Base collapse of holographic algorithms

TL;DR

This paper proves that holographic algorithms can be simplified by collapsing their bases to smaller sizes under certain conditions, extending previous results to any domain size n.

Contribution

It generalizes the base collapse phenomenon for holographic algorithms to any domain size n, using elementary matchgate transformations.

Findings

Base collapse to size r ≤ ⌊log n⌋ for any n.

Conditions include the original problem being defined by a symmetric function.

Utilizes elementary matchgate transformations instead of matchgate identities.

Abstract

A holographic algorithm solves a problem in domain of size $n$, by reducing it to counting perfect matchings in planar graphs. It may simulate a $n$-value variable by a bunch of $t$ matchgate bits, which has $2^t$ values. The transformation in the simulation can be expressed as a $n \times 2^t$ matrix $M$, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size $n \times 2^{r}$, where $r<t$. Base collapse was discovered for small domain $n=2,3,4$. For $n=3, 4$, the base collapse was proved under the condition that there is a full rank generator. We prove for any $n$, the base collapse to a $r\leq \lfloor \log n \rfloor$, with some similar conditions. One of them is that the original problem is defined by one symmetric function. In the proof, we utilize elementary matchgate transformations instead of matchgate identities.