27,972 research outputs found
Learning with Errors is easy with quantum samples
Learning with Errors is one of the fundamental problems in computational
learning theory and has in the last years become the cornerstone of
post-quantum cryptography. In this work, we study the quantum sample complexity
of Learning with Errors and show that there exists an efficient quantum
learning algorithm (with polynomial sample and time complexity) for the
Learning with Errors problem where the error distribution is the one used in
cryptography. While our quantum learning algorithm does not break the LWE-based
encryption schemes proposed in the cryptography literature, it does have some
interesting implications for cryptography: first, when building an LWE-based
scheme, one needs to be careful about the access to the public-key generation
algorithm that is given to the adversary; second, our algorithm shows a
possible way for attacking LWE-based encryption by using classical samples to
approximate the quantum sample state, since then using our quantum learning
algorithm would solve LWE
Ring Learning With Errors: A crossroads between postquantum cryptography, machine learning and number theory
The present survey reports on the state of the art of the different
cryptographic functionalities built upon the ring learning with errors problem
and its interplay with several classical problems in algebraic number theory.
The survey is based to a certain extent on an invited course given by the
author at the Basque Center for Applied Mathematics in September 2018.Comment: arXiv admin note: text overlap with arXiv:1508.01375 by other
authors/ comment of the author: quotation has been added to Theorem 5.
Learning hard quantum distributions with variational autoencoders
Studying general quantum many-body systems is one of the major challenges in
modern physics because it requires an amount of computational resources that
scales exponentially with the size of the system.Simulating the evolution of a
state, or even storing its description, rapidly becomes intractable for exact
classical algorithms. Recently, machine learning techniques, in the form of
restricted Boltzmann machines, have been proposed as a way to efficiently
represent certain quantum states with applications in state tomography and
ground state estimation. Here, we introduce a new representation of states
based on variational autoencoders. Variational autoencoders are a type of
generative model in the form of a neural network. We probe the power of this
representation by encoding probability distributions associated with states
from different classes. Our simulations show that deep networks give a better
representation for states that are hard to sample from, while providing no
benefit for random states. This suggests that the probability distributions
associated to hard quantum states might have a compositional structure that can
be exploited by layered neural networks. Specifically, we consider the
learnability of a class of quantum states introduced by Fefferman and Umans.
Such states are provably hard to sample for classical computers, but not for
quantum ones, under plausible computational complexity assumptions. The good
level of compression achieved for hard states suggests these methods can be
suitable for characterising states of the size expected in first generation
quantum hardware.Comment: v2: 9 pages, 3 figures, journal version with major edits with respect
to v1 (rewriting of section "hard and easy quantum states", extended
discussion on comparison with tensor networks
A generative modeling approach for benchmarking and training shallow quantum circuits
Hybrid quantum-classical algorithms provide ways to use noisy
intermediate-scale quantum computers for practical applications. Expanding the
portfolio of such techniques, we propose a quantum circuit learning algorithm
that can be used to assist the characterization of quantum devices and to train
shallow circuits for generative tasks. The procedure leverages quantum hardware
capabilities to its fullest extent by using native gates and their qubit
connectivity. We demonstrate that our approach can learn an optimal preparation
of the Greenberger-Horne-Zeilinger states, also known as "cat states". We
further demonstrate that our approach can efficiently prepare approximate
representations of coherent thermal states, wave functions that encode
Boltzmann probabilities in their amplitudes. Finally, complementing proposals
to characterize the power or usefulness of near-term quantum devices, such as
IBM's quantum volume, we provide a new hardware-independent metric called the
qBAS score. It is based on the performance yield in a specific sampling task on
one of the canonical machine learning data sets known as Bars and Stripes. We
show how entanglement is a key ingredient in encoding the patterns of this data
set; an ideal benchmark for testing hardware starting at four qubits and up. We
provide experimental results and evaluation of this metric to probe the trade
off between several architectural circuit designs and circuit depths on an
ion-trap quantum computer.Comment: 16 pages, 9 figures. Minor revisions. As published in npj Quantum
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Advantages of versatile neural-network decoding for topological codes
Finding optimal correction of errors in generic stabilizer codes is a
computationally hard problem, even for simple noise models. While this task can
be simplified for codes with some structure, such as topological stabilizer
codes, developing good and efficient decoders still remains a challenge. In our
work, we systematically study a very versatile class of decoders based on
feedforward neural networks. To demonstrate adaptability, we apply neural
decoders to the triangular color and toric codes under various noise models
with realistic features, such as spatially-correlated errors. We report that
neural decoders provide significant improvement over leading efficient decoders
in terms of the error-correction threshold. Using neural networks simplifies
the process of designing well-performing decoders, and does not require prior
knowledge of the underlying noise model.Comment: 11 pages, 6 figures, 2 table
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