1 Erratum to: Algorithmica (2011) 61:779–816 DOI 10.1007/s00453-009-9383-3
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.
2. As pointed out (correctly) on page 806 in [4] that for a signature \(G \in \{0, 1\}^n\), a necessary and sufficient condition for \(\left[ {\begin{matrix} 1 &{} x \\ 1 &{} -x \end{matrix}}\right] ^{\otimes n} G\) to be a standard matchgate signature for all \( x \not =0\) is that
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\(G^S = 0\), for all \(|S| \not = n/2\), and
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\(H_2^{\otimes n}G\) is a standard matchgate signature, where \(H_2 = \left[ {\begin{matrix} 1 &{} 1 \\ 1 &{} -1 \end{matrix}}\right] \).
3. In the following, we give a characterization of the realizability of G under \(H_2\).
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 \(n \times n\) skew-symmetric 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 i-th and j-th 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.
Consider the transformation of \(\Gamma \) by the Hadamard matrix \(H_2\) which is orthogonal up to a scalar \(\frac{1}{\sqrt{2}}\). Let \(G = H_2^{\otimes n} \Gamma \), then for all \(S \subseteq [n]\),
If B is any \(n \times n\) skew-symmetric matrix, and \(\lambda _1, \lambda _2, \ldots , \lambda _n\) are n indeterminants, then Valiant [5] defined the Pfaffian Sum \(\mathrm{PfS}(B)\) to be a polynomial \(\sum _{T \subseteq [n]} \left( \prod _{i \in T} \lambda _i \right) \mathrm{Pf} (B[T])\).
Now for any \(S \subseteq [n]\) we define a sequence of values
then for these values of \(\lambda _i\), and the B defined from \(\Gamma \),
By the Pfaffian Sum Theorem of [5] this can be expressed as a single Pfaffian. Suppose n is even, then define an \(n \times n\) skew-symmetric matrix \(\Lambda ^{(n)}(S)\) with its (i, j) entry \((-1)^{j-i -1} \lambda _i \lambda _j = (-1)^{j-i -1 + \chi _S(i) + \chi _S(j)}\) for \(1 \le i<j \le n\). This matrix has the form
This is the skew-symmetric matrix with the strict alternating pattern of \(\pm 1\) starting each row with \(+1\) in the upper triangular half, modified by \(-1\) at every (i, j) with \(|\{i, j \} \oplus S | = 1\). Valiant’s Pfaffian Sum Theorem in [5] states that \(G^S= \mathrm{PfS}(B) = \mathrm{Pf}(B + \Lambda ^{(n)}(S))\).
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 \(G^S= \mathrm{PfS}(B) = \mathrm{Pf}(B^{+} + \Lambda ^{(n+1)}(S))\).
Finally, suppose \(\Gamma ^{11 \ldots 1} =0\). If \(\Gamma \) is identically 0, then it can be represented by the Pfaffian of an all zero matrix. Suppose \(\Gamma \) is not identically 0, and \(\Gamma ^{T_0} = \lambda \not = 0\), for some \(T_0 \subset [n]\). We can define a modified matchgate \(\tilde{\Gamma }\) extending the i-th external node \(x_i\) by an edge \((x_i, x'_i)\) of weight one, making \(x'_i\) the new external node, for every \(i \not \in T_0\). Also add one isolated edge with weight \(1/\lambda \). Then the signature of \(\tilde{\Gamma }\) satisfies \(\Gamma ^{T} = \lambda \tilde{\Gamma }^{T \oplus {T_0^c}}\), and \(\tilde{\Gamma }^{11 \ldots 1} = \Gamma ^{T_0}/\lambda = 1\). Now we can apply the construction above to matchgate \(\tilde{\Gamma }\). More specifically, define \(\tilde{B}\) from \(\tilde{\Gamma }\), and \(\Lambda ^{(n)}(S)\) as before. Let \(G = H_2^{\otimes n} \Gamma \), then for n even,
For n odd, the same result follows using \(\tilde{B}^{+}\) and \(\Lambda ^{(n+1)}(S)\) with \(\lambda _{n+1} =1\).
We summarize the discussion as follows:
Theorem 0.1
A signature G is realizable as a matchgate signature under the Hadamard transformation \(H_2\) iff it can be parameterized by an \(n \times n\) skew-symmetric matrix as follows. If n is even, then there exist \(\lambda \in \mathbb {C}\), \(T_0 \subseteq [n]\), and \(n \times n\) skew-symmetric matrix B, such that for all \(S \subseteq [n]\),
where \(\Lambda ^{(n)}(S)\) is defined in (2). If n is odd, then it is suitably modified as described above
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J.-Y. Cai: Supported by NSF CCF-1217549.
The online version of the original article can be found under doi:10.1007/s00453-009-9383-3.
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Cai, JY., Lu, P. Erratum to: Signature Theory in Holographic Algorithms. Algorithmica 74, 1473–1476 (2016). https://doi.org/10.1007/s00453-015-0090-y
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DOI: https://doi.org/10.1007/s00453-015-0090-y