Space-Time Circuit-to-Hamiltonian Construction and Its Applications

Understand

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 fermionic model (see http://link.aps.org/doi/10.1103/PhysRevA.63.040302) We prove that for one-dimensional quantum circuits the gap of the circuit Hamiltonian is appropriately lower-bounded, partially using results on mixing times of Markov chains, so that the applications of this construction for QMA (and partially for quantum adiabatic computation) go through.