Adiabatic Gate Teleportation

The difficulty in producing precisely timed and controlled quantum gates is a significant source of error in many physical implementations of quantum computers. Here we introduce a simple universal primitive, adiabatic gate teleportation, which is robust to timing errors and many control errors and maintains a constant energy gap throughout the computation above a degenerate ground state space. This construction allows for geometric robustness based upon the control of two independent qubit interactions. Further, our piecewise adiabatic evolution easily relates to the quantum circuit model, enabling the use of standard methods from fault-tolerance theory for establishing thresholds.

Figure 1

By adiabatically dragging, cyclically, among three Hamiltonians $H_1$, $H_2$, and $H_3$, we can perform universal QC. Here we diagram how this works for a single-qubit circuit (circuit below, the Hamiltonians at different times diagramed from top to bottom). Each circle represents a qubit, and a bar represents a two-qubit Hamiltonian rotated by a labeled unitary $U_i$. Notice how, in each step to the next Hamiltonian, the qubit is swapped over two qubits (the arrows) and a gate is applied to this qubit. Thus the gates to be applied are encoded spatially across the three Hamiltonians. The $i$th gate thus depends on the Hamiltonian $H_{(i-1)\mathrm{mod}3+1}$ with the gate being applied changing the interaction between qubits $2i$ and $2i+1$ in this Hamiltonian. By generalizing to more than one qubit, this proves that universal holonomic QC can be done by interpolation among only three Hamiltonians.