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# Quantum Computing

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In the last decades, an exponential growth of computing power has taken place, because the number of transistors on a chip doubled approximately every two years as predicted by Moore's law. However, as we are approaching the physical limits of how small a transistor can be built, we need to investigate into alternative technologies, namely quantum computers, which harness quantum effects to perform computations.

Adiabatic quantum computing (AQC) is a branch of quantum computing which has gained significant interest in the last decade. Given the problem of finding a non-trivial ground state of a physical system, AQC relies on the idea that there exists another system, the ground state of which can be easily prepared. Hence, starting from the ground state of the latter, if the state is evolved adiabatically slowly under a Hamiltonian which is initially the one of the second system and in the end the one of the first, the state can be shown to always stay in the ground state of the instantaneous Hamiltonian. This way, the ground state of our initial Hamiltonian can be prepared.

Similar to the classic gate-based approach to quantum computing, AQC relies on the quantum nature of the system and lives in an exponentially large Hilbert space. Quantum evolution can hence be exactly simulated on a classical computer only for very small system. To simulate larger, much more interesting systems, we hence rely on approximations. One such approximation is simulated quantum annealing (SQA). In essence, it is an extension of the celebrated classical annealing algorithm with an additional dimension in the imaginary time direction. SQA allows us to simulate much larger systems by overcoming the exponential barrier of exact quantum evolution. However, its approximation and how close it resembles the physical system is not yet understood and further studies are hence critical to be able to use SQA reliably. Gaining a deeper understanding of this approximation and using SQA to simulate large quantum systems is another research direction in our group.

Up to now, there exist only a very limited number of quantum algorithms which scale faster on a quantum computer than on a classical computer. We develop new algorithms for quantum computers and show how they compare to classical algorithms. New quantum algorithms should scale better than classical algorithms because building a quantum computer is much more difficult than building classical logic gates. Moreover, we optimize existing quantum algorithms in order to reduce the required number of gates. Similar to how a compiler on our computer can optimize program code to increase the speed of a program, we use our expertise to reduce the required number of quantum gates for a given quantum algorithm.