Speaker
Description
The homogeneous Bethe-Salpeter equation (hBSE) [1], which models a bound system within a fully relativistic quantum field theory, has been solved for the first time using a D-Wave quantum annealer [2]. Following standard discretization methods, the hBSE in the ladder approximation can be reformulated as a generalized eigenvalue problem (GEVP) involving two square matrices, one symmetric and the other non-symmetric ( see Ref. [3] for details). This problem is of significant interest in various scientific fields, making the results broadly impactful. The non-symmetric matrix presents a challenge for a formal approach to solving the GEVP on a quantum annealer, as it needs to be converted into a quadratic unconstrained binary optimization (QUBO) problem. We have developed a hybrid algorithm. First, we reduce the non-symmetric GEVP to a standard eigenvalue problem classically. Then, we employ the QA to solve the variational problem. Drawing inspiration from approaches for symmetric matrices [4], we generalize the algorithm to accommodate the non-symmetric case, which involves complex eigenvalues (see Ref. [5] for details). A thorough numerical evaluation of the proposed algorithms, applied to matrices of up to 64 dimensions, was conducted using the proprietary simulated annealing package and the D-Wave Advantage 4.1 system thanks to the D-Wave-CINECA agreement[6], as part of an international project approved by Q@TN (INFN-UNITN-FBK-CNR)[7]. The results show excellent agreement with classical algorithms and reveal promising scalability properties.
[1] E. E. Salpeter and H. A. Bethe, A relativistic equation for bound-state problems, Phys. Rev. 84, 1232 (1951)
[2] F.Fornetti., A.Gnech, T. Frederico, F.Pederiva, M.Rinaldi, A.Roggero, G.Salmè, S.Scopetta, and M.Viviani, Solving the homogeneous Bethe-Salpeter equation with a quantum annealer, Phys. Rev. D 110 (2024) 5, 056012
[3]T. Frederico, G. Salmè, and M. Viviani, Quantitative studies of the homogeneous Bethe-Salpeter equation in Minkowski space, Phys. Rev. D 89, 016010 (2014)
[4] B. Krakoff, S. M. Mniszewski, and C. F. A. Negre, A QUBO algorithm to compute eigenvectors of symmetric matrices, (2021), arXiv:2104.11
[5] S. Alliney, F. Laudiero, and M. Savoia, A variational technique for the computation of the vibration frequencies of mechanical systems governed by nonsymmetric matrices, Applied mathematical modeling 16, 148 (1992)
[6]https://www.quantumcomputinglab.cineca.it/en/2021/05/12/collaboration-agreement-between-cineca-and-d-wave-for-the-distribution-in-italy-of-quantum-computing-resources/
[7] https://quantumtrento.eu/
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