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Martin Vesely's answer is the way to go in general, and especially if you know more than one column. However, if you're given just one column, there's an easier trick for generating a suitable unitary. Note that $V=2|v\rangle\langle v|-I$ is a unitary ($V=V^\dagger$ and $V^2=I$). So, the question is whether you can select a $|v\rangle$ such that the first ...
Take your vector $\frac{1}{\sqrt{5}}(0, 1, 1, 1, 1, 1)^T$ and five other arbitrary ones but at the same time these vectors have to be linearly independent. After that apply Gram-Schmidt process which produces orthonormal vectors. Put these vectors to a matrix and you will get a unitary matrix with the first column equal to $\frac{1}{\sqrt{5}}(0, 1, 1, 1, 1, ... 1 There are multiple ways to show that$W_j\$ is not, in general, unitary. The easiest might be to look at the determinant. A basic property of unitary matrices is that their determinant has unit modulus. In this case the determinant reduces to \text{det}(W_j)=4 \, (\vert \alpha \vert^2 - \vert \beta \vert ^2 + i2 \, \vert \alpha \vert \vert \beta \vert \... 1 The straightforward method is to compute  W W^\dagger = W^\dagger W = I  and to get constraint over your parameters solving this system. I'm going to show you how to do it only for  W W^\dagger = I but it should be very similar for  W^\dagger W. I assume here that all your parameters are real. first some notation for ease of reading, let's pose : ... 4 Hadamard gate can be interpreted as a rotation in 3D Euclidean space (on Bloch sphere) by angle \pi around X+Z axis. The qubit rotation by angle \theta around axis pointed by unit vector \textbf{n}=\{n_x,n_y,n_z\} is described by rotation operator (X, Y and Z are Pauli matrices) \begin{align} R_{\textbf{n}}(\theta)=&n_xe^{-i\frac{\theta}{2}X}... 1 The first postulate of quantum mechanics that can be found in the M. Nielsen and I. Chuang textbook: Postulate 1: Associated to any isolated physical system is a complex vector space with the inner product (that is, a Hilbert space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the ... 2 I'm not an expert on this, so there could easily be other ways (I get the impression you'd like to keep it to a d^2 dimensional calculation, where I'm taking it up to d^4), but what I'd do is calculate: J(\Phi_{U\circ V})=\left(I_d\otimes \langle\Omega|\otimes I_d\right)\left(J(\Phi_V)\otimes J(\Phi_U)\right)\left(I_d\otimes |\Omega\rangle\otimes I_d\...