[Paper Review] Optimizing quantum circuit parameters via SDP

Eunou Lee|arXiv (Cornell University)|Sep 2, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a novel framework for optimizing parameters in parameterized quantum circuits using semidefinite programming (SDP) rounding. By rounding SDP solutions to circuit parameters, the authors develop a polynomial-time 0.562-approximation algorithm for the generic Quantum Max Cut problem, improving upon the previous best ratio of less than 0.54.

ABSTRACT

In recent years, parameterized quantum circuits have become a major tool to design quantum algorithms for optimization problems. The challenge in fully taking advantage of a given family of parameterized circuits lies in finding a good set of parameters in a non-convex landscape that can grow exponentially to the number of parameters. We introduce a new framework for optimizing parameterized quantum circuits: round SDP solutions to circuit parameters. Within this framework, we propose an algorithm that produces approximate solutions for a quantum optimization problem called Quantum Max Cut. The rounding algorithm runs in polynomial time to the number of parameters regardless of the underlying interaction graph. The resulting 0.562-approximation algorithm for generic instances of Quantum Max Cut improves on the previously known best algorithms, which give approximation ratios of less than 0.54.

Motivation & Objective

• Address the challenge of optimizing quantum circuit parameters in a non-convex, exponentially growing parameter space.
• Overcome limitations of existing rigorous algorithms that restrict parameters to equal values per layer or require special graph structures.
• Develop a general-purpose, efficient, and provably good approximation algorithm for the Quantum Max Cut problem without restricting the interaction graph or parameter space.
• Leverage the Goemans-Williamson SDP rounding paradigm in a quantum context to derive a new quantum approximation algorithm.
• Achieve a provable approximation ratio improvement over existing algorithms for Quantum Max Cut, a QMA-hard problem.

Proposed method

• Formulate the Quantum Max Cut problem as a 2-local Hamiltonian optimization task using Pauli matrices.
• Use the level-1 quantum Lasserre SDP hierarchy to represent the optimization problem, providing a relaxation of the quantum state space.
• Apply a novel rounding procedure: map each SDP solution vector associated with an edge in the interaction graph to a corresponding quantum circuit parameter via trigonometric functions.
• Ensure the rounding process runs in polynomial time relative to the number of parameters, regardless of the graph structure.
• Derive analytical expressions for the expected energy contribution of each edge in the Hamiltonian based on circuit parameters and graph connectivity.
• Use Pauli matrix commutation relations and state trace calculations to compute the expectation value of the Hamiltonian under the parameterized quantum state.

Experimental results

Research questions

• RQ1 Can SDP relaxation and rounding be adapted to optimize general parameterized quantum circuits for quantum optimization problems?
• RQ2 Can a provably good approximation ratio be achieved for the generic Quantum Max Cut problem without restricting the interaction graph or parameter symmetry?
• RQ3 Does the proposed SDP-based rounding framework outperform existing approximation algorithms for Quantum Max Cut in terms of approximation ratio?
• RQ4 What is the achievable approximation ratio for the Quantum Max Cut problem using this new SDP rounding framework?
• RQ5 How does the method scale with the number of parameters and graph connectivity in the interaction graph?

Key findings

• The proposed algorithm achieves a 0.562-approximation ratio for the generic Quantum Max Cut problem, improving upon the previous best-known ratio of less than 0.54.
• The rounding algorithm runs in polynomial time relative to the number of parameters, making it scalable regardless of the underlying interaction graph structure.
• The method does not require symmetry assumptions on parameters (e.g., equal parameters per layer), enabling full parameter space exploration.
• The framework generalizes the Goemans-Williamson approach to quantum optimization, applying SDP rounding to quantum circuit parameters.
• The analytical derivation of the energy expectation value shows that contributions depend on trigonometric functions of parameters and local graph neighborhoods.
• The result holds for arbitrary weighted graphs, demonstrating broad applicability beyond regular or bounded-degree instances.