Hierarchy of universal entanglement in 2D measurement-based quantum computation

Jacob Miller,Akimasa Miyake

doi:10.1038/npjqi.2016.36

Abstract

AbstractMeasurement-based quantum computation (MQC) is a paradigm for studying quantum computation using many-body entanglement and single-qubit measurements. Although MQC has inspired wide-ranging discoveries throughout quantum information, our understanding of the general principles underlying MQC seems to be biased by its historical reliance upon the archetypal 2D cluster state. Here we utilise recent advances in the subject of symmetry-protected topological order (SPTO) to introduce a novel MQC resource state, whose physical and computational behaviour differs fundamentally from that of the cluster state. We show that, in sharp contrast to the cluster state, our state enables universal quantum computation using only measurements of single-qubit Pauli X, Y, and Z operators. This novel computational feature is related to the ‘genuine’ 2D SPTO possessed by our state, and which is absent in the cluster state. Our concrete connection between the latent computational complexity of many-body systems and macroscopic quantum orders may find applications in quantum many-body simulation for benchmarking classically intractable complexity.

Highlights

The idea of measurement-based quantum computation (MQC), where computation is carried out solely through single-qubit measurements on a fixed many-body resource state and classical feed-forward of measurement outcomes,[1,2,3] is quite surprising
-called universal resource states, the states that are capable of efficiently implementing universal MQC, represent a class of maximal entanglement in the classification of many-body entanglement,[4] so that the structure and complexity of their entanglement is of great interest for advancing the understanding of quantum computation
The earliest resource states for MQC were found in short-range correlated states described as somewhat artificial tensor network states,[4,6,7,8,9,10] a new insight has been that a class of short-ranged entangled states structured by symmetry, endowed with so-called symmetry-protected topological order (SPTO),[16,17,18,19,20,21,22,23,24] make excellent candidate resource states systematically

Summary

INTRODUCTION

The idea of measurement-based quantum computation (MQC), where computation is carried out solely through single-qubit measurements on a fixed many-body resource state and classical feed-forward of measurement outcomes,[1,2,3] is quite surprising. We introduce our ‘Union Jack’ state, which in contrast possesses SPTO entirely of a 2D nature, and demonstrate that it is a universal resource state but is ‘Pauli universal,’ in that it can implement arbitrary quantum computation using only single-qubit measurements in the Pauli bases. As elaborated later, this feature is forbidden in the 2D cluster state on account of the Gottesman-Knill theorem,[32] which proves the efficient classical simulability of certain quantum gates. We will conclude with the outlook that our proof of principle result about Pauli universality may be true for more general 2D SPTO resource states, which we connect to a possible deep connection between a hierarchy of SPTO in condensed matter physics and the so-called Clifford hierarchy of quantum computation

RESULTS

The resource state with nontrivial 2D SPTO

METHODS

DISCUSSION