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Results for “Hilbert spaces”

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 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Profusion of symmetry-protected qubits from stable ergodicity breaking

We show how combining a discrete symmetry with topological Hilbert space fragmentation can give rise to exponentially many topologically stable qubits protected by a single discrete symmetry. We illustrate this explicitly with the example of the CZ𝑝 model, where the encoded qubits are prethermally stable to arbitrary symmetry-respecting perturbations for parametrically long times, substantially enhancing the robustness of a recently proposed construction based on nontopological fragmentation. In this model, the encoded qubits naturally come in pairs for which a universal set of transversal logical gates can be performed, ruling out (by the Eastin-Knill theorem) the possibility of using them for quantum error correction. We also comment on the combination of symmetry enrichment and topological fragmentation more generally, and the implications for use of systems exhibiting Hilbert space fragmentation as quantum memories.

kinetically constrained models↗
DOE OSTI2024

How to Partition a Quantum Observable

We present a partition of quantum observables in an open quantum system that is inherited from the division of the underlying Hilbert space or configuration space. It is shown that this partition leads to the definition of an inhomogeneous continuity equation for generic, non-local observables. This formalism is employed to describe the local evolution of the von Neumann entropy of a system of independent quantum particles out of equilibrium. Crucially, we find that all local fluctuations in the entropy are governed by an entropy current operator, implying that the production of entanglement entropy is not measured by this partitioned entropy. For systems linearly perturbed from equilibrium, it is shown that this entropy current is equivalent to a heat current, provided that the system-reservoir coupling is partitioned symmetrically. Finally, we show that any other partition of the coupling leads directly to a divergence of the von Neumann entropy. Thus, we conclude that Hilbert-space partitioning is the only partition of the von Neumann entropy that is consistent with the laws of thermodynamics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Matrix ensembles with global symmetries and ’t Hooft anomalies from 2d gauge theory

The Hilbert space of a quantum system with internal global symmetry $\textit{G}$ decomposes into sectors labelled by irreducible representations of $\textit{G}$. If the system is chaotic, the energies in each sector should separately resemble ordinary random matrix theory. We show that such “sector-wise” random matrix ensembles arise as the boundary dual of two- dimensional gravity with a $\textit{G}$ gauge field in the bulk. Within each sector, the eigenvalue density is enhanced by a nontrivial factor of the dimension of the representation, and the ground state energy is determined by the quadratic Casimir. We study the consequences of ’t Hooft anomalies in the matrix ensembles, which are incorporated by adding specific topological terms to the gauge theory action. The effect is to introduce projective representations into the decomposition of the Hilbert space. Finally, we consider ensembles with $\textit{G}$ symmetry and time reversal symmetry, and analyze a simple case of a mixed anomaly between time reversal and an internal $\mathbb{Z}$ 2 symmetry.

2D Gravity↗
DOE OSTI2023

Quantum error correction in the black hole interior

We study the quantum error correction properties of the black hole interior in a toy model for an evaporating black hole: Jackiw-Teitelboim gravity entangled with a non-gravitational bath. After the Page time, the black hole interior degrees of freedom in this system are encoded in the bath Hilbert space. We use the gravitational path integral to show that the interior density matrix is correctable against the action of quantum operations on the bath which (i) do not have prior access to details of the black hole microstates, and (ii) do not have a large, negative coherent information with respect to the maximally mixed state on the bath, with the lower bound controlled by the black hole entropy and code subspace dimension. Thus, the encoding of the black hole interior in the radiation is robust against generic, low-rank quantum operations. For erasure errors, gravity comes within an O (1) distance of saturating the Singleton bound on the tolerance of error correcting codes. For typical errors in the bath to corrupt the interior, they must have a rank that is a large multiple of the bath Hilbert space dimension, with the precise coefficient set by the black hole entropy and code subspace dimension.

2D gravity↗
DOE OSTI2026

Spontaneously Broken Noninvertible Symmetries in Transverse-Field Ising Qudit Chains

Recent developments have revealed that symmetries need not form a group, but instead can be noninvertible. Here we use analytical arguments and numerical evidence to illuminate how spontaneous symmetry breaking of a noninvertible symmetry is similar yet distinct from ordinary, invertible, symmetry breaking. We consider one-dimensional chains of group-valued qudits, whose local Hilbert space is spanned by elements of a finite group 𝐺 (reducing to ordinary qubits when 𝐺=ℤ 2 ). We construct Ising-type transverse-field Hamiltonians with Rep⁡(𝐺) symmetry whose generators multiply according to the tensor product of irreducible representations (irreps) of the group 𝐺 . For non-Abelian 𝐺 , the symmetry is noninvertible. In the symmetry broken phase there is one ground state per irrep on a closed chain. The symmetry breaking can be detected by local order parameters but, unlike the invertible case, different ground states have distinct entanglement patterns. We show that for each irrep of dimension greater than one the corresponding ground state exhibits string order, entanglement spectrum degeneracies, and has gapless edge modes on an open chain—features usually associated with symmetry-protected topological order. Consequently, domain wall excitations behave as one-dimensional non-Abelian anyons with nontrivial internal Hilbert spaces and fusion rules. Our Letter identifies properties of noninvertible symmetry breaking that existing quantum hardware can probe.

