5,959 research outputs found
Quantum-Inspired Classical Algorithms for Singular Value Transformation
A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the
quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS
2017). The resulting algorithm, classical but "quantum-inspired", efficiently
computes a low-rank approximation of the users' preference matrix. Subsequent
works have shown how to construct efficient quantum-inspired algorithms for
approximating the pseudo-inverse of a low-rank matrix as well, which can be
used to (approximately) solve low-rank linear systems of equations. In the
present paper, we pursue this line of research and develop quantum-inspired
algorithms for a large class of matrix transformations that are defined via the
singular value decomposition of the matrix. In particular, we obtain classical
algorithms with complexity polynomially related (in most parameters) to the
complexity of the best quantum algorithms for singular value transformation
recently developed by Chakraborty, Gily\'{e}n and Jeffery (ICALP 2019) and
Gily\'{e}n, Su, Low and Wiebe (STOC19).Comment: 19 page
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
Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension
We construct an efficient classical analogue of the quantum matrix inversion
algorithm (HHL) for low-rank matrices. Inspired by recent work of Tang,
assuming length-square sampling access to input data, we implement the
pseudoinverse of a low-rank matrix and sample from the solution to the problem
using fast sampling techniques. We implement the pseudo-inverse by
finding an approximate singular value decomposition of via subsampling,
then inverting the singular values. In principle, the approach can also be used
to apply any desired "smooth" function to the singular values. Since many
quantum algorithms can be expressed as a singular value transformation problem,
our result suggests that more low-rank quantum algorithms can be effectively
"dequantised" into classical length-square sampling algorithms.Comment: 10 page
Quantum Computing in the NISQ era and beyond
Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the
near future. Quantum computers with 50-100 qubits may be able to perform tasks
which surpass the capabilities of today's classical digital computers, but
noise in quantum gates will limit the size of quantum circuits that can be
executed reliably. NISQ devices will be useful tools for exploring many-body
quantum physics, and may have other useful applications, but the 100-qubit
quantum computer will not change the world right away --- we should regard it
as a significant step toward the more powerful quantum technologies of the
future. Quantum technologists should continue to strive for more accurate
quantum gates and, eventually, fully fault-tolerant quantum computing.Comment: 20 pages. Based on a Keynote Address at Quantum Computing for
Business, 5 December 2017. (v3) Formatted for publication in Quantum, minor
revision
The Road to Quantum Computational Supremacy
We present an idiosyncratic view of the race for quantum computational
supremacy. Google's approach and IBM challenge are examined. An unexpected
side-effect of the race is the significant progress in designing fast classical
algorithms. Quantum supremacy, if achieved, won't make classical computing
obsolete.Comment: 15 pages, 1 figur
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