5,194 research outputs found
Span Programs and Quantum Space Complexity
While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of space-bounded quantum computation. We relate the quantum space complexity of computing a function f with one-sided error to the logarithm of its span program size, a classical quantity that is well-studied in attempts to prove formula size lower bounds.
In the more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (two-sided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program size.
While no non-trivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first non-trivial lower bound on the approximate span program size of an explicit function
Doubly infinite separation of quantum information and communication
We prove the existence of (one-way) communication tasks with a subconstant
versus superconstant asymptotic gap, which we call "doubly infinite," between
their quantum information and communication complexities. We do so by studying
the exclusion game [C. Perry et al., Phys. Rev. Lett. 115, 030504 (2015)] for
which there exist instances where the quantum information complexity tends to
zero as the size of the input increases. By showing that the quantum
communication complexity of these games scales at least logarithmically in ,
we obtain our result. We further show that the established lower bounds and
gaps still hold even if we allow a small probability of error. However in this
case, the -qubit quantum message of the zero-error strategy can be
compressed polynomially.Comment: 16 pages, 2 figures. v4: minor errors fixed; close to published
version; v5: financial support info adde
Universal resources for approximate and stochastic measurement-based quantum computation
We investigate which quantum states can serve as universal resources for
approximate and stochastic measurement-based quantum computation, in the sense
that any quantum state can be generated from a given resource by means of
single-qubit (local) operations assisted by classical communication. More
precisely, we consider the approximate and stochastic generation of states,
resulting e.g. from a restriction to finite measurement settings or from
possible imperfections in the resources or local operations. We show that
entanglement-based criteria for universality obtained for the exact,
deterministic case can be lifted to the much more general approximate,
stochastic case, moving from the idealized situation considered in previous
works, to the practically relevant context of non-perfect state preparation. We
find that any entanglement measure fulfilling some basic requirements needs to
reach its maximum value on some element of an approximate, stochastic universal
family of resource states, as the resource size grows. This allows us to rule
out various families of states as being approximate, stochastic universal. We
provide examples of resources that are efficient approximate universal, but not
exact deterministic universal. We also study the robustness of universal
resources for measurement-based quantum computation under realistic assumptions
about the (imperfect) generation and manipulation of entangled states, giving
an explicit expression for the impact that errors made in the preparation of
the resource have on the possibility to use it for universal approximate and
stochastic state preparation. Finally, we discuss the relation between our
entanglement-based criteria and recent results regarding the uselessness of
states with a high degree of geometric entanglement as universal resources.Comment: 17 pages; abstract shortened with respect to the published version to
respect the arXiv limit of 1,920 character
Group-Sparse Signal Denoising: Non-Convex Regularization, Convex Optimization
Convex optimization with sparsity-promoting convex regularization is a
standard approach for estimating sparse signals in noise. In order to promote
sparsity more strongly than convex regularization, it is also standard practice
to employ non-convex optimization. In this paper, we take a third approach. We
utilize a non-convex regularization term chosen such that the total cost
function (consisting of data consistency and regularization terms) is convex.
Therefore, sparsity is more strongly promoted than in the standard convex
formulation, but without sacrificing the attractive aspects of convex
optimization (unique minimum, robust algorithms, etc.). We use this idea to
improve the recently developed 'overlapping group shrinkage' (OGS) algorithm
for the denoising of group-sparse signals. The algorithm is applied to the
problem of speech enhancement with favorable results in terms of both SNR and
perceptual quality.Comment: 14 pages, 11 figure
Logical closure properties of propositional proof systems - (Extended abstract)
In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic
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