1-dimensional spin chains↗
DOE OSTI2023

Parameterization of Direct and Doorway Processes in R-Matrix Formalism

R-matrix formalism is extended beyond compound nuclear (CN) resonant reactions to include parameterization of direct as well as doorway processes. Direct processes in the R-matrix exterior are parameterized by a unitary matrix that introduces mixing among wave function coefficients of the incoming and outgoing wave function components at the R-matrix channel surface. Doorway processes are parameterized by separating the Hilbert space of the interior R-matrix region into its doorway and CN subspaces, from which doorway state eigenenergies, reduced width amplitudes, and the strengths of their coupling to CN levels appear as new R-matrix parameters. Parameterization of generalized as well as the conventional Reich–Moore approximation for eliminated capture channels in the presence of direct, doorway, and CN processes is presented along with a complex-valued scattering length with contributions from direct, doorway, and CN capture processes. Derivation of Brune’s alternative R-matrix parameters is extended to include doorway states. This work suggests how R-matrix formalism could be extended further by adopting the concepts from related reaction formalisms.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Coupled Cluster Downfolding Theory: towards universal many-body algorithms for dimensionality reduction of composite quantum systems in chemistry and materials science

Abstract The recently introduced coupled cluster (CC) downfolding techniques for reducing the dimensionality of quantum many-body problems recast the CC formalism in the form of the renormalization procedure allowing, for the construction of effective (or downfolded) Hamiltonians in small-dimensionality sub-space, usually identified with the so-called active space, of the entire Hilbert space. The resulting downfolded Hamiltonians integrate out the external (out-of-active-space) Fermionic degrees of freedom from the internal (in-the-active-space) parameters of the wave function, which can be determined as components of the eigenvectors of the downfolded Hamiltonians in the active space. This paper will discuss the extension of non-Hermitian (associated with standard CC formulations) and Hermitian (associated with the unitary CC approaches) downfolding formulations to composite quantum systems commonly encountered in materials science and chemistry. The non-Hermitian formulation can provide a platform for developing local CC approaches, while the Hermitian one can serve as an ideal foundation for developing various quantum computing applications based on the limited quantum resources. We also discuss the algorithm for extracting the semi-analytical form of the inter-electron interactions in the active spaces.

Bauman, Nicholas P.↗
NASA NTRS1981

Rotating unconstrained modes - A more appropriate dynamic analysis of flexible spinning spacecraft

The objective of the paper is a formation of vehicle modes for gyroscopic space structures through a continuum formulation. This requires examination of adjointness of the associated eigenvalue problems. It is shown that standard conditions of self-adjointness are inadequate even for nongyroscopic structures; modifications required depend on the problem under study. On one hand, in the nongyroscopic case the frequencies appear only quadratically and it is relatively easy to devise the corresponding vehicle modes and the self-adjoint operators. On the other hand in the case of the gyroscopic structures the frequencies appear quadratically as well as linearly; this takes the vehicle mode from a configuration space to a phase space, that is, a Hilbert space in the present case. Four elastic systems of increasing complexity typifying nonspinning and spinning elastic spacecraft are investigated in detail.

Hablani, H. B.↗
DOE OSTI2021

Liouville operators over the Hardy space

The role of Liouville operators in the study of dynamical systems through the use of occupation measures has been an active area of research in control theory over the past decade. This article investigates Liouville operators over the Hardy space, which encode complex ordinary differential equations in an operator over a reproducing kernel Hilbert space.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

Spectral dilation of L(B,H)-valued measures and its application to stationary dilation for Banach space valued processes

Let B be a Banach space and H and K two Hilbert spaces. The spectral dilation of L(B,H)-valued measures is studied and it is shown that the recent results of Makagon and Salehi (1986) and Rosenberg (1982) on the dilation of L(K,H)-valued measures can be extended to hold for the general Banach space setting of L(B,H)-valued measures. These L(B,H)-valued measures are closely connected to the Banach space valued processes. This connection is recalled and as application of spectral dilation of L(B,H)-valued measures the well known stationary dilation results for scalar valued processes is extended to the case of Banach space valued processes.

Miamee, A. G.↗
NASA NTRS1989

Spectral dilation of L(B,H)-valued measures and its application to stationary dilation for Banach space valued processes

Let B be a Banach space and H and K to Hilbert spaces. The spectral dilation of L(B,H)-valued measures is studied and it is shown that the recent results of Makagon and Salehi (1986) and Rosenberg (1982) on the dilation of L(K,H)-valued measures can be extended to hold for the general Banach space setting of L(B,H)-valued measures. These L(B,H)-valued measures are closely connected to the Banach space valued processes. This connection is recalled and as application of spectral dilation of L(B,H)-valued measures the well known stationary dilation results for scalar valued processes is extended to the case of Banach space valued processes.

Miamee, A. G.↗
DOE OSTI2021

A derivation of Born’s rule from symmetry

Highlights: • A very simple proof for the rule by which we calculate probabilities in quantum mechanics. We here put forward a simple argument for Born’s rule based on the requirement that transition probabilities in quantum mechanics should not be a function of the number of degrees of freedom in the system, or the dimension of the Hilbert-space, respectively. The argument presented here can be interpreted as deriving Born’s rule from a symmetry requirement.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Deep transfer operator learning for partial differential equations under conditional shift

Transfer learning enables the transfer of knowledge gained while learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and labelling, potential computational power limitations and dataset distribution mismatches. Here, we propose a new transfer learning framework for task-specific learning (functional regression in partial differential equations) under conditional shift based on the deep operator network (DeepONet). Task-specific operator learning is accomplished by fine-tuning task-specific layers of the target DeepONet using a hybrid loss function that allows for the matching of individual target samples while also preserving the global properties of the conditional distribution of the target data. Inspired by conditional embedding operator theory, we minimize the statistical distance between labelled target data and the surrogate prediction on unlabelled target data by embedding conditional distributions onto a reproducing kernel Hilbert space. We demonstrate the advantages of our approach for various transfer learning scenarios involving nonlinear partial differential equations under diverse conditions due to shifts in the geometric domain and model dynamics. Our transfer learning framework enables fast and efficient learning of heterogeneous tasks despite considerable differences between the source and target domains.

42 ENGINEERING↗