  The decades-old Pattern Matching with Edits problem, given a length-$n$
string $T$ (the text), a length-$m$ string $P$ (the pattern), and a positive
integer $k$ (the threshold), asks to list all fragments of $T$ that are at edit
distance at most $k$ from $P$. The one-way communication complexity of this
problem is the minimum amount of space needed to encode the answer so that it
can be retrieved without accessing the input strings $P$ and $T$.
  The closely related Pattern Matching with Mismatches problem (defined in
terms of the Hamming distance instead of the edit distance) is already well
understood from the communication complexity perspective: Clifford, Kociumaka,
and Porat [SODA 2019] proved that $\Omega(n/m \cdot k \log(m/k))$ bits are
necessary and $O(n/m \cdot k\log (m|\Sigma|/k))$ bits are sufficient; the upper
bound allows encoding not only the occurrences of $P$ in $T$ with at most $k$
mismatches but also the substitutions needed to make each $k$-mismatch
occurrence exact.
  Despite recent improvements in the running time [Charalampopoulos, Kociumaka,
and Wellnitz; FOCS 2020 and 2022], the communication complexity of Pattern
Matching with Edits remained unexplored, with a lower bound of $\Omega(n/m
\cdot k\log(m/k))$ bits and an upper bound of $O(n/m \cdot k^3\log m)$ bits
stemming from previous research. In this work, we prove an upper bound of
$O(n/m \cdot k \log^2 m)$ bits, thus establishing the optimal communication
complexity up to logarithmic factors. We also show that $O(n/m \cdot k \log m
\log (m|\Sigma|))$ bits allow encoding, for each $k$-error occurrence of $P$ in
$T$, the shortest sequence of edits needed to make the occurrence exact.
  We leverage the techniques behind our new result on the communication
complexity to obtain quantum algorithms for Pattern Matching with Edits.
