Measurement-free Implementations of Small-Scale Surface Codes for Quantum Dot Qubits

The performance of quantum error correction schemes depends sensitively on the physical realizations of the qubits and the implementations of various operations. For example, in quantum dot spin qubits, readout is typically much slower than gate operations, and conventional surface code implementations that rely heavily on syndrome measurements could therefore be challenging. However, fast and accurate reset of quantum dot qubits--without readout--can be achieved via tunneling to a reservoir. Here, we propose small-scale surface code implementations for which syndrome measurements are replaced by a combination of Toffoli gates and qubit reset. For quantum dot qubits, this enables much faster error correction than measurement-based schemes, but requires additional ancilla qubits and non-nearest-neighbor interactions. We have performed numerical simulations of two different coding schemes, obtaining error thresholds on the orders of $10^{-2}$ for a 1D architecture that only corrects bit-flip errors, and $10^{-4}$ for a 2D architecture that corrects bit- and phase-flip errors.