Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Geometry of Quantum Computing

E. Ercolessi, R. Fioresi, T. Weber

TL;DR

This expository paper focuses on quantum information geometry and ZX-calculus, establishing a connection between quantum computing questions and quantum groups, i.e. Hopf algebras.

Abstract

In this expository paper we present a brief introduction to the geometrical modeling of some quantum computing problems. After a brief introduction to establish the terminology, we focus on quantum information geometry and ZX-calculus, establishing a connection between quantum computing questions and quantum groups, i.e. Hopf algebras.

The Geometry of Quantum Computing

TL;DR

This expository paper focuses on quantum information geometry and ZX-calculus, establishing a connection between quantum computing questions and quantum groups, i.e. Hopf algebras.

Abstract

In this expository paper we present a brief introduction to the geometrical modeling of some quantum computing problems. After a brief introduction to establish the terminology, we focus on quantum information geometry and ZX-calculus, establishing a connection between quantum computing questions and quantum groups, i.e. Hopf algebras.
Paper Structure (14 sections, 9 theorems, 56 equations, 8 figures)

This paper contains 14 sections, 9 theorems, 56 equations, 8 figures.

Table of Contents

  1. Introduction
  2. An overview on Quantum computing
  3. The Qubit
  4. Quantum Logic Gates
  5. Classical and Quantum Information Geometry
  6. The Fisher Information Matrix
  7. The Quantum Information Matrix
  8. Quantum Geometric Tensor
  9. ZX-calculus and Hopf algebras
  10. Quantum circuits and ZX diagrams
  11. The ZX-calculus
  12. Unnormalized and F-Hopf Algebras
  13. Unnormalized Hopf Algebras
  14. F-Hopf algebras

Key Result

Theorem 2.1

The set of gates that consists of all 1 qubit quantum gates (namely $U(2)$ operators) and the 2 qubit gate CNOT is universal in the sense that all unitary operations on $n$ qubits (namely $U(2^n)$) can be expressed as compositions of these gates.

Figures (8)

  • Figure 1: The Bloch Sphere
  • Figure 2: Quantum logic gates for $1$ qubit
  • Figure 3: CNOT logic gate
  • Figure 4: CCNOT logic gate
  • Figure 5: $X$, $Y$, $Z$ are the Pauli matrices gates, $H$ the Hadamard gate. The $S$ gate is $e^{i\pi/2}Z$, the $T$ gate is $e^{i\pi/4}Z$, $\dagger$ as usual indicates the adjoint.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.4
  • Definition 3.6
  • Proposition 3.8
  • proof
  • ...and 12 more