### Quantum Computations on a Topologically Encoded Qubit

**From Left to Right: (**

*top row*) Daniel Nigg, Markus Müller, Esteban Martínez, Philipp Schindler, (*bottom row*) Markus Hennrich, Thomas Monz, Miguel Angel Martín-Delgado, Rainer Blatt.**Authors: Markus Müller**

^{1}and Daniel Nigg^{2}

**Affiliation:**

^{1}Departamento de Física Teórica I, Universidad Complutense, Spain.

^{2}Institut für Experimentalphysik, Universität Innsbruck, Austria.**Email: mueller@ucm.es, daniel.nigg@uibk.ac.at**

Even computers are error-prone. The slightest disturbances may alter saved information and falsify the results of calculations. To overcome these problems, computers use specific routines to continuously detect and correct errors. This also holds true for a future quantum computer, which will also require procedures for error correction. Whereas general quantum states can not be simply copied, fragile quantum information can still be protected from errors during storage and information processing by using quantum error correcting codes. Here, quantum states are encoded in entangled states that are distributed over several physical particles.

**A quantum bit encoded in seven ions**

In the experiment realized at the University of Innsbruck, Austria [1], we confined seven calcium ions in an ion trap, with one qubit stored in each of the ions. In our setup, we use lasers to cool the ion string to almost absolute zero temperature and to precisely control their quantum properties. We used the register of seven physical qubits to encode quantum states of one logical qubit in entangled states of these particles. The topological quantum error-correcting code employed in the experiment provided the program for this encoding process, and was proposed and developed in the theory group at the Universidad Complutense in Madrid, Spain. The code arranges the qubits on a two-dimensional lattice structure where they interact with the neighboring particles. The encoding of the logical qubit in the seven physical qubits was the experimentally most challenging step. It required a long sequence of laser pulses to effectively realize three entangling gate operations, each acting on subsets of four neighboring qubits belonging to one plaquette.

**Figure 1: Schematics of the string of 7 ions stored in a linear Paul trap, with each ion hosting one physical qubit. One logical qubit is encoded in entangled states of these 7 physical qubits, by using a quantum error correcting code which arranges the qubits on a two-dimensional triangular lattice of three plaquettes.**

**Detection of arbitrary errors and logical quantum gate operations**

After the encoding step, once the atoms are entangled in this specific way, the quantum correlations provide a resource for subsequent error correction and quantum computations on the encoded logical qubit. Using the available set of laser pulses we induced at purpose all types of single-qubit errors that can occur on any of the seven physical qubits. Our measurements demonstrate that the quantum code is indeed able to independently detect phase flip errors, bit flip errors as well as combinations of both, regardless on which of the qubits these occur.

**Figure 2: Schematics of error detection by the 7-qubit code: Arbitrary errors (in the shown example a phase flip error Z on qubit 5) manifests itself as excitations on one or several plaquettes (black filled circle on the blue plaquette) and by its characteristic signature, the error syndrome. The latter allows one to deduce the type, i.e. whether a bit flip, phase flip or combined error of both has occurred, as well as the location of the error in the qubit register.**

Next, we applied logical quantum gate operations onto the encoded logical qubit. The 7-qubit quantum code we used allowed us to implement individual operations and longer sequences of gate operations (the single-qubit Clifford group) on the logical qubit in a transversal way, i.e. by applying the corresponding operations bitwise on each of the 7 physical qubits.

**Towards a fault-tolerant quantum computer**

The 7-ion system we used for encoding one logical quantum bit can serve as a building block for larger quantum systems. Storing and processing logical quantum information in larger lattice systems with more physical qubits is predicted to further increase the robustness with respect to noise and errors. The required technology in the form of two-dimensional ion trap arrays, which would enable the storage and manipulation of larger numbers of qubits, are currently developed and tested at the University of Innsbruck as well as in other laboratories worldwide. Together with further theoretical progress and optimized quantum error correcting codes, the result of these developments might be a quantum computer that could reliably perform arbitrarily long quantum computations without being impeded by errors.

For further background information and explication, please watch this video:

**Funding:**

The researchers are financially supported by the Spanish Ministry of Science, the Austrian Science Fund, the U.S. Government, the European Commission and the Federation of Austrian Industries Tyrol.

**Reference:**

**[1]**Daniel Nigg, Markus Müller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel Angel Martin-Delgado, Rainer Blatt, "Quantum Computations on a Topologically Encoded Qubit". Science, 345, 302 (2014). Abstract.

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