New magic state distillation factories optimized by temporally encoded lattice surgery

Fault-tolerant quantum computers, with error correction implemented using topological codes, will most likely require lattice surgery protocols in order to implement a universal gate set. Timelike failures during lattice surgery protocols can result in logical failures during the execution of an algorithm. In addition to the spacelike distance of the topological code used to protect the qubits from errors, there is also the timelike distance which is given by the number of syndrome measurement rounds during a lattice surgery protocol. As such, a larger timelike distance requirement will result in the slowdown of an algorithm's runtime. Temporal encoding of lattice surgery (TELS) is a technique which can be used to reduce the number of syndrome measurement rounds that are required during a lattice surgery protocol. This is done by measuring an over-complete set of mutually commuting multi-qubit Pauli operators (referred to as a parallelizable Pauli set) which form codewords of a classical error correcting code. The results of the over-complete set of Pauli measurements can then be used to detect and possibly correct timelike lattice surgery failures. In this work, we introduce an improved TELS protocol and subsequently augment it with the ability to correct low-weight classical errors, resulting in greater speedups in algorithm runtimes. We also explore large families of classical error correcting codes for a wide range of parallelizable Pauli set sizes. We also apply TELS to magic state distillation protocols in the context of biased noise, where logical qubits are encoded in asymmetric surface codes. Using optimized layouts, we show improvements in the space-time cost of our magic state factories compared to previous protocols. Such improvements are achieved using computations performed in the Clifford frame.