# Quantum Optimization and Approximation Algorithms.

## Abstract

Shor's groundbreaking quantum algorithm for integer factoring provides an exponential speedup over the best-known classical algorithms. In the 20 years since Shor's algorithm was conceived, only a handful of fundamental quantum algorithmic kernels, generally providing modest polyno- mial speedups over classical algorithms, have been invented. To better understand the potential advantage quantum resources provide over their classical counterparts, one may consider other re- sources than execution time of algorithms. Quantum Approximation Algorithms direct the power of quantum computing towards optimization problems where quantum resources provide higher- quality solutions instead of faster execution times. We provide a new rigorous analysis of the recent Quantum Approximate Optimization Algorithm, demonstrating that it provably outperforms the best known classical approximation algorithm for special hard cases of the fundamental Maximum Cut graph-partitioning problem. We also develop new types of classical approximation algorithms for finding near-optimal low-energy states of physical systems arising in condensed matter by ex- tending seminal discrete optimization techniques. Our interdisciplinary work seeks to unearth new connections between discrete optimization and quantum information science.

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1492737

- Report Number(s):
- SAND2019-0739

671780

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Parekh, Ojas D., Ryan-Anderson, Ciaran, and Gharibian, Sevag.
```*Quantum Optimization and Approximation Algorithms.*. United States: N. p., 2019.
Web. doi:10.2172/1492737.

```
Parekh, Ojas D., Ryan-Anderson, Ciaran, & Gharibian, Sevag.
```*Quantum Optimization and Approximation Algorithms.*. United States. doi:10.2172/1492737.

```
Parekh, Ojas D., Ryan-Anderson, Ciaran, and Gharibian, Sevag. Tue .
"Quantum Optimization and Approximation Algorithms.". United States. doi:10.2172/1492737. https://www.osti.gov/servlets/purl/1492737.
```

```
@article{osti_1492737,
```

title = {Quantum Optimization and Approximation Algorithms.},

author = {Parekh, Ojas D. and Ryan-Anderson, Ciaran and Gharibian, Sevag},

abstractNote = {Shor's groundbreaking quantum algorithm for integer factoring provides an exponential speedup over the best-known classical algorithms. In the 20 years since Shor's algorithm was conceived, only a handful of fundamental quantum algorithmic kernels, generally providing modest polyno- mial speedups over classical algorithms, have been invented. To better understand the potential advantage quantum resources provide over their classical counterparts, one may consider other re- sources than execution time of algorithms. Quantum Approximation Algorithms direct the power of quantum computing towards optimization problems where quantum resources provide higher- quality solutions instead of faster execution times. We provide a new rigorous analysis of the recent Quantum Approximate Optimization Algorithm, demonstrating that it provably outperforms the best known classical approximation algorithm for special hard cases of the fundamental Maximum Cut graph-partitioning problem. We also develop new types of classical approximation algorithms for finding near-optimal low-energy states of physical systems arising in condensed matter by ex- tending seminal discrete optimization techniques. Our interdisciplinary work seeks to unearth new connections between discrete optimization and quantum information science.},

doi = {10.2172/1492737},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2019},

month = {1}

}