Local hidden variable and classical models for quantum systems

A computer is a physical device that is used to solve mathematical problems. When using a normal electronic computer to solve a problem, you effectively set up the initial state of an electronic circuit, allow it to run according to the laws of physics that govern the circuit's behaviour, and then finally observe the final state of the circuit to obtain the solution to your problem. Modern computers are so well engineered and user friendly that it is not necessary to know the details of this process, but ultimately that is what is happening. Thinking of computation as a physical process in this way leads to the following question - why are the laws that govern the dynamical behaviour of electronic circuits best for doing computation?

Perhaps there are other physical systems obeying different laws that could enable us to build better computers? An important series of discoveries over the last few decades has shown that this is in fact true - if we could build a computer that uses the rules of quantum physics, then there are some mathematical problems that we could solve much more efficiently than the best known method on conventional electronic computers. This has well-known implications for the field of cryptography. A number of commonly used cryptographic schemes rely on assumptions that an eavesdropper cannot solve certain mathematical problems. Some of these problems can be solved easily on quantum computers, and so understanding the computational power of quantum algorithms is important for cryptography.

It is not fully understood how and why these "quantum algorithms" work, however it is something to do with the way that quantum systems grow in complexity as their size increases. The laws of quantum theory are used to describe the smallest constituents of matter, and in many cases are needed to describe larger scale objects too. However in most situations the intricate quantum phenomena that is needed to exploit to build a quantum computer cannot be observed directly. This makes building a quantum computer extremely challenging. Although remarkable progress towards that goal has been made, we are probably still some way off from building a substantial quantum computing device.

However, behind the field of quantum computation, there is one fundamental question that remains unanswered. What is it about quantum physics that makes it able to compute somuch better than conventional (or 'classical') computers? If we could understand this question better, could it lead to further quantum algorithms for other problems? On the other hand, it is possible that quantum computers, or the imperfect error-prone ones we are likely to access in real laboratories, are not that much better than classical computers,and the quantum 'supremacy' over classical algorithms is purely because we have not dreamed up the right algorithms for classical computers.

The goal of this PhD project is to attempt to improve our understanding of questions such as these by developing new methods for simulating interesting classes of quantum systems on classical ones by considering the way in which quantum systems grow in complexity as they are increased in size. The methods rely on ideas from the study of certain types of correlations ('entanglement') in quantum systems and physical theories that are more general than it. While these methods are likely to only apply to certain quantum systems, understanding when they do and do not work will allow us to obtain further insight into the problem of what makes quantum computation special. The results could have implications for the wide variety of mathematical problems to which quantum computers may be applied, such as problems in cryptography or the study of materials.