# Using Quantum Computers for Quantum Simulation

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## Abstract

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## 1. Introduction

## 2. Universal Quantum Simulation

#### 2.1. Lloyd’s Method

#### 2.2. Errors and Efficiency

#### 2.3. Universal Hamiltonians

#### 2.4. Efficient Hamiltonian Simulation

## 3. Data Extraction

#### 3.1. Energy Gaps

#### 3.2. Eigenvalues and Eigenvectors

#### 3.3. Correlation Functions and Hermitian Operators

#### 3.4. Quantum Chaos

## 4. Initialization

#### 4.1. Direct State Construction

#### 4.2. Adiabatic Evolution

#### 4.3. Preparing Thermal Equilibrium States

## 5. Hamiltonian Evolution

#### 5.1. Quantum Pseudo-Spectral Method

#### 5.2. Lattice Gas Automata

#### 5.3. Quantum Chemistry

_{2}O and LiH, they show in detail that these methods are feasible. For the simulations, they use the Hartree-Fock approximation for the initial ground state. In some situations, however, this state has a vanishing overlap with the actual ground state. This means it may not be suitable in the dissociation limit or in the limit of large systems. A more accurate approximation of the required ground state can be prepared using adiabatic evolution, see Section 4.2. Aspuru-Guzik et al. confirm numerically that this works efficiently for molecular hydrogen. Data from experiments or classical simulations can be used to provide a good estimate of the gap during the adiabatic evolution, and hence optimise the rate of transformation between the initial and final Hamiltonians.

#### 5.4. Open Quantum Systems

## 6. Fermions and Bosons

#### 6.1. Hubbard Model

#### 6.2. The BCS Hamiltonian

#### 6.3. Initial State Preparation

#### 6.4. Lattice Gauge Theories

## 7. Overview

## 8. Proof-of-Principle Experiments

#### 8.1. NMR Experiments

#### 8.2. Photonic Systems

## 9. Atom Trap and Ion Trap Architectures

#### 9.1. Ion Trap Systems

#### 9.2. Atoms in Optical Lattices

#### 9.3. Atoms in Coupled Cavity Arrays

## 10. Electrons and Excitons

#### 10.1. Spin Lattices

#### 10.2. Quantum Dots

#### 10.3. Superconducting Architectures

## 11. Outlook

## Acknowledgments

## References

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**Figure 1.**A quantum circuit for measuring correlation functions, X is the Pauli ${\sigma}_{x}$ operator, U(t) is the time evolution of the system, and Hermitian operators $\widehat{A}$ and $\widehat{B}$ are operators (expressible as a sum of unitary operators) for which the correlation function is required. The inputs are a single qubit ancilla $|a\rangle $ prepared in the state $(|0\rangle +|1\rangle )/\sqrt{2}$ and $|\psi \rangle $, the state of the quantum system for which the correlation function is required. $\langle 2{\sigma}_{+}\rangle $ is the output obtained when the ancilla is measured in the $2{\sigma}_{+}={\sigma}_{x}+{\sigma}_{y}$ basis, which provides an estimate of the correlation function.

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Brown, K.L.; Munro, W.J.; Kendon, V.M. Using Quantum Computers for Quantum Simulation. *Entropy* **2010**, *12*, 2268-2307.
https://doi.org/10.3390/e12112268

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Brown KL, Munro WJ, Kendon VM. Using Quantum Computers for Quantum Simulation. *Entropy*. 2010; 12(11):2268-2307.
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**Chicago/Turabian Style**

Brown, Katherine L., William J. Munro, and Vivien M. Kendon. 2010. "Using Quantum Computers for Quantum Simulation" *Entropy* 12, no. 11: 2268-2307.
https://doi.org/10.3390/e12112268