The Hilbert space in quantum computation is used to express quantum states. In particular, they are vectors in $\mathbb{C} ^ {2^n}$, where $n$ is number of qubits the state is composed of. Any mapping from one state to another one is expressed by a unitary matrix.
This is (a very rough) description of basics behind mathematical model of quantum computation.
So, if you have a circuit, you can describe its action on different input basis states, i.e. how input is transformed to output. If you express these states as a vectors in a Hilbert space, in the end you come to matrix representation of the circuit. In other words, you can construct a matrix describing the circuit based on its action on different input states.
Hope this helps, as the question is a little bit unclear.
EDIT: An example how to convert circuit into matrix. Let's consider CNOT gate. This gate is two qubit gate. First qubit is called control, second target. The gate performs negation on the target if control is in state $|1\rangle$, so input-output map has this form:
- $|00\rangle \rightarrow |00\rangle$ (control in state $|1\rangle$, target is not negated)
- $|01\rangle \rightarrow |01\rangle$ (control in state $|1\rangle$, target is not negated)
- $|10\rangle \rightarrow |11\rangle$ (control in state $|1\rangle$, target is negated)
- $|11\rangle \rightarrow |10\rangle$ (control in state $|1\rangle$, target is negated)
Since
$$
|00\rangle = \begin{pmatrix}1 \\ 0 \\ 0 \\ 0 \end{pmatrix}
|01\rangle = \begin{pmatrix}0 \\ 1 \\ 0 \\ 0 \end{pmatrix}
|10\rangle = \begin{pmatrix}0 \\ 0 \\ 1 \\ 0 \end{pmatrix}
|11\rangle = \begin{pmatrix}0 \\ 0 \\ 0 \\ 1 \end{pmatrix}
$$
a matrix describing CNOT gate is
$$
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{pmatrix},
$$
i.e. outputs are simply put one beside another. The first column of the matrix represent reactio of the gate on $|00\rangle$ input, second reaction on $|01\rangle$ etc.
Concerning software, I use IBM Q Experience environment and Qiskit language.