# FOPRA Experiment 39: Universal Gate Sets for Quantum Computation

## Course 0000100039 in SS 2021

### General Data

Course Type | practical training |
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Semester Weekly Hours | 1 SWS |

Organisational Unit | Chair of Mathematical physics (Prof. Wolf) |

Lecturers |
Robert König Simone Warzel Michael Marc Wolf Assistants: Silvia Schulz |

Dates |

### Further Information

Courses are together with exams the building blocks for modules. Please keep in mind that information on the contents, learning outcomes and, especially examination conditions are given on the module level only – see section "Assignment to Modules" above.

additional remarks | In the standard (circuit) model of quantum computation, a typical computation proceeds by application of an n-qubit unitary U to a certain initial state and subsequent measurement of each qubit in the computational basis. The hardware of a universal quantum computer should therefore provide the capability of applying an arbitrary unitary U. In practice, however, control of quantum systems is severely limited by the speciﬁcs of the system under consideration and/or experimental limitations. For example, control pulses (realizing unitary evolutions) may only be applied along a certain subset of directions with a restricted range of frequencies and/or durations. Is such hardware su cient to reach the desired goal of performing universal quantum computation? If so, how many operations are required to realize a given unitary U? In this project, you will be given a (ﬁcticious) device that implements a single-qubit unitary speciﬁed by a discrete set of control parameter values. Your task is to use (several copies of) this device to (approximately) realize an arbitrary n-qubit unitary U. In other words, you will generate an instruction set (a quantum circuit) for executing the corresponding quantum computation using the given hardware. A key notion of interest here is that of a universal gate set, a ﬁnite family of (typically single- and two-qubit) unitaries (called gates) such that any unitary U can be approximately written as a product of a certain number L of gates. Given such a universal gate set, we may ask about the relationship between the length L of the gate sequence (ultimately corresponding to the runtime of your computation) and the quality of approximation. In addition, we want to efficiently (by classical computation) ﬁnd a suitable sequence. The so-called Solovay-Kitaev theorem provides answers to this problem and is the main topic of this project. The project will involve a collection of mathematical exercises. You will also gain experience programming in python by developing and implementing a compilation procedure as described above. |
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Links |
Course documents TUMonline entry |