Quantum-inspired Tensor Network for QUBO, QUDO and Tensor QUDO Problems with k-neighbors

Abstract

This work presents a novel tensor network algorithm for solving Quadratic Unconstrained Binary Optimization (QUBO) problems, Quadratic Unconstrained Discrete Optimization (QUDO) problems, and Tensor Quadratic Unconstrained Discrete Optimization (T-QUDO) problems. The proposed algorithm is based on the MeLoCoToN methodology, which solves combinatorial optimization problems by employing superposition, imaginary time evolution, and projective measurements. Additionally, two different approaches are presented to solve QUBO and QUDO problems with k-neighbors interactions in a lineal chain, one based on 4-order tensor contraction and the other based on matrix-vector multiplication, including sparse computation and a new technique called "Waterfall". Furthermore, the performance of both implementations is compared with a quadratic optimization solver to demonstrate the performance of the method, showing advantages in several problem instances.

Figures (12)

• Figure 1: Tensor network for a dense QUDO problem with 5 variables. The mathematical description of each tensor is in Appendix \ref{['appendix:section-1']}.
• Figure 2: a) Contraction scheme of the last row MPS. b) Contraction scheme between the last row tensor and the MPO of the layer above. c) Contraction scheme between a row tensor and the MPO of the layer above in the k-neighbours case.
• Figure 3: Tensor network for a $k$-neighbors QUDO problem with 5 variables and $k=2$. The mathematical description of each tensor is in Appendix \ref{['appendix:section-1']}.
• Figure 4: Schematic representation of the 5 variable $k$-neighbors QUDO tensor network for a) 1, b) 2 and c) 3-neighbors.
• Figure 5: Schematic representation of the 5 variable $k$-neighbors QUDO tensor network for 1 a), 2 b) and 3 c) neighbors and its representation in the form of a succession of matrices d).