Practical sampling schemes for quantum phase estimation

In this work we consider practical implementations of Kitaev's algorithm for quantum phase estimation. We analyze the use of phase shifts that simplify the estimation of successive bits in the estimation of unknown phase $\varphi$. By using increasingly accurate shifts we reduce the number of measurements to the point where only a single measurements in needed for each additional bit. This results in an algorithm that can estimate $\varphi$ to an accuracy of $2^{-(m+2)}$ with probability at least $1-\epsilon$ using $N_{\epsilon} + m$ measurements, where $N_{\epsilon}$ is a constant that depends only on $\epsilon$ and the particular sampling algorithm. We present different sampling algorithms and study the exact number of measurements needed through careful numerical evaluation, and provide theoretical bounds and numerical values for $N_{\epsilon}$

1-Bit Matrix Completion

In this paper we develop a theory of matrix completion for the extreme case of noisy 1-bit observations. Instead of observing a subset of the real-valued entries of a matrix M, we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M. The central question we ask is whether or not it is possible to obtain an accurate estimate of M from this data. In general this would seem impossible, but we show that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M when ||M||_{\infty} <= \alpha, and rank(M) <= r. If the log-likelihood is a concave function (e.g., the logistic or probit observation models), then we can obtain this maximum likelihood estimate by optimizing a convex program. In addition, we also show that if instead of recovering M we simply wish to obtain an estimate of the distribution generating the 1-bit measurements, then we can eliminate the requirement that ||M||_{\infty} <= \alpha. For both cases, we provide lower bounds showing that these estimates are near-optimal. We conclude with a suite of experiments that both verify the implications of our theorems as well as illustrate some of the practical applications of 1-bit matrix completion. In particular, we compare our program to standard matrix completion methods on movie rating data in which users submit ratings from 1 to 5. In order to use our program, we quantize this data to a single bit, but we allow the standard matrix completion program to have access to the original ratings (from 1 to 5). Surprisingly, the approach based on binary data performs significantly better