qFHRR: Rethinking Fourier Holographic Reduced Representations through Quantized Phase and Integer Arithmetic

Abstract

Fourier Holographic Reduced Representations (FHRR) provide a compositional framework for encoding structured information with complex-valued hypervectors. FHRR rely on floating-point arithmetic, which limits their efficiency and applicability on resource-constrained hardware. We introduce qFHRR, a quantized phase formulation of FHRR. In this representation, each dimension is encoded as a discrete phase index, enabling integer-only implementations of binding, unbinding, similarity, and bundling through modular arithmetic and lookup tables. We show that qFHRR preserves the algebraic properties of complex FHRR while significantly reducing the number of bits per dimension, from 64-bit complex representations to as few as 3--4 bits. Across a range of phase resolutions, qFHRR maintains high fidelity to the complex baseline, achieving strong performance even at low bit-widths. We further demonstrate that qFHRR preserves the spatial similarity structure induced by fractional binding. This enables accurate multi-object memory representations despite significant quantization. These results indicate that qFHRR provides an efficient and scalable alternative to complex FHRR, preserving the algebraic operations and similarity structure of the representation. It also reduces memory footprint and enables hardware-friendly implementations.