Fault-tolerant quantum computation versus Gaussian noise

We study the robustness of a fault-tolerant quantum computer subject to Gaussian non-Markovian quantum noise, and we show that scalable quantum computation is possible if the noise power spectrum satisfies an appropriate "threshold condition." Our condition is less sensitive to very-high-frequency noise than previously derived threshold conditions for non-Markovian noise.Comment: 30 pages, 6 figure

PSL(2,7) Representations and their relevance to Neutrino Physics

The investigation of the role of finite groups in flavor physics and particularly, in the interpretation of the neutrino data has been the subject of intensive research. Motivated by this fact, in this work we derive the three-dimensional unitary representations of the projective linear group PSL_2(7). Based on the observation that the generators of the group exhibit a latin square pattern, we use available computational packages on discrete algebra to determine the generic properties of the group elements. We present analytical expressions and discuss several examples which reproduce the neutrino mixing angles in accordance with the experimental data.Comment: 21 pages, 3 figure

Optimal control of a leaking qubit

Physical implementations of quantum bits can contain coherent transitions to energetically close non-qubit states. In particular, for anharmonic oscillator systems such as the superconducting phase qubit and the transmon a two-level approximation is insufficient. We apply optimal control theory to the envelope of a resonant Rabi pulse in a qubit in the presence of a single, weakly off-resonant leakage level. The gate error of a spin flip operation reduces by orders of magnitude compared to simple pulse shapes. Near-perfect gates can be achieved for any pulse duration longer than an intrinsic limit given by the nonlinearity. The pulses can be understood as composite sequences that refocus the leakage transition. We also discuss ways to improve the pulse shapes.Comment: 4 pages, 2 figure

Fault-Tolerant Thresholds for Encoded Ancillae with Homogeneous Errors

I describe a procedure for calculating thresholds for quantum computation as a function of error model given the availability of ancillae prepared in logical states with independent, identically distributed errors. The thresholds are determined via a simple counting argument performed on a single qubit of an infinitely large CSS code. I give concrete examples of thresholds thus achievable for both Steane and Knill style fault-tolerant implementations and investigate their relation to threshold estimates in the literature.Comment: 14 pages, 5 figures, 3 tables; v2 minor edits, v3 completely revised, submitted to PR

A precise CNOT gate in the presence of large fabrication induced variations of the exchange interaction strength

We demonstrate how using two-qubit composite rotations a high fidelity controlled-NOT (CNOT) gate can be constructed, even when the strength of the interaction between qubits is not accurately known. We focus on the exchange interaction oscillation in silicon based solid-state architectures with a Heisenberg Hamiltonian. This method easily applies to a general two-qubit Hamiltonian. We show how the robust CNOT gate can achieve a very high fidelity when a single application of the composite rotations is combined with a modest level of Hamiltonian characterisation. Operating the robust CNOT gate in a suitably characterised system means concatenation of the composite pulse is unnecessary, hence reducing operation time, and ensuring the gate operates below the threshold required for fault-tolerant quantum computation.Comment: 9 pages, 8 figure

Asymmetric quantum error correction via code conversion

In many physical systems it is expected that environmental decoherence will exhibit an asymmetry between dephasing and relaxation that may result in qubits experiencing discrete phase errors more frequently than discrete bit errors. In the presence of such an error asymmetry, an appropriately asymmetric quantum code - that is, a code that can correct more phase errors than bit errors - will be more efficient than a traditional, symmetric quantum code. Here we construct fault tolerant circuits to convert between an asymmetric subsystem code and a symmetric subsystem code. We show that, for a moderate error asymmetry, the failure rate of a logical circuit can be reduced by using a combined symmetric asymmetric system and that doing so does not preclude universality.Comment: 5 pages, 8 figures, presentation revised, figures and references adde

Scaling the neutral atom Rydberg gate quantum computer by collective encoding in Holmium atoms

We discuss a method for scaling a neutral atom Rydberg gate quantum processor to a large number of qubits. Limits are derived showing that the number of qubits that can be directly connected by entangling gates with errors at the $10^{-3}$ level using long range Rydberg interactions between sites in an optical lattice, without mechanical motion or swap chains, is about 500 in two dimensions and 7500 in three dimensions. A scaling factor of 60 at a smaller number of sites can be obtained using collective register encoding in the hyperfine ground states of the rare earth atom Holmium. We present a detailed analysis of operation of the 60 qubit register in Holmium. Combining a lattice of multi-qubit ensembles with collective encoding results in a feasible design for a 1000 qubit fully connected quantum processor.Comment: 6 figure

High-fidelity quantum operations on superconducting qubits in the presence of noise

We present a scheme for implementing quantum operations with superconducting qubits. Our approach uses a "coupler" qubit to mediate a controllable, secular interaction between "data" qubits, pulse sequences which strongly mitigate the effects of 1/f flux noise, and a high-Q resonator-based local memory. We develop a Monte-Carlo simulation technique capable of describing arbitrary noise-induced dephasing and decay, and demonstrate in this system a set of universal gate operations with O(10^-5) error probabilities in the presence of experimentally measured levels of 1/f noise. We then add relaxation and quantify the decay times required to maintain this error level

Scalability of quantum computation with addressable optical lattices

We make a detailed analysis of error mechanisms, gate fidelity, and scalability of proposals for quantum computation with neutral atoms in addressable (large lattice constant) optical lattices. We have identified possible limits to the size of quantum computations, arising in 3D optical lattices from current limitations on the ability to perform single qubit gates in parallel and in 2D lattices from constraints on laser power. Our results suggest that 3D arrays as large as 100 x 100 x 100 sites (i.e., $\sim 10^6$ qubits) may be achievable, provided two-qubit gates can be performed with sufficiently high precision and degree of parallelizability. Parallelizability of long range interaction-based two-qubit gates is qualitatively compared to that of collisional gates. Different methods of performing single qubit gates are compared, and a lower bound of $1 \times 10^{-5}$ is determined on the error rate for the error mechanisms affecting $^{133}$Cs in a blue-detuned lattice with Raman transition-based single qubit gates, given reasonable limits on experimental parameters.Comment: 17 pages, 5 figures. Accepted for publication in Physical Review