Variational Quantum Algorithms for Semidefinite Programming

TL;DR

This work develops variational quantum algorithms to solve semidefinite programs (SDPs) by reformulating constrained problems into unconstrained objectives realized as quantum-expectation values. It tackles three SDP forms—GF, ECSF, and ICSF—with corresponding VQAs: iVQAGF, iVQAEC, and iVQAIC; the ECSF variant includes a convergence-rate result under a weak-constraint regime, while ICSF is shown to be smooth but without a proven rate. Numerical experiments on MaxCut and random SDPs demonstrate convergence and resilience to noise on a QASM‑style, near-term quantum simulator, supporting practical potential for NISQ SDP solving. The study highlights a viable path toward quantum-assisted SDP solvers, leveraging the parameter-shift gradient and unconstrained reformulations, while leaving gradient-variance effects as an avenue for future work.

Abstract

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., $N\gg M$, where $N$ is the dimension of the input matrices and $M$ is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.

Figures (6)

• Figure 1: This figure depicts iVQAGF algorithm where we utilize two parameterized quantum circuits, i.e., $U_{R_1S_1}(\boldsymbol{\theta}_{1})$ and $U_{R_2S_2}(\boldsymbol{\theta}_{2})$.
• Figure 2: This figure depicts the iVQAEC algorithm in which we utilize one parameterized quantum circuit, i.e., $U_{RS}(\boldsymbol{\theta})$.
• Figure 3: This figure depicts the iVQAIC algorithm in which we utilize one parameterized quantum circuit, i.e., $U_{RS}(\boldsymbol{\theta})$.
• Figure 4: Convergence of iVQAGF for three randomly generated MaxCut-SDP instances with different numbers of vertices in the graph: $N \in \{8, 16, 32\}$.
• Figure 5: Convergence of iVQAEC for three separate cases of randomly generated equality constrained semidefinite programs and nonempty feasbile regions: $N \in \{8, 16, 32\}$.

Theorems & Definitions (28)

• Definition 1: Linear map
• Definition 2: Adjoint of a linear map
• Definition 3: Hermiticity-preserving linear map
• Definition 4: Choi representation of a linear map
• Lemma 5
• proof
• Definition 6: Lipschitz continuity
• Lemma 7: Lipschitz constant for a multivariate function
• proof
• Lemma 8: Lipschitz constant for a multivariate vector-valued function