Marek Sasura

We consider quantum gates for trapped ions using state-selective displacement of the ions. We generalize earlier work in order to treat arbitrary separations between the traps. This requires the impact of anharmonicity arising from the Coulomb interaction to be estimated. We show that its effects are always small enough to allow high fidelity. In particular, the method can be applied to two ions in the same trap. We also show that gates between non-neighbour ions, and hence a Toffoli (three-qubit controlled-NOT) gate, can be achieved. We discuss how the gate can be applied to logical qubits encoded in the decoherence-free-subspace {|01>,|10>}, where each pair of ions stores a single qubit. We also suggest alternatives to the spin-echo method to suppress unwanted terms in the evolution.

The "pushing gate" proposed by Cirac and Zoller in 2000 for quantum logic in ion traps is discussed, in which a force is used to give a controlled push to a pair of trapped ions and thus realize a phase gate. The original proposal had a weakness in that it involved a hidden extreme sensitivity to the size of the force. Also, the physical origin of this force was not fully addressed. Here, we discuss the sensitivity and present a way to avoid it by choosing the spatial form of the pushing force in an optimal way. We also analyse the effect of imperfections in a pair of pi pulses which are used to implement a "spin-echo" to cancel correlated errors. We present a physical model for the force, namely the dipole force, and discuss the impact of unwanted photon scattering, and of finite temperature of the ions. The main effect of the temperature is to blur the phase of the gate owing to the ions exploring a range of values of the force. When the distance scale of the force profile is smaller than the ion separation, this effect is more important than the high-order terms in the Coulomb repulsion which were originally discussed. Overall, we find that whereas the "pushing gate" is not as resistant to imperfections as was supposed, it remains a significant candidate for ion trap quantum computing since it does not require ground state cooling, and in some cases it does not require the Lamb-Dicke limit, while the gate rate is fast, close to (rather than small compared to) the trap vibrational frequency.

In this tutorial we review physical implementation of quantum computing using a system of cold trapped ions. We discuss systematically all the aspects for making the implementation possible. Firstly, we go through the loading and confining of atomic ions in the linear Paul trap, then we describe the collective vibrational motion of trapped ions. Further, we discuss interactions of the ions with a laser beam. We treat the interactions in the travelling-wave and standing-wave configuration for dipole and quadrupole transitions. We review different types of laser cooling techniques associated with trapped ions. We address Doppler cooling, sideband cooling in and beyond the Lamb-Dicke limit, sympathetic cooling and laser cooling using electromagnetically induced transparency. After that we discuss the problem of state detection using the electron shelving method. Then quantum gates are described. We introduce single-qubit rotations, two-qubit controlled-NOT and multi-qubit controlled-NOT gates. We also comment on more advanced multi-qubit logic gates. We describe how quantum logic networks may be used for the synthesis of arbitrary pure quantum states. Finally, we discuss the speed of quantum gates and we also give some numerical estimations for them. A discussion of dynamics on off-resonant transitions associated with a qualitative estimation of the weak coupling regime and of the Lamb-Dicke regime is included in Appendix.

We show how to construct a multi-qubit control gate on a quantum register of an arbitrary size N. This gate performs a single-qubit operation on a specific qubit conditioned by the state of other N-1 qubits. We provide an algorithm how to build up an array of networks consisting of single-qubit rotations and multi-qubit control-NOT gates for the synthesis of an arbitrary entangled quantum state of N qubits. We illustrate the algorithm on a system of cold trapped ions. This example illuminates the efficiency of the direct implementation of the multi-qubit CNOT gate compared to its decomposition into a network of two-qubit CNOT gates.