60 research outputs found
A Full Characterization of Quantum Advice
We prove the following surprising result: given any quantum state rho on n
qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of
two-qubit interactions), such that any ground state of H can be used to
simulate rho on all quantum circuits of fixed polynomial size. In terms of
complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which
supersedes the previous result of Aaronson that BQP/qpoly is contained in
PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in
power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools --
including a result of Alon et al. on learning of real-valued concept classes, a
result of Aaronson on the learnability of quantum states, and a result of
Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new
ones. The main new tool is a so-called majority-certificates lemma, which is
closely related to boosting in machine learning, and which seems likely to find
independent applications. In its simplest version, this lemma says the
following. Given any set S of Boolean functions on n variables, any function f
in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm
in S, such that each fi is the unique function in S compatible with O(log|S|)
input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines
needed to be changed to preserve our results. The revised definition is more
natural and has the same intuitive interpretation. 2. We needed properties of
Local Hamiltonian reductions going beyond those proved in previous works
(whose results we'd misstated). We now prove the needed properties. See p. 6
for more on both point
Ab initio symmetry-adapted no-core shell model
A multi-shell extension of the Elliott SU(3) model, the SU(3) symmetry-adapted version of the no-core shell model (SA-NCSM), is described. The significance of this SA-NCSM emerges from the physical relevance of its SU(3)-coupled basis, which - while it naturally manages center-of-mass spuriosity - provides a microscopic description of nuclei in terms of mixed shape configurations. Since typically configurations of maximum spatial deformation dominate, only a small part of the model space suffices to reproduce the low-energy nuclear dynamics and hence, offers an effective symmetry-guided framework for winnowing of model space. This is based on our recent findings of low-spin and high-deformation dominance in realistic NCSM results and, in turn, holds promise to significantly enhance the reach of ab initio shell models
Symplectic No-core Shell-model Approach to Intermediate-mass Nuclei
We present a microscopic description of nuclei in an intermediate-mass
region, including the proximity to the proton drip line, based on a no-core
shell model with a schematic many-nucleon long-range interaction with no
parameter adjustments. The outcome confirms the essential role played by the
symplectic symmetry to inform the interaction and the winnowing of shell-model
spaces. We show that it is imperative that model spaces be expanded well beyond
the current limits up through fifteen major shells to accommodate particle
excitations that appear critical to highly-deformed spatial structures and the
convergence of associated observables.Comment: 9 pages, 8 figure
Symmetry-adapted Ab initio theory for many-body correlations in nuclei
We demonstrate that no-core shell-model results for low-lying states of light and medium mass nuclei, whether they are dilute or dense systems, reveal a strong dominance of low-spin and high-deformation configurations. This result is independent of whether the system Hamiltonian is phenomenological in nature or derived from a realistic interaction. It implies that only a small fraction of the complete model space is required for a description of such states, and this in turn points to the importance of using a symmetry-adapted, no-core shell-model framework for describing such nuclei, one based on an LS coupling scheme with the associated spatial configurations organized according to deformation. These results confirm that the pioneering work of early developers of the field, J. P. Elliott with his SU(3) model and M. Moshinsky with his U(3) many-body oscillator work, extends to more open, multi-shell environments. Specifically, algebraic methods are both relevant in such an environment and they can be used to quell the combinatorial growth in dimensionality that comes with the addition of oscillator shells to a model space. Indeed, our findings demonstrate the utility of a symmetry-adapted, no-core shell-model approach, one that takes advantage of group theoretical as well as advanced computational methods. And importantly, what at first glance appear to be a daunting task - casting complex algebraic expressions of a symmetry-adapted scheme into a user-friendly and efficient shell-mode code, turns out to be not only doable, but a logical framework that embraces constructs that can be made to execute efficiently on massively parallel, multi-processor (and core) systems. Early results for some light p-shell nuclei are presented. In addition, we will show that the method can be extended to heavier nuclei of the sd-shell and beyond, including some cases of special astrophysical interest in the upper fp- and lower gds-shells, like isotopes of Ge, Se, and even Kr. © Published under licence by IOP Publishing Ltd
Regression on fixed-rank positive semidefinite matrices: a Riemannian approach
The paper addresses the problem of learning a regression model parameterized
by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear
nature of the search space and on scalability to high-dimensional problems. The
mathematical developments rely on the theory of gradient descent algorithms
adapted to the Riemannian geometry that underlies the set of fixed-rank
positive semidefinite matrices. In contrast with previous contributions in the
literature, no restrictions are imposed on the range space of the learned
matrix. The resulting algorithms maintain a linear complexity in the problem
size and enjoy important invariance properties. We apply the proposed
algorithms to the problem of learning a distance function parameterized by a
positive semidefinite matrix. Good performance is observed on classical
benchmarks
Symmetry-adapted ab initio shell model for nuclear structure calculations
An innovative concept, the symmetry-adapted ab initio shell model, that capitalizes on partial as well as exact symmetries that underpin the structure of nuclei, is discussed. This framework is expected to inform the leading features of nuclear structure and reaction data for light and medium mass nuclei, which are currently inaccessible by theory and experiment and for which predictions of modern phenomenological models often diverge. We use powerful computational and group-theoretical algorithms to perform ab initio CI (configuration-interaction) calculations in a model space spanned by SU(3) symmetry-adapted many-body configurations with the JISP16 nucleon-nucleon interaction. We demonstrate that the results for the ground states of light nuclei up through A = 16 exhibit a strong dominance of low-spin and high-deformation configurations together with an evident symplectic structure. This, in turn, points to the importance of using a symmetry-adapted framework, one based on an LS coupling scheme with the associated spatial configurations organized according to deformation
Ab initio open core shell model for nuclear structure
The SU(3) symmetry-adapted version of the No-Core Shell Model (NCSM), which reduces to the Elliott SU(3) Model in its 0ω limit, is described and shown to be effective in providing an efficient description of low-lying eigenstates of 12C and 16O. A symmetry-guided framework is suggested based on our recent findings of low-spin and high-deformation dominance in realistic NCSM results. This holds promise to significantly enhance the reach of ab initio shell models. © 2010 IOP Publishing Ltd
What a wonderful world - Simplicity within complexity
In this lecture, Louis Armstrong\u27s transformational Jazz, as exemplified by his signature recital of What a Wonderful World, will take us to the ever evolving World of Nuclear Physics. In particular, I will focus on the discovery of simplicities, or symmetry patterns, in complex nuclear systems. And as Jazz is to music, so too Nuclear Physics is a restless endeavor-the dissonance of forlorn flattened notes tracking with the intrigue of symmetry and its breaking in nuclei. I will also discuss some recent approaches to nuclear structure at this dawn of the 21st century, linking the intrigue of quarks and gluons with the fundamental science of the strong and weak interactions to the emergence of simplicity within complexity unveiled in atomic nuclei. © Published under licence by IOP Publishing Ltd
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