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Results for “HILBERT SPACE”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

Towards space from Hilbert space: finding lattice structure in finite-dimensional quantum systems

We report that field theories place one or more degrees of freedom at every point in space. Hilbert spaces describing quantum field theories, or their finite-dimensional discretizations on lattices, therefore have large amounts of structure: they are isomorphic to the tensor product of a smaller Hilbert space for each lattice site or point in space. Local field theories respecting this structure have interactions which preferentially couple nearby points. The emergence of classicality through decoherence relies on this framework of tensor-product decomposition and local interactions. We explore the emergence of such lattice structure from Hilbert-space considerations alone. We point out that the vast majority of finite-dimensional Hilbert spaces cannot be isomorphic to the tensor product of Hilbert-space subfactors that describes a lattice theory. A generic Hilbert space can only be split into a direct sum corresponding to a basis of state vectors spanning the Hilbert space, we consider setups in which the direct sum is naturally decomposed into two pieces. We define a notion of direct-sum locality which characterizes states and decompositions compatible with Hamiltonian time evolution. Finally, we illustrate these notions for a toy model that is the finite-dimensional discretization of the quantum-mechanical double-well potential. We discuss their relevance in cosmology and field theory, especially for theories which describe a landscape of vacua with different spacetime geometries.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Hilbert-space fragmentation, multifractality, and many-body localization

Investigating many-body localization (MBL) using exact numerical methods is limited by the exponential growth of the Hilbert space. However, localized eigenstates display multifractality and only extend over a vanishing fraction of the Hilbert space. Here, building on this remarkable property, we develop a simple yet efficient decimation scheme to discard the irrelevant parts of the Hilbert space of the random-field Heisenberg chain. This leads to a Hilbert space fragmentation in small clusters, allowing to access larger systems at strong disorder. The MBL transition is quantitatively predicted, together with a geometrical interpretation of MBL multifractality as a shattering of the Hilbert space.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1989

Absolute Stability And Hyperstability In Hilbert Space

Theorems on stabilities of feedback control systems proved. Paper presents recent developments regarding theorems of absolute stability and hyperstability of feedforward-and-feedback control system. Theorems applied in analysis of nonlinear, adaptive, and robust control. Extended to provide sufficient conditions for stability in system including nonlinear feedback subsystem and linear time-invariant (LTI) feedforward subsystem, state space of which is Hilbert space, and input and output spaces having finite numbers of dimensions. (In case of absolute stability, feedback subsystem memoryless and possibly time varying. For hyperstability, feedback system dynamical system.)

Wen, John Ting-Yung↗
DOE OSTI2020

Hilbert-Space Fragmentation from Strict Confinement

In this work, we study one-dimensional spin-1/2 models in which strict confinement of Ising domain walls leads to the fragmentation of Hilbert space into exponentially many disconnected subspaces. Whereas most previous works emphasize dipole moment conservation as an essential ingredient for such fragmentation, we instead require two commuting U(1) conserved quantities associated with the total domain-wall number and the total magnetization. The latter arises naturally from the confinement of domain walls. Remarkably, while some connected components of the Hilbert space thermalize, others are integrable by Bethe ansatz. We further demonstrate how this Hilbert-space fragmentation pattern arises perturbatively in the confining limit of $\mathbb{Z}_2$ gauge theory coupled to fermionic matter, leading to a hierarchy of timescales for motion of the fermions. This model can be realized experimentally in two complementary settings.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Generic Hilbert space fragmentation in Kogut-Susskind lattice gauge theories

At the heart of quantum many-body physics lies the understanding of mechanisms that avoid quantum thermalization in an isolated system quenched far from equilibrium. A prominent example is Hilbert space fragmentation, which has recently emerged as an ergodicity-breaking mechanism in constrained spin models. Here, we show that Kogut-Susskind formulations of lattice gauge theories in d+1D (d spatial and one temporal dimensions) give rise to Hilbert space fragmentation, and discuss possible implications for understanding continuum physics. Lastly, our findings not only prove that lattice gauge theories are a natural platform for Hilbert space fragmentation, they also serve as a guide to the conditions under which these models can be faithfully used to infer the thermalization properties of quantum chromodynamics.

Eigenstate thermalization↗
DOE OSTI2018

Reduced scaling Hilbert space variational Monte Carlo

We show that for both single-Slater-Jastrow and Jastrow geminal power wave functions the formal cost scaling of Hilbert space variational Monte Carlo can be reduced from fifth to fourth order in the system size, thus bringing it in line with the long-standing scaling of its real space counterpart. While traditional quantum chemistry methods can reduce costs related to the two-electron integral tensor through various tensor decomposition methods, we show that such approaches are ineffective in the presence of Hilbert space Jastrow factors. Instead, we develop a simple semi-stochastic approach that can take similar advantage of the near-sparsity of this four-index tensor. Through demonstrations on alkanes of increasing length, we show that accuracy and overall statistical uncertainty are not meaningfully affected and that a total cost crossover is reached as early as 50 electrons when using a minimal basis. Further study will be needed to assess where the crossover occurs in more compact molecular geometries and larger basis sets and to explore how in that context the crossover can be accelerated.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Interfaces and the extended Hilbert space of Chern-Simons theory

The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Hilbert space multireference coupled cluster tailored by matrix product states

In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants—the state universal one and the Brillouin–Wigner and Mukherjee’s state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO–LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations.

