122 research outputs found
Efficient and Robust Methods for Quantum Tomography
The development of large-scale platforms that implement quantum information processing protocols requires new methods for verification and validation of quantum behavior. Quantum tomography (QT) is the standard tool for diagnosing quantum states, process, and readout devices by providing complete information about each. However, QT is limited since it is expensive to not only implement experimentally, but also requires heavy classical post-processing of experimental data. In this dissertation, we introduce new methods for QT that are more efficient to implement and robust to noise and errors, thereby making QT a more widely practical tool for current quantum information experiments. The crucial detail that makes these new, efficient, and robust methods possible is prior information about the quantum system. This prior information is prompted by the goals of most experiments in quantum information. Most quantum information processing protocols require pure states, unitary processes, and rank-1 POVM operators. Therefore, most experiments are designed to operate near this ideal regime, and have been tested by other methods to verify this objective. We show that when this is the case, QT can be accomplished with significantly fewer resources, and produce a robust estimate of the state, process, or readout device in the presence of noise and errors. Moreover, the estimate is robust even if the state is not exactly pure, the process is not exactly unitary, or the POVM is not exactly rank-1. Such compelling methods are only made possible by the positivity constraint on quantum states, processes, and POVMs. This requirement is an inherent feature of quantum mechanics, but has powerful consequences to QT. Since QT is necessarily an experimental tool for diagnosing quantum systems, we discuss a test of these new methods in an experimental setting. The physical system is an ensemble of laser-cooled cesium atoms in the laboratory of Prof. Poul Jessen. The atoms are prepared in the hyperfine ground manifold, which provides a large, 16-dimensional Hilbert space to test QT protocols. Experiments were conducted by Hector Sosa-Martinez et al. to demonstrate different QT protocols. We compare the results, and conclude that the new methods are effective for QT
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
Suppose we wish to recover a signal x in C^n from m intensity measurements of
the form ||^2, i = 1, 2,..., m; that is, from data in which phase
information is missing. We prove that if the vectors z_i are sampled
independently and uniformly at random on the unit sphere, then the signal x can
be recovered exactly (up to a global phase factor) by solving a convenient
semidefinite program---a trace-norm minimization problem; this holds with large
probability provided that m is on the order of n log n, and without any
assumption about the signal whatsoever. This novel result demonstrates that in
some instances, the combinatorial phase retrieval problem can be solved by
convex programming techniques. Finally, we also prove that our methodology is
robust vis a vis additive noise
Rank-based model selection for multiple ions quantum tomography
The statistical analysis of measurement data has become a key component of
many quantum engineering experiments. As standard full state tomography becomes
unfeasible for large dimensional quantum systems, one needs to exploit prior
information and the "sparsity" properties of the experimental state in order to
reduce the dimensionality of the estimation problem. In this paper we propose
model selection as a general principle for finding the simplest, or most
parsimonious explanation of the data, by fitting different models and choosing
the estimator with the best trade-off between likelihood fit and model
complexity. We apply two well established model selection methods -- the Akaike
information criterion (AIC) and the Bayesian information criterion (BIC) -- to
models consising of states of fixed rank and datasets such as are currently
produced in multiple ions experiments. We test the performance of AIC and BIC
on randomly chosen low rank states of 4 ions, and study the dependence of the
selected rank with the number of measurement repetitions for one ion states. We
then apply the methods to real data from a 4 ions experiment aimed at creating
a Smolin state of rank 4. The two methods indicate that the optimal model for
describing the data lies between ranks 6 and 9, and the Pearson test
is applied to validate this conclusion. Additionally we find that the mean
square error of the maximum likelihood estimator for pure states is close to
that of the optimal over all possible measurements.Comment: 24 pages, 6 figures, 3 table
Phase Retrieval Using Unitary 2-Designs
We consider a variant of the phase retrieval problem, where vectors are
replaced by unitary matrices, i.e., the unknown signal is a unitary matrix U,
and the measurements consist of squared inner products |Tr(C*U)|^2 with unitary
matrices C that are chosen by the observer. This problem has applications to
quantum process tomography, when the unknown process is a unitary operation.
We show that PhaseLift, a convex programming algorithm for phase retrieval,
can be adapted to this matrix setting, using measurements that are sampled from
unitary 4- and 2-designs. In the case of unitary 4-design measurements, we show
that PhaseLift can reconstruct all unitary matrices, using a near-optimal
number of measurements. This extends previous work on PhaseLift using spherical
4-designs.
In the case of unitary 2-design measurements, we show that PhaseLift still
works pretty well on average: it recovers almost all signals, up to a constant
additive error, using a near-optimal number of measurements. These 2-design
measurements are convenient for quantum process tomography, as they can be
implemented via randomized benchmarking techniques. This is the first positive
result on PhaseLift using 2-designs.Comment: 21 pages; v3: minor revisions, to appear at SampTA 2017; v2:
rewritten to focus on phase retrieval, with new title, improved error bounds,
and numerics; v1: original version, titled "Quantum Compressed Sensing Using
2-Designs
Phase Retrieval via Matrix Completion
This paper develops a novel framework for phase retrieval, a problem which
arises in X-ray crystallography, diffraction imaging, astronomical imaging and
many other applications. Our approach combines multiple structured
illuminations together with ideas from convex programming to recover the phase
from intensity measurements, typically from the modulus of the diffracted wave.
We demonstrate empirically that any complex-valued object can be recovered from
the knowledge of the magnitude of just a few diffracted patterns by solving a
simple convex optimization problem inspired by the recent literature on matrix
completion. More importantly, we also demonstrate that our noise-aware
algorithms are stable in the sense that the reconstruction degrades gracefully
as the signal-to-noise ratio decreases. Finally, we introduce some theory
showing that one can design very simple structured illumination patterns such
that three diffracted figures uniquely determine the phase of the object we
wish to recover
A comparative study of estimation methods in quantum tomography
As quantum tomography is becoming a key component of the quantum engineering toolbox, there is a need for a deeper understanding of the multitude of estimation methods available. Here we investigate and compare several such methods: maximum likelihood, least squares, generalised least squares, positive least squares, thresholded least squares and projected least squares. The common thread of the analysis is that each estimator projects the measurement data onto a parameter space with respect to a specific metric, thus allowing us to study the relationships between different estimators. The asymptotic behaviour of the least squares and the projected least squares estimators is studied in detail for the case of the covariant measurement and a family of states of varying ranks. This gives insight into the rank-dependent risk reduction for the projected estimator, and uncovers an interesting non-monotonic behaviour of the Bures risk. These asymptotic results complement recent non-asymptotic concentration bounds of [36] which point to strong optimality properties, and high computational efficiency of the projected linear estimators. To illustrate the theoretical methods we present results of an extensive simulation study. An app running the different estimators has been made available online
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