# Heidelberg 2015 – wissenschaftliches Programm

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# Q: Fachverband Quantenoptik und Photonik

## Q 18: Quantum Information: Concepts and Methods III

### Q 18.1: Vortrag

### Dienstag, 24. März 2015, 11:00–11:15, K/HS1

**Random Hamiltonians, random circuits and unitary designs** — •Emilio Onorati^{1}, Martin Kliesch^{1}, Albert Werner^{1}, Winton Brown^{2}, and Jens Eisert^{1} — ^{1}Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany — ^{2}Département de Physique, Université de Sherbrooke

Randomness has proven to be a useful tool for a large variety of tasks in quantum information and physics, e.g. tomography and gate benchmarking, decoupling and thermalization making use of unitary designs, i.e. the application of random unitaries chosen according to the uniform Haar distribution. A known method to realise such unitary designs is provided by suitable random quantum circuits, where two-qubit gates are taken from a given distribution and repeatedly applied to a quantum state.

In this work, we provide a general picture of such designs, and specifically show that a continuous-time process with a local fluctuating Hamiltonian is able to mimic the properties of the Haar measure. In this spirit, we divide the total time in small steps; for each of these we generate a Hamiltonian composed of local terms weighted by gaussian coefficients, then we apply the corresponding unitary evolution.

The principal mathematical tool to the proof involves Markov chain theory. In particular, the continuous-time process defines a random walk over Pauli matrices, which converges to the same stationary distribution of the analogous walk induced by the Haar measure.