Towards fault-tolerant quantum computing with trapped ions
Choosing the rules of quantum physics as the physical basis for constructing models of computation allows for solving certain computational problems more efficiently as in models based on classical physics. In the quantum circuit model, information is encoded in quantum bits and manipulated by applying appropriate quantum operations acting on the joint state space of the qubits. Similar to what is done in classical computation, these quantum operations can be decomposed into a sequence of gate operations, consisting of single-qubit operations and entangling operations acting on pairs of qubits.
Currently, the experimental effort to implement in these ideas in practice have just taken the first steps demonstrating key concepts needed for the development of a full-grown quantum computer. It is generally agreed upon that such a device would have to make intensitive use of quantum error correction techniques to preserve the all-important quantum features in a system consisting of thousands of quantum bits. To build a reliable quantum computer from faulty components in a fault-tolerant way requires quantum gates operations that may not be perfect but that still have to be extremely reliable. The estimates for error thresholds fault-tolerant computation stongly depend on assumptions regarding the computational model but are generally believed to be on the order of 10-4 per gate operation.
Ion trap quantum gates
At the moment, experimental implementations of gate operations are still falling short of fulfilling this requirement. While the implementation of single-qubit gates as well as the preparation and detection of qubits have been demonstrated with error as low as 0.1%, entangling two-qubit gate operations so far exhibited errors ranging typically between 3% and 10%. Here, our most recent work published in Nature Physics  makes a significant progress by demonstrating a gate operation capable of entangling a pair of ions with an error as low as 0.7%.
All entangling operations on strings of ions demonstrated so far create entanglement by laser pulses that couple the ions' internal state with the state of motion of the ion string. In this way, the Coulomb interaction between the ions serves to mediate the entanglement generation between the internal states of the ions. In all previous experiments, our group has entangled ions by gate operations relying on strongly focussed laser beams that interact with a single ion at a time. In our latest experiment, we realize a gate between two ions by illuminating them at the same time with a laser pulse composed of two frequencies that induces correlated changes of the internal states of the ions. This technique was proposed by A. Sorensen and K. Molmer  already in 1999, and it has been implemented by groups encoding quantum information in the hyperfine states of an ion.
Entangling ions with a fidelity F> 99%
In contrast, in our experiments with 40Ca+, quantum information is stored in the ground state and a long-lived metastable state. The coupling of the laser to the ion-motion is much weaker than in experiments with hyperfine quantum bits which makes the gate more difficult to implement as the laser pulse also tends to change the internal state of the ions without coupling them to the motional degree of freedom.
In our experiments, we could demonstrate that this coupling can be overcome by replacing laser pulses that are abruptly switched on by pulses with smoothly changing light intensities [1,3]. Starting with two ions in the state |S>|S>, a pulse as short as 50 mus was used to prepare the ions in the entangled state |SS> + i |DD> with a fidelity as high as 99.3%. Moreover, for the characterization of the gate, we investigated the performance of multiple gate operations and could show that even after 21 gates the change of creating an entangled state was still 80\%. Another encouraging aspect of the gate operations is related to the fact, that in principle, it does not require the ions' motional state to be prepared in the lowest quantum state of the confining potential which greatly relaxes the requirements for laser cooling.
In a next step, we hope to extend the gate mechanism to more than two ions for the generation of GHZ states and the realization of more complicated quantum operation.
We are financially supported by Österreichische Akademie der Wissenschaften, Universität Innsbruck, Fonds zur Förderung der wissenschaftlichen Forschung (FWF) within the program "Control and Measurement of Coherent Quantum Systems", the European network "SCALA" as well as IQI and IARPA.