1 Erratum to: Algorithmica (2011) 61:779–816 DOI 10.1007/s0045300993833
1. Theorem 6.2 of [4] contains an error. Specifically the condition in Theorem 6.2 is stated as both necessary and sufficient for a signature \(G \in \{0, 1\}^n\) to be realizable as a matchgate signature under all basis transformations \(\left[ {\begin{matrix} 1 &{} x \\ 1 &{} x \end{matrix}}\right] \). The condition is indeed sufficient, and the proof in Theorem 6.2 for this is valid; the error is the claim that it is also necessary.
On lines 16–17 of page 810, it claims that there must exist some i and \(i+1\), “both in X or both out of X, and one is in D and the other is out of D.” The reason given is that \(D \not = \emptyset , [n], X, X^c\). This is incorrect. It is possible that \(D \not = \emptyset , [n], X, X^c\), and yet all changes in membership from i to \(i+1\) in D are also changes in membership in X. This does not contradict \(D \not = \emptyset , [n], X, X^c\) because there can be changes in membership in X that are not changes in membership in D. Here is a counterexample: \(n=8\), \(X = \{2,4,6,8\}\), \(S = \{1,2,4,5\}\), \(S' = \{1,4,5,6\}\), \(D = S \oplus S' = \{2,6\}\). Because of this error, we cannot prove the orthogonality of the coefficient vectors in (41) and thus we cannot deduce \(G^S G^{S \oplus X} =0\) in (41).
As the condition in Theorem 6.2 is still sufficient, Corollary 6.1 is still valid. All other parts of the paper [4] are correct, and to our knowledge, no other subsequent results of ours and others depend on this part of Theorem 6.2.

\(G^S = 0\), for all \(S \not = n/2\), and

\(H_2^{\otimes n}G\) is a standard matchgate signature, where \(H_2 = \left[ {\begin{matrix} 1 &{} 1 \\ 1 &{} 1 \end{matrix}}\right] \).
Let \(\Gamma \) be a matchgate of arity n satisfying \(\Gamma ^{11 \ldots 1} \not = 0\). We may normalize it to \(\Gamma ^{11 \ldots 1}=1\). Define an \(n \times n\) skewsymmetric matrix B where its (i, j) entry, for \(1 \le i < j \le n\), is \(\Gamma ^{11 \ldots 1 \oplus e_i \oplus e_j}\), the signature value of \(\Gamma \) on the bit pattern that has two 1’s at the ith and jth bit positions and 0 elsewhere. The theory of matchgate signatures ([3], see also [1, 2]) implies that for any \(T \subseteq [n]\), \(\Gamma ^{T} = \mathrm{Pf}(B[T])\), where \(\Gamma ^{T}\) is the signature value for the bit pattern T denoted by its characteristic sequence, \(\mathrm{Pf}\) denotes Pfaffian, B[T] is the principal minor of B with all rows and columns in T removed.
If n is odd, then \(G = H_2^{\otimes n} \Gamma \) is also expressed as a Pfaffian Sum. We define \(B^{+}\) to be the \((n+1) \times (n+1)\) matrix of which the first n rows and columns equal B, and the \((n+1)\)st row and column are all 0. Extend \(\Lambda ^{(n)}(S)\) to \(\Lambda ^{(n+1)}(S)\) with \(\lambda _{n+1} =1\). Then \(G^S= \mathrm{PfS}(B) = \mathrm{Pf}(B^{+} + \Lambda ^{(n+1)}(S))\).
We summarize the discussion as follows:
Theorem 0.1
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