3,441 research outputs found
Supervised learning with quantum enhanced feature spaces
Machine learning and quantum computing are two technologies each with the
potential for altering how computation is performed to address previously
untenable problems. Kernel methods for machine learning are ubiquitous for
pattern recognition, with support vector machines (SVMs) being the most
well-known method for classification problems. However, there are limitations
to the successful solution to such problems when the feature space becomes
large, and the kernel functions become computationally expensive to estimate. A
core element to computational speed-ups afforded by quantum algorithms is the
exploitation of an exponentially large quantum state space through controllable
entanglement and interference. Here, we propose and experimentally implement
two novel methods on a superconducting processor. Both methods represent the
feature space of a classification problem by a quantum state, taking advantage
of the large dimensionality of quantum Hilbert space to obtain an enhanced
solution. One method, the quantum variational classifier builds on [1,2] and
operates through using a variational quantum circuit to classify a training set
in direct analogy to conventional SVMs. In the second, a quantum kernel
estimator, we estimate the kernel function and optimize the classifier
directly. The two methods present a new class of tools for exploring the
applications of noisy intermediate scale quantum computers [3] to machine
learning.Comment: Fixed typos, added figures and discussion about quantum error
mitigatio
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
The Measurement Calculus
Measurement-based quantum computation has emerged from the physics community
as a new approach to quantum computation where the notion of measurement is the
main driving force of computation. This is in contrast with the more
traditional circuit model which is based on unitary operations. Among
measurement-based quantum computation methods, the recently introduced one-way
quantum computer stands out as fundamental.
We develop a rigorous mathematical model underlying the one-way quantum
computer and present a concrete syntax and operational semantics for programs,
which we call patterns, and an algebra of these patterns derived from a
denotational semantics. More importantly, we present a calculus for reasoning
locally and compositionally about these patterns.
We present a rewrite theory and prove a general standardization theorem which
allows all patterns to be put in a semantically equivalent standard form.
Standardization has far-reaching consequences: a new physical architecture
based on performing all the entanglement in the beginning, parallelization by
exposing the dependency structure of measurements and expressiveness theorems.
Furthermore we formalize several other measurement-based models:
Teleportation, Phase and Pauli models and present compositional embeddings of
them into and from the one-way model. This allows us to transfer all the theory
we develop for the one-way model to these models. This shows that the framework
we have developed has a general impact on measurement-based computation and is
not just particular to the one-way quantum computer.Comment: 46 pages, 2 figures, Replacement of quant-ph/0412135v1, the new
version also include formalization of several other measurement-based models:
Teleportation, Phase and Pauli models and present compositional embeddings of
them into and from the one-way model. To appear in Journal of AC
A Faster Quantum Algorithm for Semidefinite Programming via Robust IPM Framework
This paper studies a fundamental problem in convex optimization, which is to
solve semidefinite programming (SDP) with high accuracy. This paper follows
from existing robust SDP-based interior point method analysis due to [Huang,
Jiang, Song, Tao and Zhang, FOCS 2022]. While, the previous work only provides
an efficient implementation in the classical setting. This work provides a
novel quantum implementation.
We give a quantum second-order algorithm with high-accuracy in both the
optimality and the feasibility of its output, and its running time depending on
on well-conditioned instances. Due to the limitation of
quantum itself or first-order method, all the existing quantum SDP solvers
either have polynomial error dependence or low-accuracy in the feasibility
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