Fault-tolerant quantum computation is a technique that is necessary to build a scalable quantum computer from noisy physical building blocks. Key for the implementation of fault-tolerant computations is the ability to perform a universal set of quantum gates that act on the code space of an underlying quantum code. To implement such a universal gate set fault-tolerantly is an expensive task in terms of physical operations, and any possible shortcut to save operations is potentially beneficial and might lead to a reduction in overhead for fault-tolerant computations. We show how the automorphism group of a quantum code can be used to implement some operators on the encoded quantum states in a fault-tolerant way by merely permuting the physical qubits. We derive conditions that a code has to satisfy in order to have a large group of operations that can be implemented transversally when combining transversal CNOT with automorphisms. We give several examples for quantum codes with large groups, including codes with parameters [[8,3,3]], [[15,7,3]], [[22,8,4]], and [[31,11,5]].

}, author = {M. Grassl and M. R{\"o}tteler}, booktitle = {Proceedings of the 2013 IEEE International Symposium on Information Theory (ISIT'13), Istanbul, Turkey}, note = {See also arXiv preprint arXiv:1302.1035}, pages = {534–538}, title = {Leveraging automorphisms of quantum codes for fault-tolerant quantum computation}, url = {http://research.microsoft.com/apps/pubs/default.aspx?id=209848}, year = {2013}, }