Flag Gadgets based on Classical Codes

Fault-tolerant quantum error correction is essential for full-scale quantum computing due to high noise levels; however, fault-tolerance in general requires many expensive resources, particularly qubits. Recently it has been shown that by using flag gadgets it is possible to perform fault-tolerant syndrome extraction, a key subroutine of quantum error correction, using fewer resources than conventional methods. Although flag gadgets have already been used in several experiments, a framework that applies to general quantum codes and does not require fast physical operations to achieve a resource reduction has been missing.

We develop a framework to design flag gadgets using classical codes, which is applicable to all stabilizer codes. Using this framework we show how to perform fault-tolerant syndrome extraction using exponentially fewer qubits than conventional methods. We also show how to fault-tolerantly measure multiple stabilizers using a single gadget. Using the developed framework we perform numerical simulations to find several small examples where our constructions reduce the number of qubits required. These small examples may be relevant to near-term experiments on small-scale quantum computers.