title: Thresholds for Universal Concatenated Quantum Codes.
abstract: Quantum error correction and fault tolerance make it possible to perform quantum computations in the presence of imprecision and imperfections of realistic devices. An important question is to find the noise rate at which errors can be arbitrarily suppressed. By concatenating the 7-qubit Steane and 15-qubit Reed-Muller codes, the 105-qubit code enables a universal set of fault-tolerant gates despite not all of them being transversal. Importantly, the cnot gate remains transversal in both codes, and as such has increased error protection relative to the other single qubit logical gates. We show that while the level-1 pseudothreshold for the concatenated scheme is limited by the logical Hadamard gate, the error suppression of the logical cnot gates allows for the asymptotic threshold to increase by orders of magnitude at higher levels. We establish a lower bound of 1.28×10^{-3} for the asymptotic threshold of this code, which is competitive with known concatenated models and does not rely on ancillary magic state preparation for universal computation.
authors.affiliation: Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
authors.forename: Christopher
authors.initials: C
authors.lastname: Chamberland
authors.affiliation: Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
authors.forename: Tomas
authors.initials: T
authors.lastname: Jochym-O'Connor
authors.affiliation: Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.|Perimeter Institute, Waterloo, Ontario N2L 2Y5, Canada.|Canadian Institute For Advanced Research, Toronto, Ontario M5G 1Z8, Canada.
authors.forename: Raymond
authors.initials: R
authors.lastname: Laflamme