The circuit-to-Hamiltonian construction translates dynamics (a quantum circuit and its output) into statics (the groundstate of a circuit Hamiltonian) by explicitly defining a quantum register for a clock. The standard Feynman–Kitaev construction uses one global clock for all qubits while we consider a different construction in which a clock is assigned to each interacting qubit. This makes it possible to capture the spatio-temporal structure of the original quantum circuit into features of the circuit Hamiltonian. The construction is inspired by the original two-dimensional interacting fermion model in Mizel et al (2001 Phys. Rev. A 63 040302). We prove that for one-dimensional quantum circuits the gap of the circuit Hamiltonian is appropriately lowerbounded so that the applications of this construction for quantum Merlin–Arthur (and partially for quantum adiabatic computation) go through. For one-dimensional quantum circuits, the dynamics generated by the circuit Hamiltonian corresponds to the diffusion of a string around the torus.