We report on an experiment to detect non-classical correlations in a highly mixed state. The correlations are characterized by the quantum discord and are observed using four qubits in a liquid state nuclear magnetic resonance quantum information processor. The state analyzed is the output of a DQC1 computation, whose input is a single quantum bit accompanied by n maximally mixed qubits. This model of computation outperforms the best known classical algorithms, and although it contains vanishing entanglement it is known to have quantum correlations characterized by the quantum discord. This experiment detects non-vanishing quantum discord, ensuring the existence of non-classical correlations as measured by the quantum discord.

Quantum gates in experiment are inherently prone to errors that need to be characterized before they can be corrected. Full characterization via quantum process tomography is impractical and often unnecessary. For most practical purposes, it is enough to estimate more general quantities such as the average fidelity. Here we use a unitary 2-design and twirling protocol for efficiently estimating the average fidelity of Clifford gates, to certify a 7-qubit entangling gate in a nuclear magnetic resonance quantum processor. Compared with more than 10 8 experiments required by full process tomography, we conducted 1656 experiments to satisfy a statistical confidence level of 99%. The average fidelity of this Clifford gate in experiment is 55.1%, and rises to 87.5% if the infidelity due to decoherence is removed. The entire protocol of certifying Clifford gates is efficient and scalable, and can easily be extended to any general quantum information processor with minor modifications. . Introduction. Benchmarking protocols for characterizing the level of coherent control are fundamental in evaluating potential quantum information processing (QIP) devices. They provide an objective comparison of quantum control capabilities between diverse QIP devices, and also indicate the prospects of a given platform with respect to fault-tolerant quantum computation [1]. The traditional approach of using quantum process tomography (QPT) [2,3] is useful for completely characterizing a quantum channel, and has been applied to at most 3-qubit systems in experiment [4][5][6][7][8][9][10][11]. However, QPT requires number of measurements that scale exponentially with number of qubits n (≈ 2 4n ), making it impractical even in relatively small systems. Moreover,for many practical purposes, such as benchmarking, the full description of a particular quantum channel is not necessary and more accessible properties of the gates are sufficient. To benchmark a gate it is enough to estimate the distance between the implemented channel and the ideal gate. Several methods such as randomized benchmarking [12][13][14], twirling [15][16][17], and Monte Carlo estimations [18,19] have been proposed to evaluate a particular quantum channel in an efficient manner, each with its own restrictions and drawbacks. Here, in order to benchmark our coherent controls on a 7-qubit nuclear magnetic resonance (NMR) system, we adopted the twirling protocol [17] to estimate the average fidelity of an important Clifford gate in QIP. The gate of interest generates maximal coherence from single coherence with the aid of lo-

of chaos in the dynamics of quantum discord", Physical Review E 91, 032906 (2015).PHYSICAL REVIEW E 91, 032906 (2015) Signatures of chaos in the dynamics of quantum discord We identify signatures of chaos in the dynamics of discord in a multiqubit system collectively modelled as a quantum kicked top. The evolution of discord between any two qubits is quasiperiodic in regular regions, while in chaotic regions the quasiperiodicity is lost. As the initial wave function is varied from the regular regions to the chaotic sea, a contour plot of the time-averaged discord remarkably reproduces the structures of the classical stroboscopic map. We also find surprisingly opposite behavior of two-qubit discord versus entanglement of the two qubits as measured by the concurrence. Our results provide evidence of signatures of chaos in dynamically generated discord.

We describe the package MSGARCH, which implements Markov-switching GARCH (generalized autoregressive conditional heteroscedasticity) models in R with efficient C++ object-oriented programming. Markov-switching GARCH models have become popular methods to account for regime changes in the conditional variance dynamics of time series. The package MSGARCH allows the user to perform simulations as well as maximum likelihood and Bayesian Markov chain Monte Carlo estimations of a very large class of Markov-switching GARCH-type models. The package also provides methods to make single-step and multi-step ahead forecasts of the complete conditional density of the variable of interest. Risk management tools to estimate conditional volatility, value-at-risk, and expected-shortfall are also available. We illustrate the broad functionality of the MSGARCH package using exchange rate and stock market return data. ∼ D(0, 1, ξ k ).2 Furthermore, assuming a non zero state dependent mean would generally imply E[ytyt−1] = 0, which is against the model we specified.3 As explained below, the parametric formulation of the conditional distribution D(0, h k,t , ξ k ) can be different across regimes. In this case, the notation D k (0, h k,t , ξ k ) would be more appropriate. The same applies for the h(·) function in (2). We keep the simpler notation to improve readability. Also, for t = 1, we initialize the regime probabilities and the conditional variances at their unconditional levels. To simplify exposition, we use henceforth for t = 1 the same notation as for general t, since there is no confusion possible. Conditional variance dynamicsAs in Haas et al. (2004a), the conditional variance of y t is assumed to follow a GARCHtype model. Hence, conditionally on regime s t = k, h k,t is available as a function of the past observation, y t−1 , past variance h k,t−1 , and the additional regime-dependent vector of parameters θ k :where h(·) is a I t−1 -measurable function that defines the filter for the conditional variance and also ensures its positiveness. In the MSGARCH package, the initial value of the variance recursions, that is h k,1 (k = 1, . . . , K), are set equal to the unconditional variance in regime k.Depending on the form of h(·), we obtain different scedastic specifications. In the R package MSGARCH, we follow this specification in order to reduce model complexity. Finally, when K = 1, we recover single-regime GARCH-type models identified by the form of h(·). 4 The mixture of GARCH model presented in Haas et al. (2004b) allows for interactions between the mixture component variances. Here, we report the case referred to as "Diagonal" by Haas et al. (2004b). Journal of Statistical Software 5Below we briefly present the scedastic specifications available in the R package MSGARCH. Each of them is identified with a label used in the code for defining a model specification. Similarly, model coefficients are also identified with labels. ARCH modelThe ARCH model of Engle (1982) is given by:for k = 1, . . . , K. In this case, we have θ...

A method to hedge variable annuities in the presence of basis risk is developed. A regime-switching model is considered for the dynamics of market assets. The approach is based on a local optimization of risk and is therefore very tractable and flexible. The local optimization criterion is itself optimized to minimize capital requirements associated with the variable annuity policy, the latter being quantified by the Conditional Value-at-Risk (CVaR) risk metric. In comparison to benchmarks, our method is successful in simultaneously reducing capital requirements and increasing profitability. Indeed the proposed local hedging scheme benefits from a higher exposure to equity risk and from time diversification of risk to earn excess return and facilitate the accumulation of capital. A robust version of the hedging strategies addressing model risk and parameter uncertainty is also provided.

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