An Improved Approximation Algorithm for Quantum Max-Cut on Triangle-Free Graphs

Robbie King

Department of Computing and Mathematical Sciences, Caltech

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Abstract

We give an approximation algorithm for Quantum Max-Cut which works by rounding an SDP relaxation to an entangled quantum state. The SDP is used to choose the parameters of a variational quantum circuit. The entangled state is then represented as the quantum circuit applied to a product state. It achieves an approximation ratio of 0.582 on triangle-free graphs. The previous best algorithms of Anshu, Gosset, Morenz, and Parekh, Thompson achieved approximation ratios of 0.531 and 0.533 respectively. In addition, we study the EPR Hamiltonian, which we argue is a natural intermediate problem which isolates some key quantum features of local Hamiltonian problems. For the EPR Hamiltonian, we give an approximation algorithm with approximation ratio $1 / \sqrt{2}$ on all graphs.

How can we round an SDP relaxation to an entangled quantum state? And how should we optimize a variational circuit ansatz? In this work, we solve these two problems by combining them: Use the SDP solution to choose the parameters of the variational circuit.

► BibTeX data

► References

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[2] Ilya Kull, Norbert Schuch, Ben Dive, and Miguel Navascués, "Lower Bounds on Ground-State Energies of Local Hamiltonians through the Renormalization Group", Physical Review X 14 2, 021008 (2024).

[3] Nicolas PD Sawaya, Daniel Marti-Dafcik, Yang Ho, Daniel P Tabor, David E Bernal Neira, Alicia B Magann, Shavindra Premaratne, Pradeep Dubey, Anne Matsuura, Nathan Bishop, Wibe A de Jong, Simon Benjamin, Ojas D Parekh, Norm Tubman, Katherine Klymko, and Daan Camps, "HamLib: A library of Hamiltonians for benchmarking quantum algorithms and hardware", arXiv:2306.13126, (2023).

[4] Dmitry Grinko and Maris Ozols, "Linear programming with unitary-equivariant constraints", arXiv:2207.05713, (2022).

[5] Sujit Rao, "Analysis of sum-of-squares relaxations for the quantum rotor model", arXiv:2311.09010, (2023).

[6] Adam Bene Watts, Anirban Chowdhury, Aidan Epperly, J. William Helton, and Igor Klep, "Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators", arXiv:2307.15661, (2023).

[7] Charlie Carlson, Zackary Jorquera, Alexandra Kolla, Steven Kordonowy, and Stuart Wayland, "Approximation Algorithms for Quantum Max-$d$-Cut", arXiv:2309.10957, (2023).

[8] Andrew Zhao and Nicholas C. Rubin, "Expanding the reach of quantum optimization with fermionic embeddings", arXiv:2301.01778, (2023).

[9] Eunou Lee and Ojas Parekh, "An improved Quantum Max Cut approximation via matching", arXiv:2401.03616, (2024).

[10] Steven Heilman, "Sphere Valued Noise Stability and Quantum MAX-CUT Hardness", arXiv:2306.03912, (2023).

[11] Vorapong Suppakitpaisarn and Jin-Kao Hao, "Utilizing Graph Sparsification for Pre-processing in Maxcut QUBO Solver", arXiv:2401.13004, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-19 08:42:47) and SAO/NASA ADS (last updated successfully 2024-04-19 08:42:48). The list may be incomplete as not all publishers provide suitable and complete citation data.