As we know we can construct unitary matrix as $H=H_1H_2\dots,H_k$, by stacking householder matrices $H_i\in \mathbb{R}^{d\times d}$. The number of householder matrices we use, i.e., $k$, determines the expressivity of construction.
Conversely, for a given unitary matrix $H$, how can I know how many householder matrices I need at least for building this $H$? I mean how can I know the minimal $k$.
A Householder decomposition of a $d \times d$ unitary $H$ can be achieved with at most $d$ Householder matrices (see Theorem 1) and a tighter upper bound is $d-m$ Householder matrices where $m = \dim( \operatorname{ker}(H-I_d) )$ (see Theorem 2).
Uhlig, Frank, Constructive ways for generating (generalized) real orthogonal matrices as products of (generalized) symmetries, Linear Algebra Appl. 332-334, No. 1-3, 459-467 (2001). ZBL0982.65049.