Chemistry↗
DOE OSTI2020

Localization from Hilbert space shattering: From theory to physical realizations

We show how a finite number of conservation laws can globally “shatter” Hilbert space into exponentially many dynamically disconnected subsectors, leading to an unexpected dynamics with features reminiscent of both many-body localization and quantum scars. A crisp example of this phenomenon is provided by a “fractonic” model of quantum dynamics constrained to conserve both charge and dipole moment. We show how the Hilbert space of the fractonic model dynamically fractures into disconnected emergent subsectors within a particular charge and dipole symmetry sector. This shattering can occur in arbitrary spatial dimensions. A large number of the emergent subsectors, exponentially many in system volume, have dimension one and exhibit strictly localized quantum dynamics—even in the absence of spatial disorder and in the presence of temporal noise. Other emergent subsectors display nontrivial dynamics and may be constructed by embedding finite-sized nontrivial blocks into the localized subspace. While “fractonic” models provide a particularly clean realization, the shattering phenomenon is more general, as we discuss. We also discuss how the key phenomena may be readily observed in near term ultracold atom experiments. In experimental realizations, the conservation laws are approximate rather than exact, so the localization only survives up to a prethermal timescale that we estimate. We comment on the implications of these results for recent predictions of Bloch/Stark many-body localization.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS2003

Quantum Search in Hilbert Space

A proposed quantum-computing algorithm would perform a search for an item of information in a database stored in a Hilbert-space memory structure. The algorithm is intended to make it possible to search relatively quickly through a large database under conditions in which available computing resources would otherwise be considered inadequate to perform such a task. The algorithm would apply, more specifically, to a relational database in which information would be stored in a set of N complex orthonormal vectors, each of N dimensions (where N can be exponentially large). Each vector would constitute one row of a unitary matrix, from which one would derive the Hamiltonian operator (and hence the evolutionary operator) of a quantum system. In other words, all the stored information would be mapped onto a unitary operator acting on a quantum state that would represent the item of information to be retrieved. Then one could exploit quantum parallelism: one could pose all search queries simultaneously by performing a quantum measurement on the system. In so doing, one would effectively solve the search problem in one computational step. One could exploit the direct- and inner-product decomposability of the unitary matrix to make the dimensionality of the memory space exponentially large by use of only linear resources. However, inasmuch as the necessary preprocessing (the mapping of the stored information into a Hilbert space) could be exponentially expensive, the proposed algorithm would likely be most beneficial in applications in which the resources available for preprocessing were much greater than those available for searching.

Zak, Michail↗
NASA NTRS1972

Davidon-Broyden rank-one minimization methods in Hilbert space with application to optimal control problems

The Davidon-Broyden class of rank one, quasi-Newton minimization methods is extended from Euclidean spaces to infinite-dimensional, real Hilbert spaces. For several techniques of choosing the step size, conditions are found which assure convergence of the associated iterates to the location of the minimum of a positive definite quadratic functional. For those techniques, convergence is achieved without the problem of the computation of a one-dimensional minimum at each iteration. The application of this class of minimization methods for the direct computation of the solution of an optimal control problem is outlined. The performance of various members of the class are compared by solving a sample optimal control problem. Finally, the sample problem is solved by other known gradient methods, and the results are compared with those obtained with the rank one quasi-Newton methods.

Straeter, T. A.↗
DOE OSTI2019

Higher spin de Sitter Hilbert space

We propose a complete microscopic definition of the Hilbert space of minimal higher spin de Sitter quantum gravity and its Hartle-Hawking vacuum state. The fundamental degrees of freedom are 2N bosonic fields living on the future conformal boundary, where N is proportional to the de Sitter horizon entropy. The vacuum state is normalizable. The model agrees in perturbation theory with expectations from a previously proposed dS-CFT description in terms of a fermionic Sp(N) model, but it goes beyond this, both in its conceptual scope and in its computational power. In particular it resolves the apparent pathologies affecting the Sp(N) model, and it provides an exact formula for late time vacuum correlation functions. We illustrate this by computing probabilities for arbitrarily large field excursions, and by giving fully explicit examples of vacuum 3- and 4-point functions. We discuss bulk reconstruction and show the perturbative bulk QFT canonical commutations relations can be reproduced from the fundamental operator algebra, but only up to a minimal error term ~e -O(N) , and only if the operators are coarse grained in such a way that the number of accessible “pixels” is less than O(N). Independent of this, we show that upon gauging the higher spin symmetry group, one is left with 2N physical degrees of freedom, and that all gauge invariant quantities can be computed by a 2N × 2N matrix model. This suggests a concrete realization of the idea of cosmological complementarity.

1/N expansion↗