Fault-tolerant syndrome extraction and cat state preparation with fewer qubits

Abstract

We reduce the extra qubits needed for two fault-tolerant quantum computing protocols: error correction, specifically syndrome bit measurement, and cat state preparation. For distance-three fault-tolerant syndrome extraction, we show an exponential reduction in qubit overhead over the previous best protocol. For a weight-w stabilizer, we demonstrate that stabilizer measurement tolerating one fault needs at most ⌈log2 w⌉ + 1 ancilla qubits. If qubits reset quickly, four ancillas suffice. We also study the preparation of entangled cat states, and prove that the overhead for distance-three fault tolerance is logarithmic in the cat state size. These results apply both to near-term experiments with a few qubits, and to the general study of the asymptotic resource requirements of syndrome measurement and state preparation. With a flag qubits, previous methods use O(a) flag patterns to identify faults. In order to use the same flag qubits more efficiently, we show how to use nearly all 2a possible flag patterns, by constructing maximal-length paths through the a-dimensional hypercube.