52 research outputs found
Embedding classical dynamics in a quantum computer
We develop a framework for simulating measure-preserving, ergodic dynamical
systems on a quantum computer. Our approach provides a new operator-theoretic
representation of classical dynamics by combining ergodic theory with quantum
information science. The resulting quantum embedding of classical dynamics
(QECD) enables efficient simulation of spaces of classical observables with
exponentially large dimension using a quadratic number of quantum gates. The
QECD framework is based on a quantum feature map for representing classical
states by density operators on a reproducing kernel Hilbert space, , and an embedding of classical observables into self-adjoint operators on
. In this scheme, quantum states and observables evolve unitarily
under the lifted action of Koopman evolution operators of the classical system.
Moreover, by virtue of the reproducing property of , the quantum
system is pointwise-consistent with the underlying classical dynamics. To
achieve an exponential quantum computational advantage, we project the state of
the quantum system to a density matrix on a -dimensional tensor product
Hilbert space associated with qubits. By employing discrete Fourier-Walsh
transforms, the evolution operator of the finite-dimensional quantum system is
factorized into tensor product form, enabling implementation through a quantum
circuit of size . Furthermore, the circuit features a state preparation
stage, also of size , and a quantum Fourier transform stage of size
, which makes predictions of observables possible by measurement in the
standard computational basis. We prove theoretical convergence results for
these predictions as . We present simulated quantum circuit
experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum
System One.Comment: 42 pages, 9 figure
Data Assimilation in Operator Algebras
We develop an algebraic framework for sequential data assimilation of
partially observed dynamical systems. In this framework, Bayesian data
assimilation is embedded in a non-abelian operator algebra, which provides a
representation of observables by multiplication operators and probability
densities by density operators (quantum states). In the algebraic approach, the
forecast step of data assimilation is represented by a quantum operation
induced by the Koopman operator of the dynamical system. Moreover, the analysis
step is described by a quantum effect, which generalizes the Bayesian
observational update rule. Projecting this formulation to finite-dimensional
matrix algebras leads to new computational data assimilation schemes that are
(i) automatically positivity-preserving; and (ii) amenable to consistent
data-driven approximation using kernel methods for machine learning. Moreover,
these methods are natural candidates for implementation on quantum computers.
Applications to data assimilation of the Lorenz 96 multiscale system and the El
Nino Southern Oscillation in a climate model show promising results in terms of
forecast skill and uncertainty quantification.Comment: 46 pages, 4 figure
Bayesian algorithms for recovering structure from single-particle diffraction snapshots of unknown orientation: a comparison
The advent of X-ray Free Electron Lasers promises the possibility to
determine the structure of individual particles such as microcrystallites,
viruses and biomolecules from single-shot diffraction snapshots obtained before
the particle is destroyed by the intense femtosecond pulse. This program
requires the ability to determine the orientation of the particle giving rise
to each snapshot at signal levels as low as ~10-2 photons/pixel. Two apparently
different approaches have recently demonstrated this capability. Here we show
they represent different implementations of the same fundamental approach, and
identify the primary factors limiting their performance.Comment: 10 pages, 2 figure
Retrieving functional pathways of biomolecules from single-particle snapshots
A primary reason for the intense interest in structural biology is the fact that knowledge of structure can elucidate macromolecular functions in living organisms. Sustained effort has resulted in an impressive arsenal of tools for determining the static structures. But under physiological conditions, macromolecules undergo continuous conformational changes, a subset of which are functionally important. Techniques for capturing the continuous conformational changes underlying function are essential for further progress. Here, we present chemically-detailed conformational movies of biological function, extracted data-analytically from experimental single-particle cryo-electron microscopy (cryo-EM) snapshots of ryanodine receptor type 1 (RyR1), a calcium-activated calcium channel engaged in the binding of ligands. The functional motions differ substantially from those inferred from static structures in the nature of conformationally active structural domains, the sequence and extent of conformational motions, and the way allosteric signals are transduced within and between domains. Our approach highlights the importance of combining experiment, advanced data analysis, and molecular simulations
Coherent diffraction of single Rice Dwarf virus particles using hard X-rays at the Linac Coherent Light Source
Single particle diffractive imaging data from Rice Dwarf Virus (RDV) were recorded using the Coherent X-ray Imaging (CXI) instrument at the Linac Coherent Light Source (LCLS). RDV was chosen as it is a wellcharacterized model system, useful for proof-of-principle experiments, system optimization and algorithm development. RDV, an icosahedral virus of about 70 nm in diameter, was aerosolized and injected into the approximately 0.1 mu m diameter focused hard X-ray beam at the CXI instrument of LCLS. Diffraction patterns from RDV with signal to 5.9 angstrom ngstrom were recorded. The diffraction data are available through the Coherent X-ray Imaging Data Bank (CXIDB) as a resource for algorithm development, the contents of which are described here.11Ysciescopu
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