# Optimization

Imagine you are building a house, and have a list of things you want to have in your house, but you can’t afford everything on your list because you are constrained by a budget. What you really want to work out is the combination of items which gives you the best value for your money.

This is an example of a optimization problem, where you are trying to find the best combination of things given some constraints. Typically, these are very hard problems to solve because of the huge number of possible combinations. With just 270 on/off switches, there are more possible combinations than atoms in the universe!

These types of optimization problems exist in many different domains - systems design, mission planning, airline scheduling, financial analysis, web search, cancer radiotherapy and many more. They are some of the most complex problems in the world, with potentially enormous benefits to businesses, people and science if optimal solutions can be readily computed.

## Optimization problems are some of the most complex problems to solve.

## Radiotherapy Optimization

There are many examples of problems where a quantum computer can complement an HPC (high performance computing) system. While the quantum computer is well suited to discrete optimization, the HPC system is much better at large scale numerical simulations. Problems like optimizing cancer radiotherapy, where a patient is treated by injecting several radiation beams into the patient intersecting at the tumor, illustrates how the two systems can work together.

The goal when devising a radiation plan is to minimize the collateral damage to the surrounding tissue and body parts – a very complicated optimization problem with thousands of variables. To arrive at the optimal radiation plan requires many simulations until an optimal solution is determined. With a quantum computer, the horizon of possibilities that can be considered between each simulation is much broader. But HPC is still the more powerful computation tool for running simulations. Using the quantum computer with an HPC system will allow faster convergence on an optimal design than is attainable by using HPC alone.

## Protein Folding

Simulating the folding of proteins could lead to a radical transformation of our understanding of complex biological systems and our ability to design powerful new drugs.

This application looks into how to use the quantum computer to explore the possible folding configurations of these interesting molecules. With an astronomical number of possible structural arrangements, protein folding is an enormously complex computational problem. Scientific research indicates that nature optimizes the amino acid sequences to create the most stable protein - which correlates well to the search for the lowest energy solutions.

With researchers at Harvard, we designed a system for predicting the folding patterns for lattice protein folding models and successfully ran small protein folding problems in hardware.

## Water Network Optimization

This is an example of using a quantum computer with a conventional or HPC system. EPANET is public domain numerical software that simulates water movement and water quality within pressurized pipe networks. It can model the flow of water in each pipe, the pressure at each node, the height of the water in each tank, the type of chemical concentration throughout the network during a simulation period, water age, source, and tracing. EPANET can compute properties of a water network given discrete choices for the design of network.

The quantum computer gives us a tool for designing the optimal network, by penalizing undesirable outcomes in the network such as low pressure or the presence of chemical contaminant levels, while rewarding desirable outcomes such as low cost, low risk, safety, etc. This Quantum-Classical hybrid solution quickly hones in on good solutions by asking the conventional system to evaluate far fewer possibilities.

# Machine Learning

When you look at a photograph it is very easy for you to pick out the different objects in the image: trees, mountains, velociraptors etc. This task is almost effortless for humans, but is in fact a hugely difficult task for computers to achieve. This is because programmers don’t know how to define the essence of a ‘tree’ in computer code.

Machine learning is the most successful approach to solving this problem, by which programmers write algorithms that automatically learn to recognize the ‘essences’ of objects by detecting recurring patterns in huge amounts of data. Because of the amount of data involved in this process, and the immense number of potential combinations of data elements, this is a very computationally-expensive optimization problem. As with other optimization problems, these can be mapped to the native capability of the D-Wave QPU.

## Machines learn to recognize objects by detecting recurring patterns.

## Object Detection

Quantum hardware, trained using a binary classification algorithm, is able to detect whether or not an image contains a car.

Together with researchers at Google, we built software for determining whether or not there is a car in an image using a binary classification algorithm run in hardware. In excess of 500,000 discrete optimization problems were solved during the learning phase, with Google developers accessing the D-Wave system remotely.

## Labeling News Stories

We built software for automatically applying category labels to news stories and images. We found that our approach provided better labeling accuracy than a state of the art conventional approach.

The labeling of news stories can be difficult for computers as they can see the keywords but don’t understand the meaning of the words when combined. For labeling news stories the corpus we used for training and testing performance was the REUTERS corpus, a well-known data set for testing multiple label assignment algorithms.

We took a similar approach to labeling images and used the SCENE corpus for training and testing performance, a well-known data set for testing multiple label assignment algorithms.

We found that our approach worked extremely well on these problems, demonstrating the quantum computer's ability to do multiple label assignment and to label images.

## Video Compression

Using unsupervised machine learning approaches, one can automate the discovery of a very sparse way to represent objects. This technique can be used for incredibly efficient compression.

The algorithm works by finding a concise representation of the objects being fed into the computer. The techniques involved are closely related to those in compressive sensing. As a test of the unsupervised feature learning algorithm, we discovered an extremely sparse representation of the ‘Frey faces’ data set, and demonstrated the ability to perform video compression on the quantum computer.

# Monte Carlo Simulation

Many things in the world are uncertain, and governed by the rules of probability. We have, in our heads, a model of how things will turn out in the future, and the better our model is, the better we are at predicting the future. We can also build computer models to try and capture the statistics of reality. These tend to be very complicated, involving a huge number of variables.

In order to check to see if a computer’s statistical model represents reality we need to be able to draw samples from it, and check that the statistics of our model match the statistics of real world data. Monte Carlo simulation, which relies on repeated random sampling to approximate the probability of certain outcomes, is an approach used in many industries such as finance, energy, manufacturing, engineering oil & gas and the environment. For a complex model, with many different variables, this is a difficult task to do quickly.

## The better our model, the better we are at predicting the future.