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Algorithmica

, Volume 74, Issue 4, pp 1473–1476 | Cite as

Erratum to: Signature Theory in Holographic Algorithms

  • Jin-Yi Cai
  • Pinyan Lu
Erratum

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

  • \(H_2^{\otimes n}G\) is a standard matchgate signature, where \(H_2 = \left[ {\begin{matrix} 1 &{} 1 \\ 1 &{} -1 \end{matrix}}\right] \).

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 (ij) 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 \(S \subseteq [n]\),
$$\begin{aligned} G^S = \sum _{T \subseteq [n]} (-1)^{|S \cap T|} \Gamma ^{T}. \end{aligned}$$
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
$$\begin{aligned} \lambda _i = \left\{ \begin{array}{cl} -1 &{} \text{ if } i \in S\\ 1 &{} \text{ otherwise } \end{array} \right. \end{aligned}$$
then for these values of \(\lambda _i\), and the B defined from \(\Gamma \),
$$\begin{aligned} \mathrm{PfS}(B) = \sum _{T \subseteq [n]} (-1)^{|S \cap T|} \mathrm{Pf} (B[T]) = \sum _{T \subseteq [n]} (-1)^{|S \cap T|} \Gamma ^{T} = G^S. \end{aligned}$$
(1)
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 (ij) 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
$$\begin{aligned} \begin{bmatrix} (-1)^{\chi _S(1)}&0&\ldots&0\\ 0&(-1)^{\chi _S(2)}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\ldots&(-1)^{\chi _S(n)} \end{bmatrix} \begin{bmatrix} 0&1&-1&\ldots&1\\ -1&0&1&\ldots&-1\\ \vdots&\vdots&\vdots&\ddots&\vdots \\ -1&1&-1&\ldots&0 \end{bmatrix} \begin{bmatrix} (-1)^{\chi _S(1)}&0&\ldots&0\\ 0&(-1)^{\chi _S(2)}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\ldots&(-1)^{\chi _S(n)} \end{bmatrix} \end{aligned}$$
(2)
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 (ij) 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,
$$\begin{aligned} G^S= & {} \sum _{T \subseteq [n]} (-1)^{|S \cap T|} \Gamma ^{T}\\= & {} \lambda \sum _{T \subseteq [n]} (-1)^{|S \cap T|} \tilde{\Gamma }^{T \oplus T_0^c}\\= & {} \lambda \sum _{T' \subseteq [n]} (-1)^{|S \cap T'|} (-1)^{|S \cap T_0^c|} \tilde{\Gamma }^{T'}\\= & {} \lambda (-1)^{|S \cap T_0^c|} \mathrm{Pf}(\tilde{B} + \Lambda ^{(n)}(S)). \end{aligned}$$
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 \(S \subseteq [n]\),
$$\begin{aligned} G^S = \lambda (-1)^{|S \cap T_0^c|} \mathrm{Pf}({B} + \Lambda ^{(n)}(S)), \end{aligned}$$
where \(\Lambda ^{(n)}(S)\) is defined in (2). If n is odd, then it is suitably modified as described above
$$\begin{aligned} G^S = \lambda (-1)^{|S \cap T_0^c|} \mathrm{Pf}({B}^+ + \Lambda ^{(n+1)}(S)). \end{aligned}$$

References

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    Cai, J-Y., Choudhary, V.: Some results on matchgates and holographic algorithms. In: Proceedings of ICALP 2006, Part I. Lecture Notes in Computer Science vol. 4051. pp 703–714. Int. J. Softw. Info. 1(1), 3–36 (2007)Google Scholar
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    Cai, J-Y., Choudhary, V., Lu, P.: On the theory of matchgate computations. In Proceedings of CCC ’07: Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, 2007, pp. 305–318. Theory Comput. Syst. 45(1):108–132 (2009)Google Scholar
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    Cai, J-Y.: Aaron Gorenstein. Matchgates Revisited. Theory of Computing (ToC) Volume 10, Article 868, pp. 401–430 (2014)Google Scholar
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    Cai, J.-Y., Lu, P.: Signature theory in holographic algorithms. Algorithmica 61(4), 779–816 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
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    Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput. 31(4), 1229–1254 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA
  2. 2.Microsoft Research AsiaBeijingChina

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