On the computational complexity of exterior products

where ∧ denotes exterior product. The above expression is the exterior product of M two-forms of rank two, which produces a volume form D2,2(v1, . . . , v4M ) in the 2M -dimensional linear space. If we fix a basis e1, . . . , e2M in the linear space, then each of the vectors vi can be expressed as a sequence of 2M numbers, and D2,2(v1, . . . , v4M ) is also a number (the coefficient at e1∧. . .∧e2M ). In other terms, D2,2(v1, . . . , v4M ) is a sum of 2 determinants of 2M × 2M matrices composed of vectors vi: the matrix is split into M pair of rows, and, for each pair of rows, two pairs of vectors vi are considered (for the first pair of rows, either v1 and v2 or v3 and v4, and so on). It is well known that computing the determinant of a matrix can be done in a time polynomial in the matrix dimension (see, e.g., the textbook [1]). One may therefore pose a question if a calculation of D2,2(v1, . . . , v4M ) could also be optimized to a polynomial in M number of operations. The result of this paper answers this question negatively, assuming that #P-hard problems cannot be solved in polynomial time. More precisely, I show that computing D2,2(v1, . . . , v4M ) is at least as difficult as computing a permanent of a matrix, which is, in turn, known to be #P-complete [2]. The proof is done by re-expressing the permanent of an arbitrary N × N matrix as D2,2(v1, . . . , v4M ) with M = N for a specially chosen set of vectors vi. The details of the proof are presented in Section II. An alternative proof was communicated to me by L. Gurvits [3] based on a relation to mixed discriminants [4–6]. This proof is outlined in Section III. Section IV contains a brief discussion of the result, including some simple generalizations. Finally, the last section provides a physical motivation for considering the function (1). Namely, it emerges in one of the attempts to generalize the BosonSampling proposal for quantum computing [7] to the sampling of noninteracting fermions with the initial state entangled in a special way.