Speaker
Description
The purpose of this work is to use a Quantum Annealer (QA) to solve the homogeneous Bethe-Salpeter equation (hBSE)[1] for two massive scalars interacting via the exchange of a massive scalar, a problem previously addressed with classical computation [2]. To achieve this, we transform the hBSE,by a suitable discretization, into a non-symmetric generalized eigenvalue problem (GEVP) (see Ref. [2] for details) from which we need to determine the maximum real eigenvalues along with their corresponding eigenvectors. This involves solving a quadratic minimization problem, which, after transformation into a Quadratic Unconstrained Binary Optimization (QUBO) form, becomes manageable by the QA.
We have developed a hybrid algorithm for this task. 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 [3], we generalize the algorithm to accommodate the non-symmetric case, which involves complex eigenvalues (see Ref. [4] for details). Notably, the GEVP is a problem of broad interest across various fields, thus the results obtained could have wide-reaching implications.
We benchmark and analyze the statistical distribution of results using different parameters of the algorithm, employing a simulated annealing sampler (SA)[5].After that, very nice results for the target eigenpair have obtained by using a quantum annealer provided by D-Wave Systems, thanks to the D-Wave-CINECA agreement[6], as part of an international project approved by Q@TN (INFN-UNITN-FBK-CNR)[7]. We investigate how the algorithm's performance scales with the dimension of the matrices involved by comparing results obtained with QA and SA.
[1] E. E. Salpeter and H. A. Bethe, A Relativistic Equation for Bound-State Problems, Phys. Rev. 84, 1232 (195)
[2] T. Frederico, G. Salmè, and M. Viviani, Quantitative studies of the homogeneous Bethe-Salpeter equation in Minkowski space, Phys. Rev. D 89, 016010 (2014)
[3] B. Krakoff, S. M. Mniszewski, and C. F. A. Negre, A QUBO algorithm to compute eigenvectors of symmetric matrices, (2021), arXiv:2104.11
[4] S. Alliney, F. Laudiero, and M. Savoia, A variationaltechnique for the computation of the vibration frequencies of mechanical systems governed by nonsymmetric matrices, Applied mathematical modelling 16, 148 (1992)
[5] Neal, Radford M. "Annealed importance sampling." Statistics and computing 11 (2001): 125-139.
[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/
| Abstract category | Numerical Methods |
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