Southwest Quantum Information and Technology

Universal logical gate sets with constant-depth circuits for topological and hyperbolic quantum codes

Presenting Author: Guanyu Zhu, IBM; Joint Quantum Institute (JQI): University of Maryland
Contributing Author(s): Ali Lavasani, Maissam Barkeshli

A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. To date, common approaches for implementing a universal logical gate set, such as schemes utilizing magic state distillation, require a substantial space-time overhead. In this work, we show that braids and Dehn twists, which generate the mapping class group of a generic high genus surface and correspond to logical gates on encoded qubits in arbitrary topological codes, can be performed through a constant depth circuit acting on the physical qubits. In particular, the circuit depth is independent of code distance d and system size. The constant depth circuit is composed of a local quantum circuit, which implements a local geometry deformation, and a permutation of qubits. When applied to anyon braiding or Dehn twists in the Fibonacci Turaev-Viro code based on the Levin-Wen model, our results demonstrate that a universal logical gate set can be implemented on encoded qubits in O(1) time through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. Our results for Dehn twists can be extended to the context of hyperbolic Turaev-Viro codes as well, which have constant space overhead (constant rate encoding). This implies the possibility of achieving a space-time overhead of O(d/log d).

Read this article online: https://quantum-journal.org/papers/q-2019-08-26-180/, https://arxiv.org/abs/1806.02358, https://arxiv.org/abs/1806.06078

(Session 5 : Saturday from 5:00pm - 7:00pm)