Using superconducting qubit circuits to engineer exotic lattice systems

We propose an architecture based on superconducting qubits and resonators for the implementation of a variety of exotic lattice systems, such as spin and Hubbard models in higher or fractal dimensions and higher-genus topologies. Spin systems are realized naturally using qubits, while superconducting resonators can be used for the realization of Bose-Hubbard models. Fundamental requirements for these designs, such as controllable interactions between arbitrary qubit pairs, have recently been implemented in the laboratory, rendering our proposals feasible with current technology.

Figure 1

Schematic diagrams of (a) a simple three-junction flux qubit, (b) a flux qubit with a tunable gap, (c) two flux qubits interacting through a tunable coupler, and (d) two resonators interacting through a tunable coupler. In (a) the external magnetic flux ${\Phi }_{\mathrm{ext}}$ threading the superconducting loop controls the parameter $\epsilon$ in the single-qubit Hamiltonian [Eq. (1)]. In (b) the magnetic flux ${\Phi }_{\epsilon }$ controls the parameter $\epsilon$ in the Hamiltonian, while ${\Phi }_{\alpha }$ controls the parameter $\Delta$. In (c) two flux qubits are coupled inductively to a common coupler, resulting in an effective coupling between the two qubits. The effective coupling strength $J$ can be tuned through the magnetic flux ${\Phi }_{\mathrm{C}}$. In (d) two $\mathit{LC}$ circuits, i.e., resonators, are effectively coupled to each other through a tunable coupler.

Figure 2

Engineering the effective dimensionality of a lattice system by changing the connections between qubits. (a) Nine qubits can be connected so as to form a linear chain [purple (black) connections] or a $3\times 3$ square lattice [purple (black) and orange (gray) connections]. (b) Eight qubits can be connected into a three-dimensional cube.

Figure 3

First few generations of a Sierpinski carpet with dimension $\log [8]/\log [3]\approx 1.9$: one starts from a square of eight qubits, a $3\times 3$ lattice with the middle qubit removed; then each site is replaced by the original, first-generation square; and so on. Any dimension between 1 and 2 can be obtained by adjusting the total size and missing fraction in the first-generation square: if the length of the square is $n$ qubits and the length of the missing core is $l$ qubits, the effective dimension is $\log [n^2-l^2]/\log [n]$. Starting from a three-dimensional cube, one can obtain any dimension between 2 and 3, and so on.