Here is used representation initially introduced for perfect state transfer. The chain with $n$ nodes is modeled by $n$ qubits. The more compact representation with $n$ qubits for a chain $N=2^n$ nodes is discussed elsewhere, e.g., see arXiv:1710.03615 [quant-ph](2017) with tutorial about topological quantum walk.
The nodes chain are mapped into states with single unit in computational basis
$$|100\dots 0\rangle,|010\dots 0\rangle,\dots,|000\dots1\rangle.$$Let us denote
$$|{\underline k}\rangle \equiv |{\underbrace{0\ldots 0}_{k-1}\,}1\underbrace{0\ldots 0}_{n-k}\rangle, \quad k=1,\ldots,n.$$The method also may be extended for $m>1$ particles using states with $m$ units in computational basis.
Let us start with with one particle on a chain with two nodes. By definition, the evolution may affect only states $|{\underline 1}\rangle \equiv |10\rangle$ and $|{\underline 2}\rangle \equiv |01\rangle$. The two-gate for such evolution is represented by some matrix
$$ M_u \equiv \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & u_{11} & u_{12} & 0 \\ 0 & u_{21} & u_{22} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, $$where $u_{jk}$ is unitary matrix.
The chain with two nodes corresponds to a link in a qubit chain and such a representation is directly extended into a sequence of two-qubit gates on a chain.
A matchgate is defined by two unitary matrices $u$ and $u'$ with $\det{u'}=\det{u}$
$$ M(u,u') \equiv \begin{pmatrix} u'_{11} & 0 & 0 & u'_{12} \\ 0 & u_{11} & u_{12} & 0 \\ 0 & u_{21} & u_{22} & 0 \\ u'_{21} & 0 & 0 & u'_{22} \end{pmatrix} $$The $M_u$ is particular case of matchgate $M(u,{\bf 1})$ with $u' = \begin{pmatrix}1&0\\0&1\end{pmatrix}.$
A quantum circuit with matchgates acting on neighboring qubits (aka matchcircuit) may be effectively modeled on a classical computer. So, such circuits may be used for testing of quantum computers with many qubits.
The $M_u$ also respects number of units in computational basis and so can be used for modeling of distribution of states or quantum walks along a chains in agreement with model defined earlier. State distribution along a quantum network with $n$ qubits in such a case may be mapped into corresponding scalar chain with $n$ nodes.
Because of $\det(u)=1$ the gate $M_u$ may be described with three angles similar with $u3$ and $C_{u3}$ gate in QISKit. Let us denote that as
$$ M_u \equiv M_{u3}(\theta, \phi, \lambda) \equiv \begin{pmatrix}\ 1 & 0 & 0 & 0\\ 0 & e^{-i(\phi+\lambda)/2}\cos(\theta/2) & -e^{-i(\phi-\lambda)/2}\sin(\theta/2) & 0\\ 0 & e^{i(\phi-\lambda)/2}\sin(\theta/2) & e^{i(\phi+\lambda)/2}\cos(\theta/2) & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}. $$The gate $M_{u3}$ may be written as a procedure on OpenQASM 2.0
mu3 (theta, phi, lambda) a,b {
cx a,b;
cu3 (theta,phi,lambda) b,a;
cx a,b;
}
Similar method is used further with Qiskit, however it may be not very optimal for hardware.
Let us consider as an example the staggered walk described earlier in simpler model of scalar quantum chain. The sequences of gates representing modifications of partitions $S'_1$ and $S_2$ now should use $4 \times 4$ analogues of $2 \times 2$ matrices applied earlier to neighboring nodes of chain
$$ m_2 \equiv M_{i S_2} = \begin{pmatrix}\ 1 & 0 & 0 & 0\\ 0 & 0 & i & 0\\ 0 & i & 0 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix} = M_{u3}(\pi/2,0,0). $$$$ m_1 \equiv M_{S'_1} = \begin{pmatrix}\ 1 & 0 & 0 & 0\\ 0 & 1/\sqrt{2} & -1/\sqrt{2} & 0\\ 0 & 1/\sqrt{2} & 1/\sqrt{2} & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix} = M_{u3}(\pi,\pi,0). $$See notebook with modeling of qubit chain using Qiskit.