3 research outputs found
Optimizing Quantum Circuit Parameters via SDP
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 by Anshu, Gosset, and Morenz with a ratio 0.531 and by Parekh and Thompson with a ratio 0.533
An improved Quantum Max Cut approximation via matching
Finding a high (or low) energy state of a given quantum Hamiltonian is a
potential area to gain a provable and practical quantum advantage. A line of
recent studies focuses on Quantum Max Cut, where one is asked to find a high
energy state of a given antiferromagnetic Heisenberg Hamiltonian. In this work,
we present a classical approximation algorithm for Quantum Max Cut that
achieves an approximation ratio of 0.584 given a generic input, and a ratio of
0.595 given a triangle-free input, outperforming the previous best algorithms
of Lee \cite{Lee22} (0.562, generic input) and King \cite{King22} (0.582,
triangle-free input). The algorithm is based on finding the maximum weighted
matching of an input graph and outputs a product of at most 2-qubit states,
which is simpler than the fully entangled output states of the previous best
algorithms. --v2 update: Ojas Parekh added as an author, triangle free
condition removed
An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem
We present a classical approximation algorithm for the MAX-2-Local Hamiltonian problem. This is a maximization version of the QMA-complete 2-Local Hamiltonian problem in quantum computing, with the additional assumption that each local term is positive semidefinite. The MAX-2-Local Hamiltonian problem generalizes NP-hard constraint satisfaction problems, and our results may be viewed as generalizations of approximation approaches for the MAX-2-CSP problem. We work in the product state space and extend the framework of Goemans and Williamson for approximating MAX-2-CSPs. The key difference is that in the product state setting, a solution consists of a set of normalized 3-dimensional vectors rather than boolean numbers, and we leverage approximation results for rank-constrained Grothendieck inequalities. For MAX-2-Local Hamiltonian we achieve an approximation ratio of 0.328. This is the first example of an approximation algorithm beating the random quantum assignment ratio of 0.25 by a constant factor