Quantum circuits for strongly correlated quantum systems

We present an approach to gain detailed control on the quantum simulation of strongly correlated quantum many-body systems by constructing the explicit finite quantum circuits that diagonalize their dynamics. As a particularly simple instance, the full dynamics of a one-dimensional Quantum Ising model in a transverse field with four spins is shown to be reproduced using a quantum circuit of only six local gates. This opens up the possibility of experimentally producing strongly correlated states, their time evolution at zero time, and even thermal superpositions at zero temperature. Our method also allows one to uncover the exact circuits corresponding to models that exhibit topological order and to stabilizer states.

Figure 1

(Color online) Structure of the quantum circuit performing the exact diagonalization of the $XY$ Hamiltonian for eight sites. The circuit follows the structure of a Bogoliubov transformation followed by a fast Fourier transform. Three types of gates are involved: type-$B$ (responsible for the Bogoliubov transformation and depending on the external magnetic field $\lambda$ and the anisotropy parameter $\gamma$), type-fSWAP (depicted as crosses and necessary to implement the anticommuting properties of fermions), and type-$F$ (gates associated to the fast Fourier transform). Some initial gates have been eliminated since they only amount to some reordering of initial qubits.

Figure 2

(Color online) Quantum circuit that reproduces all the dynamical properties of the quantum Ising Hamiltonian in a external magnetic field with four spins. Some initial and final reordering gates can be sparsed in an experimental realization.