Quantum fault tolerance with constant-space and logarithmic-time overheads

In a model of fault-tolerant quantum computation with quick and noiseless polyloglog-time auxiliary classical computation, we construct a fault tolerance protocol with constant-space and -time overhead, where hides sub-polylog factors. Our construction utilizes constant-rate quantum locally testable codes (qLTC), new fault-tolerant gadgets on qLTCs/qLDPC codes including sub-logarithmic spacetime overhead magic state distillation, and a new analysis framework which we expect will be of further interest.