618 research outputs found
Quantum machine learning: a classical perspective
Recently, increased computational power and data availability, as well as
algorithmic advances, have led machine learning techniques to impressive
results in regression, classification, data-generation and reinforcement
learning tasks. Despite these successes, the proximity to the physical limits
of chip fabrication alongside the increasing size of datasets are motivating a
growing number of researchers to explore the possibility of harnessing the
power of quantum computation to speed-up classical machine learning algorithms.
Here we review the literature in quantum machine learning and discuss
perspectives for a mixed readership of classical machine learning and quantum
computation experts. Particular emphasis will be placed on clarifying the
limitations of quantum algorithms, how they compare with their best classical
counterparts and why quantum resources are expected to provide advantages for
learning problems. Learning in the presence of noise and certain
computationally hard problems in machine learning are identified as promising
directions for the field. Practical questions, like how to upload classical
data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde
The Power Of Quantum Walk Insights, Implementation, And Applications
In this thesis, I investigate quantum walks in quantum computing from three aspects: the insights, the implementation, and the applications. Quantum walks are the quantum analogue of classical random walks. For the insights of quantum walks, I list and explain the required components for quantizing a classical random walk into a quantum walk. The components are, for instance, Markov chains, quantum phase estimation, and quantum spectrum theorem. I then demonstrate how the product of two reflections in the walk operator provides a quadratic speed-up, in comparison to the classical counterpart. For the implementation of quantum walks, I show the construction of an efficient circuit for realizing one single step of the quantum walk operator. Furthermore, I devise a more succinct circuit to approximately implement quantum phase estimation with constant precision controlled phase shift operators. From an implementation perspective, efficient circuits are always desirable because the realization of a phase shift operator with high precision would be a costly task and a critical obstacle. For the applications of quantum walks, I apply the quantum walk technique along with other fundamental quantum techniques, such as phase estimation, to solve the partition function problem. However, there might be some scenario in which the speed-up of spectral gap is insignificant. In a situation like that that, I provide an amplitude amplification-based iii approach to prepare the thermal Gibbs state. Such an approach is useful when the spectral gap is extremely small. Finally, I further investigate and explore the effect of noise (perturbation) on the performance of quantum walk
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
Quantum computing is powerful because unitary operators describing the
time-evolution of a quantum system have exponential size in terms of the number
of qubits present in the system. We develop a new "Singular value
transformation" algorithm capable of harnessing this exponential advantage,
that can apply polynomial transformations to the singular values of a block of
a unitary, generalizing the optimal Hamiltonian simulation results of Low and
Chuang. The proposed quantum circuits have a very simple structure, often give
rise to optimal algorithms and have appealing constant factors, while usually
only use a constant number of ancilla qubits. We show that singular value
transformation leads to novel algorithms. We give an efficient solution to a
certain "non-commutative" measurement problem and propose a new method for
singular value estimation. We also show how to exponentially improve the
complexity of implementing fractional queries to unitaries with a gapped
spectrum. Finally, as a quantum machine learning application we show how to
efficiently implement principal component regression. "Singular value
transformation" is conceptually simple and efficient, and leads to a unified
framework of quantum algorithms incorporating a variety of quantum speed-ups.
We illustrate this by showing how it generalizes a number of prominent quantum
algorithms, including: optimal Hamiltonian simulation, implementing the
Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude
amplification, robust oblivious amplitude amplification, fast QMA
amplification, fast quantum OR lemma, certain quantum walk results and several
quantum machine learning algorithms. In order to exploit the strengths of the
presented method it is useful to know its limitations too, therefore we also
prove a lower bound on the efficiency of singular value transformation, which
often gives optimal bounds.Comment: 67 pages, 1 figur
Estimation and uncertainty of reversible Markov models
Reversibility is a key concept in Markov models and Master-equation models of
molecular kinetics. The analysis and interpretation of the transition matrix
encoding the kinetic properties of the model relies heavily on the
reversibility property. The estimation of a reversible transition matrix from
simulation data is therefore crucial to the successful application of the
previously developed theory. In this work we discuss methods for the maximum
likelihood estimation of transition matrices from finite simulation data and
present a new algorithm for the estimation if reversibility with respect to a
given stationary vector is desired. We also develop new methods for the
Bayesian posterior inference of reversible transition matrices with and without
given stationary vector taking into account the need for a suitable prior
distribution preserving the meta- stable features of the observed process
during posterior inference. All algorithms here are implemented in the PyEMMA
software - http://pyemma.org - as of version 2.0
Faster quantum mixing for slowly evolving sequences of Markov chains
Markov chain methods are remarkably successful in computational physics,
machine learning, and combinatorial optimization. The cost of such methods
often reduces to the mixing time, i.e., the time required to reach the steady
state of the Markov chain, which scales as , the inverse of the
spectral gap. It has long been conjectured that quantum computers offer nearly
generic quadratic improvements for mixing problems. However, except in special
cases, quantum algorithms achieve a run-time of , which introduces a costly dependence on the Markov chain size
not present in the classical case. Here, we re-address the problem of mixing of
Markov chains when these form a slowly evolving sequence. This setting is akin
to the simulated annealing setting and is commonly encountered in physics,
material sciences and machine learning. We provide a quantum memory-efficient
algorithm with a run-time of ,
neglecting logarithmic terms, which is an important improvement for large state
spaces. Moreover, our algorithms output quantum encodings of distributions,
which has advantages over classical outputs. Finally, we discuss the run-time
bounds of mixing algorithms and show that, under certain assumptions, our
algorithms are optimal.Comment: 20 pages, 2 figure
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