Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code

Many current quantum error correcting codes that achieve full fault-tolerance suffer from having low ratios of logical to physical qubits and significant overhead. This makes them difficult to implement on current noisy intermediate-scale quantum (NISQ) computers and results in the inability to perform quantum algorithms at useful scales with near-term quantum processors. Due to this, calculations are generally done without encoding. We propose a middle ground between these two approaches: constructions in the [[n,n-2,2]] quantum error detecting code that can detect any error from a single faulty gate by measuring the stabilizer generators of the code and additional ancillas at the end of the computation. This achieves what we call weak fault-tolerance. As we show, this demonstrates a significant improvement over no error correction for low enough physical error probabilities and requires much less overhead than codes that achieve full fault-tolerance. We give constructions for a set of gates that achieve universal quantum computation in this error detecting code, while satisfying weak fault-tolerance up to analog imprecision on the physical rotation gate. We will also show how one can potentially get around the issue of analog errors though teleportation into the [[8,3,2]] code. This code admits a tranvseral CCZ gate, which with additional Clifford operations may allow for a fully weakly fault-tolerant Toffoli gate. Combined with weakly fault-tolerant constructions for the Clifford group in the [[n,n-2,2]] code, such a logical gate would allow for universal weakly fault-tolerant quantum computation without analog errors.