Optimal symmetric low-rank BD-RIS configuration maximizing the determinant of a MIMO link

TL;DR

This paper proposes a low-rank, symmetric BD-RIS configuration that maximizes the MIMO channel determinant, achieving near-optimal performance with reduced computational complexity.

Contribution

It introduces a closed-form low-rank symmetric unitary scattering matrix that matches the optimal determinant, enabling efficient implementation with minimal reconfigurable circuits.

Findings

The low-rank solution attains the same determinant as the optimal unitary BD-RIS.

Rate loss diminishes at high SNR or large number of elements.

The approach is significantly faster than iterative methods and suitable for large surfaces.

Abstract

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) significantly improve wireless performance by allowing tunable interconnections among elements, but their design in multiple-input multiple-output (MIMO) systems has so far relied on complex iterative algorithms or suboptimal approximations. This work introduces a simple yet powerful approach: instead of directly maximizing the achievable rate, we maximize the absolute value of the determinant of the equivalent MIMO channel. We derive a closed-form symmetric unitary scattering matrix whose rank is exactly twice the channel's degrees of freedom ($2r$). Remarkably, this low-rank solution achieves the same determinant value as the optimal unitary BD-RIS. Using log-majorization theory, we prove that the rate loss relative to the optimal unitary BD-RIS vanishes at high signal-to-noise ratio (SNR) or when the number of BD-RIS elements becomes large. Moreover, the proposed solution can be perfectly implemented using a $q$-stem BD-RIS architecture with only $q=2r-1$ stems, requiring a minimum number of reconfigurable circuits. The resulting Max-Det solution is orders of magnitude faster to compute than existing iterative methods while achieving near-optimal rates in practical scenarios. This makes high-performance BD-RIS deployment feasible even with large surfaces and limited computational resources.