Topological Qubit Design and Leakage

Abstract

We examine how best to design qubits for use in topological quantum computation. These qubits are topological Hilbert spaces associated with small groups of anyons. Op- erations are performed on these by exchanging the anyons. One might argue that, in order to have as many simple single qubit operations as possible, the number of anyons per group should be maximized. However, we show that there is a maximal number of particles per qubit, namely 4, and more generally a maximal number of particles for qudits of dimension d. We also look at the possibility of having topological qubits for which one can perform two-qubit gates without leakage into non-computational states. It turns out that the requirement that all two-qubit gates are leakage free is very restrictive and this property can only be realized for two-qubit systems related to Ising-like anyon models, which do not allow for universal quantum computation by braiding. Our results follow directly from the representation theory of braid groups which means they are valid for all anyon models. We also make some remarks on generalizations to other exchange groups.

13 pages, 3 figures

References in corpus (10)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Braid Topologies for Quantum Computation
• Quantum groups and nonabelian braiding in quantum Hall systems
• Interferometry of non-Abelian Anyons
• The two-eigenvalue problem and density of Jones representation of braid groups
• A modular functor which is universal for quantum computation
• Topological Quantum Compiling
• Representations of the braid group B_3 and of SL(2,Z)
• Constructing Functional Braids for Low-Leakage Topological Quantum Computing

Cited by in corpus (11)

• Introduction to topological quantum computation with non-Abelian anyons
• Coping with qubit leakage in topological codes
• Understanding the effects of leakage in superconducting quantum error detection circuits
• Comparing Zeeman qubits to hyperfine qubits in the context of the surface code: ${}^{174}$Yb${}^{+}$ and ${}^{171}$Yb${}^{+}$
• Handling Leakage with Subsystem Codes
• Focus on topological quantum computation
• Topological quantum compilation of two-qubit gates
• Symmetry defects and their application to topological quantum computing
• Error metric for non-trace-preserving quantum operations
• Optimized Topological Quantum Compilation of Three-Qubit Controlled Gates in the Fibonacci Anyon Model: A Controlled-Injection Approach
• Universal Gates from Braiding and Fusing Anyons on Quantum Hardware