Universal logical gate sets with constant time overhead in quantum error correcting codes

A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. To date, all proposed schemes for implementing a universal logical gate set, such as magic state distillation or code switching, require a substantial space-time overhead, including a time overhead that necessarily diverges in the limit of vanishing logical error rate. I will present some new results in the theory of Turaev-Viro topological quantum error correcting codes that demonstrate how braids and Dehn twists can be achieved through constant depth unitary circuits, in the limit of arbitrarily large code distance. When applied to specific types of codes, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth circuit, and without increasing the asymptotic scaling of the space overhead. The resulting space-time overhead is optimal for topological codes with local syndromes.