247 research outputs found
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
Quantum machine learning: a classical perspective
Recently, increased computational power and data availability, as well as
algorithmic advances, have led machine learning techniques to impressive
results in regression, classification, data-generation and reinforcement
learning tasks. Despite these successes, the proximity to the physical limits
of chip fabrication alongside the increasing size of datasets are motivating a
growing number of researchers to explore the possibility of harnessing the
power of quantum computation to speed-up classical machine learning algorithms.
Here we review the literature in quantum machine learning and discuss
perspectives for a mixed readership of classical machine learning and quantum
computation experts. Particular emphasis will be placed on clarifying the
limitations of quantum algorithms, how they compare with their best classical
counterparts and why quantum resources are expected to provide advantages for
learning problems. Learning in the presence of noise and certain
computationally hard problems in machine learning are identified as promising
directions for the field. Practical questions, like how to upload classical
data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde
Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming
Semidefinite programming (SDP) is a central topic in mathematical
optimization with extensive studies on its efficient solvers. In this paper, we
present a proof-of-principle sublinear-time algorithm for solving SDPs with
low-rank constraints; specifically, given an SDP with constraint matrices,
each of dimension and rank , our algorithm can compute any entry and
efficient descriptions of the spectral decomposition of the solution matrix.
The algorithm runs in time
given access to a sampling-based low-overhead data structure for the constraint
matrices, where is the precision of the solution. In addition, we
apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with 1) SDP solvers based on the matrix
multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2)
sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to
compute the matrix exponential required in the MMW framework, we introduce two
new techniques that may be of independent interest:
Weighted sampling: assuming sampling access to each individual
constraint matrix , we propose a procedure that gives a
good approximation of .
Symmetric approximation: we propose a sampling procedure that gives
the \emph{spectral decomposition} of a low-rank Hermitian matrix . To the
best of our knowledge, this is the first sampling-based algorithm for spectral
decomposition, as previous works only give singular values and vectors.Comment: 37 pages, 1 figure. To appear in the Proceedings of the 45th
International Symposium on Mathematical Foundations of Computer Science (MFCS
2020
Improvements in Quantum SDP-Solving with Applications
Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore [Brandão and Svore, 2017] in 2016, rapid developments have been made on quantum optimization algorithms. In this paper we improve and generalize all prior quantum algorithms for SDP-solving and give a simpler and unified framework. We take a new perspective on quantum SDP-solvers and introduce several new techniques. One of these is the quantum operator input model, which generalizes the different input models used in previous work, and essentially any other reasonable input model. This new model assumes that the input matrices are embedded in a block of a unitary operator. In this model we give a O~((sqrt{m}+sqrt{n}gamma)alpha gamma^4) algorithm, where n is the size of the matrices, m is the number of constraints, gamma is the reciprocal of the scale-invariant relative precision parameter, and alpha is a normalization factor of the input matrices. In particular for the standard sparse-matrix access, the above result gives a quantum algorithm where alpha=s. We also improve on recent results of Brandão et al. [Fernando G. S. L. Brandão et al., 2018], who consider the special case w
A Quantum Interior Point Method for LPs and SDPs
We present a quantum interior point method with worst case running time
for
SDPs and for LPs, where the output of our algorithm is a pair of matrices
that are -optimal -approximate SDP solutions. The factor
is at most for SDPs and for LP's, and is
an upper bound on the condition number of the intermediate solution matrices.
For the case where the intermediate matrices for the interior point method are
well conditioned, our method provides a polynomial speedup over the best known
classical SDP solvers and interior point based LP solvers, which have a worst
case running time of and respectively. Our results
build upon recently developed techniques for quantum linear algebra and pave
the way for the development of quantum algorithms for a variety of applications
in optimization and machine learning.Comment: 32 page
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