Post-processing for the solutions with total sum error = 2 The absolute sum error
=2 situation has two cases. The first one is similar to the previous case; for each of the
m rows, we have either +2 or -2 (and of course 0), which we will resolve similarly. In the
second case, for each pair of rows,
r
i
, r
j
, i, j = 1, . . . , m
, among m(m − 1)/2 possible
pairs, we have four sub-groups of errors (r
i
, r
j
) that are (+1, +1), (+1, -1), (-1, +1), and (-1,
-1). For each sub-group any pairwise subtraction results to zero. Additionally, we have two
sub-groups of (+1, +1), (-1, -1) and (+1, -1), (-1, +1) that can have pairwise subtraction
to result in zero. The above-mentioned operations, for all m rows, on average roughly cost
O
3N
2
u2
4m
column pairwise additions to resolve +2, -2 errors and O
3
8
N
2
u2
pairwise additions
to resolve the eight remaining categories of various +1, -1 errors, where N
u2
is the number
of unique solutions with a total sum error of 2.
Post-processing for the solutions with total sum error = 3 This situation has three
general categories of errors. As before, if for each of m rows, in each column we have either
+3 or -3, this case is similar to the first case (and the first case in the second situation) and
is resolved similarly. The second case here has similarities with the second category of the
second case. It consists of 4 subgroups on m(m − 1)/2 pairs of rows, that have (-2, -1), (-2,
+1), (+2, -1), (+2, +1). Likewise, it needs in-group subtraction for each of the four groups
and two pairwise additions between the subgroups (-2, -1), (+2, +1) and (-2, +1), (+2, -1).
The third category consists of columns, which have one of the eight groups of (-1, -1, -1), (-1,
-1, +1), (-1, +1, -1), (-1, +1, +1), (+1, -1, -1), (+1, -1, +1), (+1, +1, -1), (+1, +1, +1). For
every triple row among
m
3
possible row combinations, we have eight in-group pairwise
column subtractions, plus four out-group pairwise additions between (-1, -1, -1), (+1, +1,
+1), (-1, -1, +1), (+1, +1, -1), (-1, +1, -1), (+1, -1, +1), (-1, +1, +1), (+1, -1, -1). These
operations, for all m rows, on average roughly cost O
3N
2
u3
4m
column pairwise additions to
resolve +3, -3 errors and O
3
8
N
2
u3
pairwise additions to resolve the eight category of +/-2,
+/-1 errors, and maximum O
2
9
mN
2
u3
for resolving 12 cases with three +/-1 error in a
column, which N
u3
is the number of unique solutions with total sum error equal to 3.
Notes:
(a) The routine used in case sum error =1 is reused (with different values) in the first cases
of sum error=2 and sum error=3. Similarly, the routine used in the second category of sum
error =2 is used in the second case of sum error =3.
(b) This systematic post-processing procedure can go to higher levels of errors, but detailed
analysis of all unique solutions using the D-Wave processor shows that the majority of sub-
optimal solutions have sum errors up to 3 (with low chainbreak rate, also while using the
minimize energy chain break strategy), so we did not implement the post-processing for the
higher error values.
15