  We present dynamic algorithms with polylogarithmic update time for estimating
the size of the maximum matching of a graph undergoing edge insertions and
deletions with approximation ratio strictly better than $2$. Specifically, we
obtain a $1+\frac{1}{\sqrt{2}}+\epsilon\approx 1.707+\epsilon$ approximation in
bipartite graphs and a $1.973+\epsilon$ approximation in general graphs. We
thus answer in the affirmative the major open question first posed in the
influential work of Onak and Rubinfeld (STOC'10) and repeatedly asked in the
dynamic graph algorithms literature. Our randomized algorithms also work
against an adaptive adversary and guarantee worst-case polylog update time,
both w.h.p.
  Our algorithms are based on simulating new two-pass streaming matching
algorithms in the dynamic setting. Our key new idea is to invoke the recent
sublinear-time matching algorithm of Behnezhad (FOCS'21) in a white-box manner
to efficiently simulate the second pass of our streaming algorithms, while
bypassing the well-known vertex-update barrier.
