Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count

摘要

We present an infinite family of protocols to distill magic states for$T$-gates that has a low space overhead and uses an asymptotic number of inputmagic states to achieve a given target error that is conjectured to be optimal.The space overhead, defined as the ratio between the physical qubits to thenumber of output magic states, is asymptotically constant, while both thenumber of input magic states used per output state and the $T$-gate depth ofthe circuit scale linearly in the logarithm of the target error $\delta$ (up to$\log \log 1/\delta$). Unlike other distillation protocols, this protocolachieves this performance without concatenation and the input magic states areinjected at various steps in the circuit rather than all at the start of thecircuit. The protocol can be modified to distill magic states for other gatesat the third level of the Clifford hierarchy, with the same asymptoticperformance. The protocol relies on the construction of weakly self-dual CSScodes with many logical qubits and large distance, allowing us to implementcontrol-SWAPs on multiple qubits. We call this code the "inner code". Thecontrol-SWAPs are then used to measure properties of the magic state and detecterrors, using another code that we call the "outer code". Alternatively, we useweakly-self dual CSS codes which implement controlled Hadamards for the innercode, reducing circuit depth. We present several specific small examples ofthis protocol.