A quantum Mirković-Vybornov isomorphism

Modular Representations of Reductive Groups and Geometry of Affine Grassmannians

By the geometric Satake isomorphism of Mirković and Vilonen, decomposition numbers for reductive groups can be interpreted as decomposition numbers for equivariant perverse sheaves on the complex affine Grassmannian of the Langlands dual group. Using a description of the minimal degenerations of the affine Grassmannian obtained by Malkin, Ostrik and Vybornov, we are able to recover geometricall...

Graph isomorphism and adiabatic quantum computing

In the Graph Isomorphism (GI) problem two N -vertex graphs G and G′ are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G → G′. If yes, then G and G′ are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important problem in computer science and is thought to be of comparable difficul...

Isomorphism and Factorization- Classical and Quantum Algorithms

Graph Non-Isomorphism Has a Succinct Quantum Certificate

This paper presents the first quantum computational characterization of the Graph Non-Isomorphism problem (GNI). We show that GNI is in a quantum non-deterministic polynomial-time class, i.e., quantum Merlin-Arthur game class (QMA) or quantum probabilistic NP (BQNP). In other words, given two graphs (G0, G1) ∈ GNI, (G0, G1) has a polynomial-size quantum certificate with which (G0, G1) ∈ GNI can...

A Quantum Observable for the Graph Isomorphism Problem

Suppose we are given two graphs on n vertices. We define an observable in the Hilbert space C[(Sn ≀ S2)m] which returns the answer “yes” with certainty if the graphs are isomorphic and “no” with probability at least 1 − n! 2 if the graphs are not isomorphic. We do not know if this observable is efficiently implementable.