1,149 research outputs found
Polynomial time quantum algorithms for certain bivariate hidden polynomial problems
We present a new method for solving the hidden polynomial graph problem
(HPGP) which is a special case of the hidden polynomial problem (HPP). The new
approach yields an efficient quantum algorithm for the bivariate HPGP even when
the input consists of several level set superpositions, a more difficult
version of the problem than the one where the input is given by an oracle. For
constant degree, the algorithm is polylogarithmic in the size of the base
field. We also apply the results to give an efficient quantum algorithm for the
oracle version of the HPP for an interesting family of bivariate hidden
functions. This family includes diagonal quadratic forms and elliptic curves.Comment: Theorem numbering changed; new subsection with a high-level
description of the main algorith
Hidden Symmetry Subgroup Problems
We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this connection, we obtain the first efficient quantum algorithm for the hidden polynomial problem for multivariate quadratic polynomials over fields of constant characteristic. We also apply the new methods to polynomial function graph problems and present an efficient quantum procedure for constant degree multivariate polynomials over any field. This result improves in several ways the currently known algorithms
Lower Bounds on Quantum Query Complexity
Shor's and Grover's famous quantum algorithms for factoring and searching
show that quantum computers can solve certain computational problems
significantly faster than any classical computer. We discuss here what quantum
computers_cannot_ do, and specifically how to prove limits on their
computational power. We cover the main known techniques for proving lower
bounds, and exemplify and compare the methods.Comment: survey, 23 page
AI Feynman: a Physics-Inspired Method for Symbolic Regression
A core challenge for both physics and artificial intellicence (AI) is
symbolic regression: finding a symbolic expression that matches data from an
unknown function. Although this problem is likely to be NP-hard in principle,
functions of practical interest often exhibit symmetries, separability,
compositionality and other simplifying properties. In this spirit, we develop a
recursive multidimensional symbolic regression algorithm that combines neural
network fitting with a suite of physics-inspired techniques. We apply it to 100
equations from the Feynman Lectures on Physics, and it discovers all of them,
while previous publicly available software cracks only 71; for a more difficult
test set, we improve the state of the art success rate from 15% to 90%.Comment: 15 pages, 2 figs. Our code is available at
https://github.com/SJ001/AI-Feynman and our Feynman Symbolic Regression
Database for benchmarking can be downloaded at
https://space.mit.edu/home/tegmark/aifeynman.htm
On Solving Systems of Diagonal Polynomial Equations Over Finite Fields
We present an algorithm to solve a system of diagonal polynomial equations
over finite fields when the number of variables is greater than some fixed
polynomial of the number of equations whose degree depends only on the degree
of the polynomial equations. Our algorithm works in time polynomial in the
number of equations and the logarithm of the size of the field, whenever the
degree of the polynomial equations is constant. As a consequence we design
polynomial time quantum algorithms for two algebraic hidden structure problems:
for the hidden subgroup problem in certain semidirect product p-groups of
constant nilpotency class, and for the multi-dimensional univariate hidden
polynomial graph problem when the degree of the polynomials is constant.Comment: A preliminary extended abstract of this paper has appeared in
Proceedings of FAW 2015, Springer LNCS vol. 9130, pp. 125-137 (2015
Exponential Quantum Speed-ups are Generic
A central problem in quantum computation is to understand which quantum
circuits are useful for exponential speed-ups over classical computation. We
address this question in the setting of query complexity and show that for
almost any sufficiently long quantum circuit one can construct a black-box
problem which is solved by the circuit with a constant number of quantum
queries, but which requires exponentially many classical queries, even if the
classical machine has the ability to postselect.
We prove the result in two steps. In the first, we show that almost any
element of an approximate unitary 3-design is useful to solve a certain
black-box problem efficiently. The problem is based on a recent oracle
construction of Aaronson and gives an exponential separation between quantum
and classical bounded-error with postselection query complexities.
In the second step, which may be of independent interest, we prove that
linear-sized random quantum circuits give an approximate unitary 3-design. The
key ingredient in the proof is a technique from quantum many-body theory to
lower bound the spectral gap of local quantum Hamiltonians.Comment: 24 pages. v2 minor correction
Testing microscopic discretization
What can we say about the spectra of a collection of microscopic variables
when only their coarse-grained sums are experimentally accessible? In this
paper, using the tools and methodology from the study of quantum nonlocality,
we develop a mathematical theory of the macroscopic fluctuations generated by
ensembles of independent microscopic discrete systems. We provide algorithms to
decide which multivariate gaussian distributions can be approximated by sums of
finitely-valued random vectors. We study non-trivial cases where the
microscopic variables have an unbounded range, as well as asymptotic scenarios
with infinitely many macroscopic variables. From a foundational point of view,
our results imply that bipartite gaussian states of light cannot be understood
as beams of independent d-dimensional particle pairs. It is also shown that the
classical description of certain macroscopic optical experiments, as opposed to
the quantum one, requires variables with infinite cardinality spectra.Comment: Proof of strong NP-hardness. Connection with random walks. New
asymptotic results. Numerous typos correcte
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
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