[quantum-info] PIQuDos seminar Wed Sept 24: Beni Yoshida

[Gus Gutoski]

Please join us on Wednesday, September 24 at 4pm in the Time room for
a quantum information seminar by Beni Yoshida.

Title:  Fault-tolerant logical gates in quantum error-correcting codes
Speaker: Beni Yoshida (Caltech)

Abstract: 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

Date: Wednesday, September 24, 2014.
Time: 16:00
Location: Time Room