Flag Gadgets based on Classical Codes

Fault-tolerant quantum error correction is essential for full-scale quantum computing due to high noise levels; however, conventional fault-tolerant schemes require high overhead. 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, with less overhead. 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. Taking advantage of the sublinear scaling of our construction, we also show how to fault-tolerantly measure several stabilizers using a single gadget, further reducing the number of ancilla qubits required. 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.