Studies on the Role of Entanglement in Mixed-state Quantum Computation

TL;DR

This thesis investigates the role of quantum entanglement and discord in mixed-state quantum computation, revealing that quantum discord may be the key resource despite minimal entanglement.

Contribution

It demonstrates the limited entanglement in DQC1 circuits and highlights quantum discord as a potential fundamental resource for mixed-state quantum computation.

Findings

Minimal bipartite entanglement in DQC1 circuits

Exponential classical resources needed for MPS simulation

Quantum discord remains significant and may drive computation

Abstract

In this thesis, I look at the role of quantum entanglement in mixed-state quantum computation. The model we consider is the DQC1 or `power of one qubit' model. I show that there is minimal bipartite entanglement in a typical instance of the DQC1 circuit and even put an upper bound on the possible amount of entanglement. This limited amount of entanglement, however, does not imply that the system is classically simulatable. A matrix product state (MPS) algorithm for a typical instance of the DQC1 system requires exponential classical resources. This exposes a gap between the amount of entanglement and the amount of purely nonclassical correlations in a quantum system. This gap, I suggest, can be filled by quantum discord. I calculate it in a typical instance of the DQC1 circuit and find that the amount of discord is a constant fraction of the maximum possible discord for a system of that size. This allows an interpretation of quantum discord as the resource that drives mixed-state quantum computation. I also study quantum discord as a quantity of independent interest. Its role in the phenomenon of entanglement distribution is studied through an easily comprehensible example. This thesis also contains discussions on the relation between the complexity classes P, BQP and DQC1. Additional material is presented on the connections between the DQC1 model, Jones polynomials and statistical mechanics. The thesis concludes with a discussion of a few open problems related to the DQC1 model, the quantum discord and their scope in quantum information science.