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In this paper, Aaronson and Arkhipov use the following lemma (lemma 67 in the paper):

Let $V \in \mathbb{C}^{n \times n}$ be a matrix of rank $k$. Then $\operatorname{Per}(V + I)$ is computable exactly in $n^{O(k)}$ time.

Unfortunately, they refer to a "forthcoming paper" which I was unable to find. Is there a publication where this or a similar statement is proven?

Alexey Uvarov
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1 Answers1

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Alexander Barvinok gave an algorithm to compute the permanent of a rank $r$ matrix in $O(n^{r-1})$ arithmetic operations; see Theorem 3.3 in:

Barvinok, Alexander I., Two algorithmic results for the traveling salesman problem, Math. Oper. Res. 21, No. 1, 65-84 (1996). ZBL0846.90115.

This is for $\text{Per}(V)$, not $\text{Per}(\text{Id}+V)$. I'm not sure how to put the identity matrix in; the first version of this answer said I could do it, but then I started writing out the details and failed.

David E Speyer
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