Fault Tolerant Quantum Computational Models from Topological Quantum Field Theories

EE376A (Winter 2019)

Milind Jagota, Alec Lau

https://www.overleaf.com/read/zhfdjyjjpkyr

INTRODUCTION:

Decoherence in many methods of quantum computation represent extremely difficult engineering and theoretical challenges. How error scales with the number of qubits could determine whether a practical quantum computer is even possible. Topological quantum computation provides a beautiful solution to this problem. Errors are most often local perturbations, but topological invariants encoding information would be immune to all but the most odious errors, as topological invariants are independent of local changes.


WHAT WE DID:

We did a literature review and did explicit computations for a model for topological quantum computation. This involved studying the physics of the involved particles’ (called anyons) systems, the mathematics of such systems, and mathematical models of such systems geared toward computation.


TECHNICAL OVERVIEW:

In the overview linked at the top we describe the particle systems necessary: how they arise, why they have the properties they do, and what properties make them fault tolerant as well as potential qubit systems.


Then we dive a bit deeper to show how topology plays a role in said properties.


Next we describe how the particles interact with each other, and the mathematics classifying these interactions, and, in particular, why information encoded is a topological invariant. We describe how a topological quantum computer would perform a calculation.


After the physics/mathematical physics of the problem, we switch gears to the language of theoretical fault tolerance as it is used in papers today. This is the introduction of categories and, more importantly, tensor categories. We describe the connection between tensor categories and our particles of interest.


With this introduction to tensor categories we can introduce the driving force between these computational models; that is, we provide an extremely brief overview of the aspects of topological quantum field theory important to computation. This involves a geometric interpretation of topological quantum field theory that is hopefully easy to visualize. For those without a background in quantum mechanics this is not necessary.

Geometric intuition for quantum mechanical calculations


After this, we describe objects called quantum groups, and how such tensor categories can be constructed from quantum groups. This is hugely important in this field, but requires a lot of tedious abstract algebra and isn’t too important for getting the gist of our model. We go further to introduce the Jones Polynomial, and its connection to #P-hard problems.


Finally, we note that a topological quantum field theory provides solutions to #P-hard instances of Jones polynomials, which shows how a physical system described by such a topological quantum field theory may be used as a computer to solve NP and even #P-hard problems.


Lastly, we describe an actual example of such a computational model. Here we explicitly calculate a result that the solution of which was presented without proof, and was to us not obvious. We conclude with some striking results from our reading about how this topological field-theoretic model can approximate an exact BQP-solving model to an arbitrary degree, and how an implementation is not too far off from an exact model.

Creating a Qubit
Our computation space as manifolds


OUTREACH:

Explaining theoretical physics concepts to elementary schoolers represented quite a challenge. Because fault tolerance uses the topology of systems, we decided to illustrate this to elementary schoolers using Play-Doh. We designed a game: we took a piece of Play-Doh curved in a certain way, and put it in the Play-Doh box and shook the box. We agreed beforehand that if it remains the same, or changes its curve to another curve we had in front of them, they should make something out of a Play-Doh color corresponding to the curve it ends up with. After shaking the box, the piece almost always changed.For those not completely focused on their Play-Doh creations, we played the game again, this time with pieces with 0, 1, 2, or 3 holes. The Play-Doh color they were to play with depended on the number of holes it ended up with. We shook the piece in the box, and the number of holes never changed, so the ‘code’ was sent ‘in the box of error’ perfectly.


We concluded the game by noting how small (local) shakes may change curves in pieces (conventional qubits), but didn’t change the number of holes (topological invariants) and had fun with our Play-Doh creations.


We hope that these kids, if they ever hope to develop codes through Play-Doh, they do not use local properties of their creations, and instead use holes.

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.