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Algorithmica

, Volume 74, Issue 4, pp 1473–1476

# 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., 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)
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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)

## Copyright information

© Springer Science+Business Media New York 2015

## Authors and Affiliations

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