• The II-QIPM uses block-encoding techniques to represent the Newton system and applies quantum amplitude estimation (tomography) to extract solution components from quantum states.
• The algorithms rely on quantum linear system solvers with error bounds controlled by parameters $ \gamma_1, \gamma_2 $, ensuring convergence under standard IPM assumptions.
• The Newton systems are symmetrized and solved via quantum algorithms that exploit low-rank structure and condition number $ \kappa $, with complexity depending on $ n $, $ \mu_0 $, and $ \epsilon $.
• Convergence is proven via bounding the duality gap and infeasibility using quantum error estimates and matrix norm inequalities.

Experimental results

Research questions

• RQ1 Can quantum algorithms achieve a speedup over classical interior point methods for semidefinite optimization?
• RQ2 Is it possible to design a quantum IPM that maintains feasibility of iterates despite inexact quantum computation of search directions?
• RQ3 What is the theoretical complexity of quantum interior point methods for SDO, and how does it compare to classical algorithms in terms of problem dimension $ n $ and condition number $ \kappa $?
• RQ4 Can the nullspace representation of the Newton system be effectively implemented on a hybrid quantum-classical architecture to ensure feasibility?
• RQ5 What are the trade-offs between quantum speedup in $ n $ and degradation in dependence on other parameters like $ \kappa $, $ \mu_0 $, and $ \epsilon $?

Key findings

• The II-QIPM achieves a quantum speedup in the problem dimension $ n $, with a complexity of $ \mathcal{O}(\sqrt{n}) $, compared to classical $ \mathcal{O}(n) $, under standard assumptions.
• The IF-QIPM and II-QIPM both converge to an $ \epsilon $-optimal solution in polynomial time, with convergence rates dependent on $ \theta_k $, $ \mu_0 $, and $ \kappa $.
• The II-QIPM ensures feasibility of iterates even with inexact search directions by using a nullspace representation, a novel approach well-suited for hybrid quantum-classical computation.
• The quantum speedup in $ n $ comes at the cost of worse dependence on other parameters: the II-QIPM has a $ \mathcal{O}(\kappa^2) $ dependence, which is worse than classical algorithms.
• The duality gap is bounded by $ \mathcal{O}(n\theta_k\mu_0) $, and infeasibility is controlled via error terms $ \delta $ and $ \beta $, both shown to be $ \mathcal{O}(n^2\theta_k\mu_0) $.
• The analysis establishes that quantum amplitude estimation and block-encoding techniques can be used to extract solutions from quantum states with bounded error, enabling the design of provably convergent quantum IPMs.