Constructing Functional Braids for Low-Leakage Topological Quantum Computing

We investigate the measurement-only topological quantum computation (MOTQC) approach proposed by Bonderson et al. [Phys. Rev. Lett. 101, 010501 (2008)] where the braiding operation is shown to be equivalent to a series of topological charge "forced measurements" of anyons. In a forced measurement, the charge measurement is forced to yield the desired outcome (e.g. charge 0) via repeatedly measuring charges in different bases. This is a probabilistical process with a certain s...

We review the q-deformed spin network approach to topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a shape suitable for computation and algebraic work.

Schemes for topological quantum computation are usually based on the assumption that the system is initially prepared in a specific state. In practice, this state preparation is expected to be challenging as it involves non-topological operations which heavily depend on the experimental realization and are not naturally robust against noise. Here we show that this assumption can be relaxed by using composite anyons: starting from an unknown state with reasonable physical prop...

This review presents an entry-level introduction to topological quantum computation -- quantum computing with anyons. We introduce anyons at the system-independent level of anyon models and discuss the key concepts of protected fusion spaces and statistical quantum evolutions for encoding and processing quantum information. Both the encoding and the processing are inherently resilient against errors due to their topological nature, thus promising to overcome one of the main o...

We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyo...

Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show tha...

Among the list of major threats to quantum computation, quantum decoherence poses one of the largest because it generates losses to the environment within a computational system which cannot be recovered via error correction methods. These methods require the assumption that the environmental interaction forces the qubit state into some linear combination of qubit eigenstates. In reality, the environment causes the qubit to enter into a mixed state where the original is no lo...

While the realization of scalable quantum computation will arguably require topological stabilization and, with it, topological-hardware-aware quantum programming and topological-quantum circuit verification, the proper combination of these strategies into dedicated topological quantum programming languages has not yet received attention. Here we describe a fundamental and natural scheme that we are developing, for typed functional (hence verifiable) topological quantum progr...

Topological quantum computing has recently proven itself to be a very powerful model when considering large- scale, fully error corrected quantum architectures. In addition to its robust nature under hardware errors, it is a software driven method of error corrected computation, with the hardware responsible for only creating a generic quantum resource (the topological lattice). Computation in this scheme is achieved by the geometric manipulation of holes (defects) within the...

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computati...