Laboratoire d'Informatique de Grenoble

Abstract: The construction of an error correcting code with a universal set of transversal gates would be a very natural way of achieving fault tolerant universal quantum computing. Indeed, it was put forward by the pioneers of quantum computing as a potential option. However, such codes were later ruled out by the Eastin-Knill theorem.  Moreover, it also ruled out codes which are covariant with respect to the action of transversal unitary operations forming continuous symmetries. In this work, starting from an arbitrary code, we construct approximate codes which are covariant with respect to local SU(d) symmetries using quantum reference frames. We show that our codes are capable of efficiently correcting different types of erasure errors. When only a small fraction of the n qudits upon which the code is built are erased, our covariant code has an error that scales as 1/n^2, which is reminiscent of the Heisenberg limit of quantum metrology. When every qudit has a chance of being erased, our covariant code has an error that scales as 1/n. We show that the error scaling is optimal in both cases. Our approach has implications for fault-tolerant quantum computing, reference frame error correction, and the AdS-CFT duality.  ArXiv preprint available here: https://arxiv.org/abs/2007.09154