$q =~ s/^.*$a//sm;
The "m" is for multiline, which is a handy mnemonic.
Update: Although, as discussed in this thread the "m" is not actually required if we understand what nikolay wants correctly. We can golf the whole thing down to:
$q =~ s/.*$a//s;
as here:
#!/usr/bin/env perl
use strict;
use warnings;
use Test::More tests => 1;
my $q = 'qqqweqwe
asdasdasd
zxczxczxc
tyutyutyi
';
my $a = 'zxczxczxc';
my $want = "\ntyutyutyi\n";
$q =~ s/.*$a//s;
is ($q, $want, "With /s only and no anchor - matched");