Optimal quantics index ordering for combined frequency–momentum/orbital vertex representations

Determine the optimal grouping and ordering of quantics bits when representing the four-point vertex F_{ijkl}(ν, ν′, ω) in a single QTT that simultaneously encodes frequency and momentum/orbital indices, so as to minimize bond dimensions and enable efficient evaluation of parquet and Bethe–Salpeter equations.

Background

Extending QTT methods to vertices with orbital and momentum dependencies introduces N^4 components, making naive per-index QTTs impractical. A unified QTT that incorporates both frequency and index dependencies could exploit low-rank structure across components, but its efficiency hinges on a judicious quantics index ordering.

Choosing an index ordering that preserves scale separation and minimizes inter-core entanglement is crucial for maintaining small bond dimensions and tractable MPO operations.