17,324 research outputs found
Improved scaling of Time-Evolving Block-Decimation algorithm through Reduced-Rank Randomized Singular Value Decomposition
When the amount of entanglement in a quantum system is limited, the relevant
dynamics of the system is restricted to a very small part of the state space.
When restricted to this subspace the description of the system becomes
efficient in the system size. A class of algorithms, exemplified by the
Time-Evolving Block-Decimation (TEBD) algorithm, make use of this observation
by selecting the relevant subspace through a decimation technique relying on
the Singular Value Decomposition (SVD). In these algorithms, the complexity of
each time-evolution step is dominated by the SVD. Here we show that, by
applying a randomized version of the SVD routine (RRSVD), the power law
governing the computational complexity of TEBD is lowered by one degree,
resulting in a considerable speed-up. We exemplify the potential gains in
efficiency at the hand of some real world examples to which TEBD can be
successfully applied to and demonstrate that for those system RRSVD delivers
results as accurate as state-of-the-art deterministic SVD routines.Comment: 14 pages, 5 figure
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
Quantum computing is powerful because unitary operators describing the
time-evolution of a quantum system have exponential size in terms of the number
of qubits present in the system. We develop a new "Singular value
transformation" algorithm capable of harnessing this exponential advantage,
that can apply polynomial transformations to the singular values of a block of
a unitary, generalizing the optimal Hamiltonian simulation results of Low and
Chuang. The proposed quantum circuits have a very simple structure, often give
rise to optimal algorithms and have appealing constant factors, while usually
only use a constant number of ancilla qubits. We show that singular value
transformation leads to novel algorithms. We give an efficient solution to a
certain "non-commutative" measurement problem and propose a new method for
singular value estimation. We also show how to exponentially improve the
complexity of implementing fractional queries to unitaries with a gapped
spectrum. Finally, as a quantum machine learning application we show how to
efficiently implement principal component regression. "Singular value
transformation" is conceptually simple and efficient, and leads to a unified
framework of quantum algorithms incorporating a variety of quantum speed-ups.
We illustrate this by showing how it generalizes a number of prominent quantum
algorithms, including: optimal Hamiltonian simulation, implementing the
Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude
amplification, robust oblivious amplitude amplification, fast QMA
amplification, fast quantum OR lemma, certain quantum walk results and several
quantum machine learning algorithms. In order to exploit the strengths of the
presented method it is useful to know its limitations too, therefore we also
prove a lower bound on the efficiency of singular value transformation, which
often gives optimal bounds.Comment: 67 pages, 1 figur
Optimized Decimation of Tensor Networks with Super-orthogonalization for Two-Dimensional Quantum Lattice Models
A novel algorithm based on the optimized decimation of tensor networks with
super-orthogonalization (ODTNS) that can be applied to simulate efficiently and
accurately not only the thermodynamic but also the ground state properties of
two-dimensional (2D) quantum lattice models is proposed. By transforming the 2D
quantum model into a three-dimensional (3D) closed tensor network (TN)
comprised of the tensor product density operator and a 3D brick-wall TN, the
free energy of the system can be calculated with the imaginary time evolution,
in which the network Tucker decomposition is suggested for the first time to
obtain the optimal lower-dimensional approximation on the bond space by
transforming the TN into a super-orthogonal form. The efficiency and accuracy
of this algorithm are testified, which are fairly comparable with the quantum
Monte Carlo calculations. Besides, the present ODTNS scheme can also be
applicable to the 2D frustrated quantum spin models with nice efficiency
Quantum Recommendation Systems
A recommendation system uses the past purchases or ratings of products by
a group of users, in order to provide personalized recommendations to
individual users. The information is modeled as an preference
matrix which is assumed to have a good rank- approximation, for a small
constant .
In this work, we present a quantum algorithm for recommendation systems that
has running time . All known classical
algorithms for recommendation systems that work through reconstructing an
approximation of the preference matrix run in time polynomial in the matrix
dimension. Our algorithm provides good recommendations by sampling efficiently
from an approximation of the preference matrix, without reconstructing the
entire matrix. For this, we design an efficient quantum procedure to project a
given vector onto the row space of a given matrix. This is the first algorithm
for recommendation systems that runs in time polylogarithmic in the dimensions
of the matrix and provides an example of a quantum machine learning algorithm
for a real world application.Comment: 22 page
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Easy implementable algorithm for the geometric measure of entanglement
We present an easy implementable algorithm for approximating the geometric
measure of entanglement from above. The algorithm can be applied to any
multipartite mixed state. It involves only the solution of an eigenproblem and
finding a singular value decomposition, no further numerical techniques are
needed. To provide examples, the algorithm was applied to the isotropic states
of 3 qubits and the 3-qubit XX model with external magnetic field.Comment: 9 pages, 3 figure
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
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