Abstract
Topological quantum computing promises error-resistant quantum computation without active error correction. However, there is a worry that during the process of executing quantum gates by braiding anyons around each other, extra anyonic excitations will be created that will disorder the encoded quantum information. Here we explore this question in detail by studying adiabatic code deformations on Hamiltonians based on topological codes, notably Kitaev's surface codes and the more recently discovered color codes. We develop protocols that enable universal quantum computing by adiabatic evolution in a way that keeps the energy gap of the system constant with respect to the computation size and introduces only simple local Hamiltonian interactions. This allows one to perform holonomic quantum computing with these topological quantum computing systems. The tools we develop allow one to go beyond numerical simulations and understand these processes analytically.
Highlights
There are many approaches to constructing a quantum computer
We develop protocols that enable universal quantum computing by adiabatic evolution in a way that keeps the energy gap of the system constant with respect to the computation size and introduces only simple local Hamiltonian interactions
We focus on the latter class of architectures and address the following question: “How does one quantum compute on a system protected from decoherence by a static Hamiltonian?” We present a solution that adiabatically interpolates between static Hamiltonians, each of which protects the quantum information stored in its ground space
Summary
There are many approaches to constructing a quantum computer. In addition to the numerous different physical substrates available, there are a plethora of different underlying computational architectures from which to choose. The second line of research relevant to our proposal is the recent use of code deformations to perform quantum computation on topological quantum error-correcting codes [8,9,10,12,32]. Oreshkov et al demonstrated a novel manner for achieving universality within the context of faulttolerant quantum computing [6] This result showed how to perform gates on information encoded into a quantum stabilizer code. We examine explicit adiabatic interpolations between Hamiltonians that simulate code deformation, as in the third line of research This is all done while keeping the energy gap in the system constant, a necessary requirement to use these techniques to maintain the topological protection offered by these systems. We prevent the environment from doing this by ensuring that it is cold, and we prevent ourselves from introducing excitations accidentally by carefully designing our procedures
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