A Fourier-analytic approach to counting partial Hadamard matrices

Abstract

Abstract. In this paper, we study a family of lattice walks which are related to the Hadamard conjecture. There is a bijection between paths of these walks which originate and terminate at the origin and equivalence classes of partial Hadamard matrices. Therefore, the existence of partial Hadamard matrices can be proved by showing that there is positive probability of a random walk returning to the origin after a specified number of steps. Moreover, the number of these designs can be approximated by estimating the return probabilities. We use the inversion formula for the Fourier transform of the random walk to provide such estimates. We also include here an upper bound, derived by elementary methods, on the number of partial Hadamard.

Entangling and assisted entangling power of bipartite unitary operations

“…⊓ ⊔ As a consequence of Lemma 21, we obtain a possible simplification for the proof of Conjecture 11: We need only consider the case that the θ j ∈ [0, 2π) in the conjecture are pairwise different. Finally, the generalized GCNOT gates may be defined as j |j j| ⊗ D j , where each D j is a diagonal unitary such that TrD † j D k = 0 for j = k. The existence of such gates is related to an open problem on the partial Hadamard matrices [36].…”

Counting Results for Thin Butson Matrices

“…As mentioned in [7], the method there should apply to more general situations. We discuss in what follows a potential extension to the partial Butson matrices: Observe that at q " 2 we obtain the PHM.…”

“…The proof in [7] uses a random walk interpretation of the partial Hadamard matrices, then the Fourier inversion formula, and then some real analysis methods. Importantly, as pointed out there, this method can be probably used for more general situations.…”

Asymptotic Enumeration of Difference Matrices over Cyclic Groups

“…Difference matrices have long been a part of the combinatorial design literature, and they are related to many other types of designs such as orthogonal arrays, transversal designs, pairwisebalanced designs, and more. For instance, a difference matrix over Z 2 is also a partial Hadamard matrix (see, for instance, [5]). A comprehensive overview of the existing literature on difference matrices and their relationships with other types of designs can be found in [4].…”

“…We pause to remark that this type of analysis is certainly not new to the study of combinatorial designs. De Launey and Levin used this tactic to study partial Hadamard matrices [5], and their enumeration results can be recognized as the particular case of Theorem 2 where g = 2, although their work shows the formula to be valid so long as the pairs (k, λ) satisfy the more generous condition k ≤ (2λ) 1/(12+ε0) . Additionally, the author of this work has used this strategy to count balanced incomplete block design incidence matrices [15,Thm 2.3].…”