Unification and limitations of error suppression techniques for adiabatic quantum computing

While adiabatic quantum computation (AQC) possesses some intrinsic robustness to noise, it is expected that a form of error control will be necessary for large scale computations. Error control ideas developed for circuit-model quantum computation do not transfer easily to the AQC model and to date there have been two main proposals to suppress errors during an AQC implementation: energy gap protection and dynamical decoupling. Here we show that these two methods are fundamentally related and may be analyzed within the same formalism. We analyze the effectiveness of such error suppression techniques and identify critical constraints on the performance of error suppression in AQC, suggesting that error suppression by itself is insufficient for fault-tolerant, large-scale AQC and that a form of error correction is needed. This manuscript has been superseded by the articles, "Error suppression and error correction in adiabatic quantum computation I: techniques and challenges," arXiv:1307.5893, and "Error suppression and error correction in adiabatic quantum computation II: non-equilibrium dynamics," arXiv:1307.5892.Comment: 9 pages. Update replaces "Equivalence" with "Unification." This manuscript has been superseded by the two-article series: arXiv:1307.5892 and arXiv:1307.589

Minimax Quantum Tomography: Estimators and Relative Entropy Bounds

© 2016 American Physical Society. A minimax estimator has the minimum possible error ("risk") in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as O(1/N) - in contrast to that of classical probability estimation, which is O(1/N) - where N is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states

Estimating the bias of a noisy coin

Optimal estimation of a coin's bias using noisy data is surprisingly different from the same problem with noiseless data. We study this problem using entropy risk to quantify estimators' accuracy. We generalize the "add Beta" estimators that work well for noiseless coins, and we find that these hedged maximum-likelihood (HML) estimators achieve a worst-case risk of O(N^{-1/2}) on noisy coins, in contrast to O(1/N) in the noiseless case. We demonstrate that this increased risk is unavoidable and intrinsic to noisy coins, by constructing minimax estimators (numerically). However, minimax estimators introduce extreme bias in return for slight improvements in the worst-case risk. So we introduce a pointwise lower bound on the minimum achievable risk as an alternative to the minimax criterion, and use this bound to show that HML estimators are pretty good. We conclude with a survey of scientific applications of the noisy coin model in social science, physical science, and quantum information science.Comment: 10 page