Fault-tolerant logical gates in quantum error-correcting codes


Yoshida, B. (2014). Fault-tolerant logical gates in quantum error-correcting codes. Perimeter Institute. https://pirsa.org/14090032


Yoshida, Beni. Fault-tolerant logical gates in quantum error-correcting codes. Perimeter Institute, Sep. 24, 2014, https://pirsa.org/14090032


          @misc{ pirsa_PIRSA:14090032,
            doi = {10.48660/14090032},
            url = {https://pirsa.org/14090032},
            author = {Yoshida, Beni},
            keywords = {Quantum Information},
            language = {en},
            title = {Fault-tolerant logical gates in quantum error-correcting codes},
            publisher = {Perimeter Institute},
            year = {2014},
            month = {sep},
            note = {PIRSA:14090032 see, \url{https://pirsa.org}}

Beni Yoshida Perimeter Institute for Theoretical Physics


Recently, Bravyi and Koenig have shown that there is a tradeoff between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant depth geometrically local circuit and are thus fault-tolerant by construction. In particular, they shown that, for local stabilizer codes in D spatial dimensions, locality preserving gates are restricted to a set of unitary gates known as the D-th level of the Clifford hierarchy. In this paper, we elaborate this idea and provide several extensions and applications of their characterization in various directions. First, we present a new no-go theorem for self-correcting quantum memory. Namely, we prove that a three-dimensional stabilizer Hamiltonian with a locality-preserving implementation of a non-Clifford gate cannot have a macroscopic energy barrier. Second, we prove that the code distance of a D-dimensional local stabilizer code with non-trivial locality-preserving m-th level Clifford logical gate is upper bounded by L^{D+1-m}. For codes with non-Clifford gates (m>2), this improves the previous best bound by Bravyi and Terhal. Third we prove that a qubit loss threshold of codes with non-trivial transversal m-th level Clifford logical gate is upper bounded by 1/m. As such, no family of fault-tolerant codes with transversal gates in increasing level of the Clifford hierarchy may exist. This result applies to arbitrary stabilizer and subsystem codes, and is not restricted to geometrically-local codes. Fourth we extend the result of Bravyi and Koenig to subsystem codes. A technical difficulty is that, unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of error threshold in a subsystem code, and the same conclusion as Bravyi-Koenig is recovered. This is a joint work with Fernando Pastawski. arXiv:1408.1720