Speaker: Guanyu Zhu (IBM)
Title: Non-Abelian quantum LDPC codes: TQFT beyond manifold and application to universal logical gates
Abstract: Quantum low-density parity-check (qLDPC) codes are promising candidates for low-
overhead fault-tolerant quantum computation due to its optimal encoding rate and
distance. They can be considered as generalization of Abelian topological stabilizer
codes, which are defined on general chain complexes instead of triangulated manifolds.
They also give rise to highly-entangled k-local quantum states of matter which extends
the notion of topological order to the so-called NLTS states, where not only the ground
states but all the low-lying excited states below certain energy density are non-trivial.
Open questions include how to define a topological quantum field theory (TQFT) to
describe such a state of matter and how to achieve a universal set of logical gates on
this type of codes.
In this talk, I’ll present a generalization of the framework of non-Abelian topological
codes on manifolds to non-Abelian qLDPC codes and the corresponding combinatorial
TQFT defined on Poincare CW complexes and certain types of general chain
complexes. This is achieved by geometrizing the quantum codes into a manifold using
the mapping by Freedman and Hastings. I’ll also show how to construct the spacetime
path integrals as topological invariants on these complexes. Remarkably, one can show
that native non-Clifford logical gates can be realized using constant-rate 2D hypergraph-
product codes and their non-Abelian variants. This is achieved by a spacetime path
integral effectively implementing the addressable gauging measurement of a new type
of 0-form subcomplex symmetries, which correspond to addressable transversal Clifford
gates and become higher-form symmetries when lifted to higher-dimensional CW
complexes or manifolds. Building on this structure, one can apply the gauging protocol
to the magic state fountain scheme for parallel preparation of O(n 1/2 ) disjoint logical CZ
magic states with code distance of O(n), using a total number of n qubits.
Reference:
[1] G. Zhu, R. Kobayashi, P.S. Hsin, arXiv:2601.06736