Introduction to topological quantum computation with non-Abelian anyons (1802.06176v2)

Abstract: Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.

• The paper provides a comprehensive review of using non-Abelian anyons, particularly Fibonacci anyons, to enable fault-tolerant quantum computation.
• It details a braiding methodology where topologically protected operations perform universal quantum algorithms, exemplified by the AJL algorithm.
• Simulations validate the practical implementation of topological quantum computation and highlight its potential to solve complex quantum problems.

Overview of Topological Quantum Computation with Non-Abelian Anyons

The paper provides a comprehensive review of topological quantum computation using non-Abelian anyons, with a focus on Fibonacci anyons. It explores the theoretical underpinnings and practical implementation of topological quantum computing, promising significant advantages in fault tolerance compared to other quantum computing paradigms.

Fundamental Concepts

Topological quantum computing leverages the unique exchange statistics of non-Abelian anyons, which differ fundamentally from the bosonic and fermionic particles in 3D space. These quasiparticles exist in two-dimensional systems and are capable of performing unitary operations via braiding. The braiding is topologically protected, offering inherent fault-tolerance by encoding information globally rather than locally, thus reducing susceptibility to errors from decoherence.

Utilizing Fibonacci Anyons

The paper primarily focuses on Fibonacci anyons, an exemplar of non-Abelian anyons, which possess a quantum dimension equal to the golden ratio. This model is shown to be universal for quantum computation, meaning it can perform any unitary operation to arbitrary accuracy merely through the braiding process. Each qubit in this model is formed by four anyons, but computationally treated as comprising three due to redundancy in braiding operations.

Developing Quantum Algorithms

The authors delve into the Aharonov-Jones-Landau (AJL) algorithm for evaluating the Jones polynomial—a topological invariant at roots of unity—demonstrating applications in simulating quantum algorithms, such as Shor’s algorithm. The AJL algorithm provides a quantum advantage by efficiently approximating solutions that are otherwise computationally prohibitive with classical methods, showcasing the potential of topological quantum computing.

Simulation and Practical Implementation

Simulations were performed using the braiding of Fibonacci anyons to validate the operation of a topological quantum computer, with a focus on calculating the Jones polynomial—a knot invariant linked to topological quantum field theory. The simulations illustrate how braided anyons can perform the AJL algorithm, and details of the simulator's functionality are provided in the paper.

Implications and Future Directions

The strong link between non-Abelian anyons and the Jones polynomial suggests a profound connection between knot theory and quantum computation. While the findings are presented as a proof-of-concept, practical implementations using existing physical systems could soon follow. Significant interest lies in using quantum algorithms to simulate complex quantum systems, process information beyond classical means, and solve otherwise intractable problems with topological quantum computers. The paper expects the ongoing search for physical systems hosting non-Abelian anyons will advance the development of topological quantum computers capable of robust, fault-tolerant quantum computation.

Conclusion

This paper serves as a foundation for understanding topological quantum computing with non-Abelian anyons, highlighting both the theoretical framework and practical strategy for its realization. The use of non-Abelian anyons, particularly Fibonacci anyons, provides a pathway toward building fault-tolerant quantum computers that leverage topological invariants in quantum algorithms, ushering in prospects for future advancements.