  In quantum computing, the connectivity of qubits placed on two-dimensional
chips limits the scalability and functionality of solid-state quantum
computers. This paper presents two approaches to constructing complex quantum
networks from simple qubit arrays, specifically grid lattices. The first
approach utilizes a subset of qubits as tunable couplers, effectively yielding
a range of non-trivial graph-based Hamiltonians. The second approach employs
dynamic graph engineering by periodically activating and deactivating couplers,
enabling the creation of effective quantum walks with longer-range couplings.
Numerical simulations verify the effective dynamics of these approaches. In
terms of these two approaches, we explore implementing various graphs,
including cubes and fullerenes, etc, on two-dimensional lattices. These
techniques facilitate the realization of analog quantum simulation,
particularly continuous-time quantum walks discussed in detail in this
manuscript, for different computational tasks on superconducting quantum chips
despite their inherent low dimensional simple architecture.
