6,548 research outputs found
The Road to Quantum Computational Supremacy
We present an idiosyncratic view of the race for quantum computational
supremacy. Google's approach and IBM challenge are examined. An unexpected
side-effect of the race is the significant progress in designing fast classical
algorithms. Quantum supremacy, if achieved, won't make classical computing
obsolete.Comment: 15 pages, 1 figur
Direct certification of a class of quantum simulations
One of the main challenges in the field of quantum simulation and computation
is to identify ways to certify the correct functioning of a device when a
classical efficient simulation is not available. Important cases are situations
in which one cannot classically calculate local expectation values of state
preparations efficiently. In this work, we develop weak-membership formulations
of the certification of ground state preparations. We provide a non-interactive
protocol for certifying ground states of frustration-free Hamiltonians based on
simple energy measurements of local Hamiltonian terms. This certification
protocol can be applied to classically intractable analog quantum simulations:
For example, using Feynman-Kitaev Hamiltonians, one can encode universal
quantum computation in such ground states. Moreover, our certification protocol
is applicable to ground states encodings of IQP circuits demonstration of
quantum supremacy. These can be certified efficiently when the error is
polynomially bounded.Comment: 10 pages, corrected a small error in Eqs. (2) and (5
The Limits of Post-Selection Generalization
While statistics and machine learning offers numerous methods for ensuring
generalization, these methods often fail in the presence of adaptivity---the
common practice in which the choice of analysis depends on previous
interactions with the same dataset. A recent line of work has introduced
powerful, general purpose algorithms that ensure post hoc generalization (also
called robust or post-selection generalization), which says that, given the
output of the algorithm, it is hard to find any statistic for which the data
differs significantly from the population it came from.
In this work we show several limitations on the power of algorithms
satisfying post hoc generalization. First, we show a tight lower bound on the
error of any algorithm that satisfies post hoc generalization and answers
adaptively chosen statistical queries, showing a strong barrier to progress in
post selection data analysis. Second, we show that post hoc generalization is
not closed under composition, despite many examples of such algorithms
exhibiting strong composition properties
An Atypical Survey of Typical-Case Heuristic Algorithms
Heuristic approaches often do so well that they seem to pretty much always
give the right answer. How close can heuristic algorithms get to always giving
the right answer, without inducing seismic complexity-theoretic consequences?
This article first discusses how a series of results by Berman, Buhrman,
Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the
early 1970s through the early 1990s, explicitly or implicitly limited how well
heuristic algorithms can do on NP-hard problems. In particular, many desirable
levels of heuristic success cannot be obtained unless severe, highly unlikely
complexity class collapses occur. Second, we survey work initiated by Goldreich
and Wigderson, who showed how under plausible assumptions deterministic
heuristics for randomized computation can achieve a very high frequency of
correctness. Finally, we consider formal ways in which theory can help explain
the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012
issue of SIGACT New
On Statistical Query Sampling and NMR Quantum Computing
We introduce a ``Statistical Query Sampling'' model, in which the goal of an
algorithm is to produce an element in a hidden set with
reasonable probability. The algorithm gains information about through
oracle calls (statistical queries), where the algorithm submits a query
function and receives an approximation to . We
show how this model is related to NMR quantum computing, in which only
statistical properties of an ensemble of quantum systems can be measured, and
in particular to the question of whether one can translate standard quantum
algorithms to the NMR setting without putting all of their classical
post-processing into the quantum system. Using Fourier analysis techniques
developed in the related context of {em statistical query learning}, we prove a
number of lower bounds (both information-theoretic and cryptographic) on the
ability of algorithms to produces an , even when the set is fairly
simple. These lower bounds point out a difficulty in efficiently applying NMR
quantum computing to algorithms such as Shor's and Simon's algorithm that
involve significant classical post-processing. We also explicitly relate the
notion of statistical query sampling to that of statistical query learning.
An extended abstract appeared in the 18th Aunnual IEEE Conference of
Computational Complexity (CCC 2003), 2003.
Keywords: statistical query, NMR quantum computing, lower boundComment: 17 pages, no figures. Appeared in 18th Aunnual IEEE Conference of
Computational Complexity (CCC 2003
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
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