A holographic algorithm solves a problem in a 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  × 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  × 2  r  , where  r  t .   Base collapse is discovered for small domain  n =2,3,4. For  n =3, 4, the base collapse is proved under the condition that there is a full rank generator. We prove for any  n , the base collapse to a  r ≤ ⌊ log n  ⌋, under some similar conditions. One of the conditions is that the original problem is defined by one symmetric function. In the proof, we utilize elementary matchgate transformations instead of matchgate identities.
['Mingji Xia']
Base collapse of holographic algorithms