126 research outputs found
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 . 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 to an accuracy of
with probability at least using
measurements, where is a constant that depends only on
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
Techniques for learning sparse Pauli-Lindblad noise models
Error-mitigation techniques such as probabilistic error cancellation and
zero-noise extrapolation benefit from accurate noise models. The sparse
Pauli-Lindblad noise model is one of the most successful models for those
applications. In existing implementations, the model decomposes into a series
of simple Pauli channels with one- and two-local terms that follow the qubit
topology. While the model has been shown to accurately capture the noise in
contemporary superconducting quantum processors for error mitigation, it is
important to consider higher-weight terms and effects beyond nearest-neighbor
interactions. For such extended models to remain practical, however, we need to
ensure that they can be learned efficiently. In this work we present new
techniques that accomplish exactly this. We introduce twirling based on Pauli
rotations, which enables us to automatically generate single-qubit learning
correction sequences and reduce the number of unique fidelities that need to be
learned. In addition, we propose a basis-selection strategy that leverages
graph coloring and uniform covering arrays to minimize the number of learning
bases. Taken together, these techniques ensure that the learning of the
extended noise models remains efficient, despite their increased complexity
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
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