Reliable channel-adapted error correction: Bacon-Shor code recovery from amplitude damping

We construct two simple error correction schemes adapted to amplitude damping noise for Bacon-Shor codes and investigate their prospects for fault-tolerant implementation. Both consist solely of Clifford gates and require far fewer qubits, relative to the standard method, to achieve correction to a desired order in the damping rate. The first, employing one-bit teleportation and single-qubit measurements, needs only one fourth as many physical qubits, while the second, using just stabilizer measurements and Pauli corrections, needs only half. We show that existing fault-tolerance methods can be employed for the latter, while the former can be made to avoid potential catastrophic errors and can easily cope with damping faults in ancilla qubits.Comment: 8 pages, 1 figur

Cointegration tests based on record counting statistics

This paper presents of number of cointegration tests that exploit the statistical properties of the records from the original time series variables. We prove their consistency and obtain their asymptotic null distributions. Among the advantages of this novel methodology, the new tests are invariant with respect to the individual series' variances and also with respect to monotonic transformations applied to these series. In addition, these tests are robust against the presence of level breaks as long as the number of these breaks increases slowly enough with the sample size. Finally, an alternative scheme is proposed to deal with additive outliers, which prevent them from causing size distortions

Information-Theoretic Analysis of Serial Dependence and Cointegration

This paper is devoted to presenting wider characterizations of memory and cointegration in time series, in terms of information-theoretic statistics such as the entropy and the mutual information between pairs of variables. We suggest a nonparametric and nonlinear methodology for data analysis and for testing the hypotheses of long memory and the existence of a cointegrating relationship in a nonlinear context. This new framework represents a natural extension of the linear-memory concepts based on correlations. Finally, we show that our testing devices seem promising for exploratory analysis with nonlinearly cointegrated time series.Publicad

Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames

Following the introduction of the task of reference frame error correction, we show how, by using reference frame alignment with clocks, one can add a continuous Abelian group of transversal logical gates to any error-correcting code. With this we further explore a way of circumventing the no-go theorem of Eastin and Knill, which states that if local errors are correctable, the group of transversal gates must be of finite order. We are able to do this by introducing a small error on the decoding procedure that decreases with the dimension of the frames used. Furthermore, we show that there is a direct relationship between how small this error can be and how accurate quantum clocks can be: the more accurate the clock, the smaller the error; and the no-go theorem would be violated if time could be measured perfectly in quantum mechanics. The asymptotic scaling of the error is studied under a number of scenarios of reference frames and error models. The scheme is also extended to errors at unknown locations, and we show how to achieve this by simple majority voting related error correction schemes on the reference frames. In the Outlook, we discuss our results in relation to the AdS/CFT correspondence and the Page-Wooters mechanism.Comment: 10+35 pages. Also see related work uploaded to the arXiv on the same day; arXiv:1902.0771

Range unit root tests

Since the seminal paper by Dickey and Fuller in 1979, unit-root tests have conditioned the standard approaches to analyse time series with strong serial dependence, the focus being placed in the detection of eventual unit roots in an autorregresive model fitted to the series. In this paper we propose a completely different method to test for the type of "long-wave" patterns observed not only in unit root time series but also in series following more complex data generating mechanism. To this end, our testing device analyses the trend exhibit by the data, without imposing any constraint on the generating mechanism. We call our device the Range Unit Root (RUR) Test since it is constructed from running ranges of the series. These statistics allow a more general characterization of a strong serial dependence in the mean behavior, thus endowing our test with a number of desirable properties. Among these properties are the invariance to nonlinear monotonic transformations of the series and the robustness to the presence of level shifts and additive outliers. In addition, the RUR test outperforms the power of standard unit root tests on near-unit-root stationary time series

Information-Theoretic Analysis of Serial Dependence and Cointegration.

This paper is devoted to presenting wider characterizations of memory and cointegration in time series, in terms of information-theoretic statistics such as the entropy and the mutual information between pairs of variables. We suggest a nonparametric and nonlinear methodology for data analysis and for testing the hypotheses of long memory and the existence of a cointegrating relationship in a nonlinear context. This new framework represents a natural extension of the linear-memory concepts based on correlations. Finally, we show that our testing devices seem promising for exploratory analysis with nonlinearly cointegrated time series.

Instrumental Variable Interpretation of Cointegration with Inference Results for Fractional Cointegration

In this paper we propose an alternative characterization of the central notion of cointegration, exploiting the relationship between the autocovariance and the cross-covariance functions of the series. This characterization leads us to propose a new estimator of the cointegrating parameter based on the instrumental variables (IV) methodology. The instrument is a delayed regressor obtained from the conditional bivariate system of nonstationary fractionally integrated processes with a weakly stationary error correction term. We prove the consistency of this estimator and derive its limiting distribution. We also show that, in the I(1) case, with a semiparametric correction simpler than the one required for the fully modified ordinary least squares (FM-OLS), our fully modified instrumental variables (FM-IV) estimator is median-unbiased, a mixture of normals, and asymptotically efficient. As a consequence, standard inference can be conducted with this new FM-IV estimator of the cointegrating parameter. We show by the use of Monte Carlo simulations that the small sample gains with the new IV estimator over OLS are remarkable.Publicad