A Lattice-Reduction Aided Vector Perturbation Precoder Relying on Quantum Annealing

TL;DR

This work addresses downlink MIMO precoding via vector perturbation by recasting the problem as the closest vector problem ($CVP$) and solving it on quantum annealing hardware through a quadratic unconstrained binary optimization ($QUBO$). The method, LRAQVP, employs Lenstra-Lenstra-Lovasz lattice reduction ($LLL$) preprocessing to obtain a reduced basis and a tractable integer encoding $\\mathbf{l}=\\mathbf{C}\\mathbf{q}$, with a QUBO matrix $\\mathbf{Q}$ and hardware-aware chain strengths to embed on the Ising model. Experiments on the D-Wave Advantage show a roughly 5 dB gain over lattice reduction zero-forcing precoding ($LRZFP$) and performance approaching the sphere-encoder lower bound (HA), indicating potential quantum advantage for larger $N_r$ and scalable nonlinear TPC. The results underscore the importance of preprocessing and hardware-aware encoding to enable quantum-assisted nonlinear downlink precoding on NISQ devices.

Abstract

Quantum annealing (QA) is proposed for vector perturbation precoding (VPP) in multiple input multiple output (MIMO) communications systems. The mathematical framework of VPP is presented, outlining the problem formulation and the benefits of lattice reduction algorithms. Lattice reduction aided quantum vector perturbation (LRAQVP) is designed by harnessing physical quantum hardware, and the optimization of hardware parameters is discussed. We observe a 5dB gain over lattice reduction zero forcing precoding (LRZFP), which behaves similarly to a quantum annealing algorithm operating without a lattice reduction stage. The proposed algorithm is also shown to approach the performance of a sphere encoder, which exhibits an exponentially escalating complexity.

Figures (4)

• Figure 1: The impact of lattice reduction on limited search spaces in 2D. The bases $\{\mathbf{g}_1, \mathbf{g}_2\}$ and $\{\mathbf{f}_1, \mathbf{f}_2\}$ describe the same lattice. The $\mathbf{u}$ vector in green is closest to the encircled lattice point. The search points in the non-reduced basis do not include this closest point, showing the necessity of lattice reduction in limited search space examples.
• Figure 2: Distributions of $\lVert \mathbf{x} \rVert^2$, calculated via the LRAQVP algorithm, with varying chain strength parameter. The reduced median values show the impact of increased chain strength counteracting the noise, ensuring better solution quality. The limited dynamic range of the quantum annealer becomes the dominant factor in noise above the optimal value of $p$. There are 150 samples in each distribution.
• Figure 3: The distributions of $\lVert \mathbf{x} \rVert^{2}$ for Zero forcing, the LRZFP protocol and the LRAQVP presented here. The HA lower bound represent a spherical approximation to what the closest vector could be. The HA is a lower bound on the median performance that is averaged over all symbol vectors that could pass through a given channel.
• Figure 4: The SER simulations for a 10x10 MIMO system. A 5 dB gain can be seen from LRZFP to LRAQVP protocols. QVP can be seen to behave similarly to LRZFP, however it scales with much higher computational complexity, and performs worse in the high SNR regime. The uniform input limits (UILs) are theoretical predictions for the zero forcing protocols used to verify the empirical results. HA is a lower bound on the performance, that LRAQVP is approaching.