105 research outputs found
Threshold-Based Quantum Optimization
We propose and study Th-QAOA (pronounced Threshold QAOA), a variation of the
Quantum Alternating Operator Ansatz (QAOA) that replaces the standard phase
separator operator, which encodes the objective function, with a threshold
function that returns a value for solutions with an objective value above
the threshold and a otherwise. We vary the threshold value to arrive at a
quantum optimization algorithm. We focus on a combination with the Grover Mixer
operator; the resulting GM-Th-QAOA can be viewed as a generalization of
Grover's quantum search algorithm and its minimum/maximum finding cousin to
approximate optimization.
Our main findings include: (i) we show semi-formally that the optimum
parameter values of GM-Th-QAOA (angles and threshold value) can be found with
iterations of the classical outer loop, where is
the number of QAOA rounds and is an upper bound on the solution value
(often the number of vertices or edges in an input graph), thus eliminating the
notorious outer-loop parameter finding issue of other QAOA algorithms; (ii)
GM-Th-QAOA can be simulated classically with little effort up to 100 qubits
through a set of tricks that cut down memory requirements; (iii) somewhat
surprisingly, GM-Th-QAOA outperforms its non-thresholded counterparts in terms
of approximation ratios achieved. This third result holds across a range of
optimization problems (MaxCut, Max k-VertexCover, Max k-DensestSubgraph,
MaxBisection) and various experimental design parameters, such as different
input edge densities and constraint sizes
A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs
A -birthday repetition of a
two-prover game is a game in which the two provers are sent
random sets of questions from of sizes and respectively.
These two sets are sampled independently uniformly among all sets of questions
of those particular sizes. We prove the following birthday repetition theorem:
when satisfies some mild conditions, decreases exponentially in where is the total number of
questions. Our result positively resolves an open question posted by Aaronson,
Impagliazzo and Moshkovitz (CCC 2014).
As an application of our birthday repetition theorem, we obtain new
fine-grained hardness of approximation results for dense CSPs. Specifically, we
establish a tight trade-off between running time and approximation ratio for
dense CSPs by showing conditional lower bounds, integrality gaps and
approximation algorithms. In particular, for any sufficiently large and for
every , we show the following results:
- We exhibit an -approximation algorithm for dense Max -CSPs
with alphabet size via -level of Sherali-Adams relaxation.
- Through our birthday repetition theorem, we obtain an integrality gap of
for -level Lasserre relaxation for fully-dense Max
-CSP.
- Assuming that there is a constant such that Max 3SAT cannot
be approximated to within of the optimal in sub-exponential
time, our birthday repetition theorem implies that any algorithm that
approximates fully-dense Max -CSP to within a factor takes
time, almost tightly matching the algorithmic
result based on Sherali-Adams relaxation.Comment: 45 page
Quantum-inspired classical algorithm for graph problems by Gaussian boson sampling
We present a quantum-inspired classical algorithm that can be used for
graph-theoretical problems, such as finding the densest -subgraph and
finding the maximum weight clique, which are proposed as applications of a
Gaussian boson sampler. The main observation from Gaussian boson samplers is
that a given graph's adjacency matrix to be encoded in a Gaussian boson sampler
is nonnegative, which does not necessitate quantum interference. We first
provide how to program a given graph problem into our efficient classical
algorithm. We then numerically compare the performance of ideal and lossy
Gaussian boson samplers, our quantum-inspired classical sampler, and the
uniform sampler for finding the densest -subgraph and finding the maximum
weight clique and show that the advantage from Gaussian boson samplers is not
significant in general. We finally discuss the potential advantage of a
Gaussian boson sampler over the proposed sampler.Comment: 11 pages, 5 figure
The power of sum-of-squares for detecting hidden structures
We study planted problems---finding hidden structures in random noisy
inputs---through the lens of the sum-of-squares semidefinite programming
hierarchy (SoS). This family of powerful semidefinite programs has recently
yielded many new algorithms for planted problems, often achieving the best
known polynomial-time guarantees in terms of accuracy of recovered solutions
and robustness to noise. One theme in recent work is the design of spectral
algorithms which match the guarantees of SoS algorithms for planted problems.
Classical spectral algorithms are often unable to accomplish this: the twist in
these new spectral algorithms is the use of spectral structure of matrices
whose entries are low-degree polynomials of the input variables. We prove that
for a wide class of planted problems, including refuting random constraint
satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community
detection in stochastic block models, planted clique, and others, eigenvalues
of degree-d matrix polynomials are as powerful as SoS semidefinite programs of
roughly degree d. For such problems it is therefore always possible to match
the guarantees of SoS without solving a large semidefinite program. Using
related ideas on SoS algorithms and low-degree matrix polynomials (and inspired
by recent work on SoS and the planted clique problem by Barak et al.), we prove
new nearly-tight SoS lower bounds for the tensor and sparse principal component
analysis problems. Our lower bounds for sparse principal component analysis are
the first to suggest that going beyond existing algorithms for this problem may
require sub-exponential time
A Comparison of Quantum Algorithms for the Maximum Clique Problem
Two of the most promising computational models for quantum computing are the qubit-based model and the continuous variable model, which result in two different computational approaches, namely the qubit gate model and boson sampling. The qubit gate model is a universal form of quantum computation that relies heavily on the principles of superposition and entanglement to solve problems using qubits based on technologies ranging from magnetic fields created from superconducting materials to the spins of valence electrons in atoms. Boson sampling is a non-universal form of quantum computation that uses bosons as continuous-variable values for its computation. Both models show promising prospects for useful quantum advantages over classical computers, but these models are fundamentally different, not only on their technologies but on their applications. Each model excels in different sets of applications.
A direct comparison for solving a problem using qubit gate models and boson sampling allows one to better understand not only the individual technologies, but how to decide which model is better suited to solving a given problem and how to start development on solving the given problem. This thesis uses the maximum clique problem to examine the application development process in the qubit gate model and boson sampling as well as a comparison of other known algorithms to the maximum clique problem. The maximum clique problem is an NP-Hard problem concerned with finding the largest fully-connected subgraph. The qubit gate model algorithm to the maximum clique problem is a novel algorithm
Rounding Sum-of-Squares Relaxations
We present a general approach to rounding semidefinite programming
relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our
approach is based on using the connection between these relaxations and the
Sum-of-Squares proof system to transform a *combining algorithm* -- an
algorithm that maps a distribution over solutions into a (possibly weaker)
solution -- into a *rounding algorithm* that maps a solution of the relaxation
to a solution of the original problem.
Using this approach, we obtain algorithms that yield improved results for
natural variants of three well-known problems:
1) We give a quasipolynomial-time algorithm that approximates the maximum of
a low degree multivariate polynomial with non-negative coefficients over the
Euclidean unit sphere. Beyond being of interest in its own right, this is
related to an open question in quantum information theory, and our techniques
have already led to improved results in this area (Brand\~{a}o and Harrow, STOC
'13).
2) We give a polynomial-time algorithm that, given a d dimensional subspace
of R^n that (almost) contains the characteristic function of a set of size n/k,
finds a vector in the subspace satisfying ,
where . Aside from being a natural relaxation, this
is also motivated by a connection to the Small Set Expansion problem shown by
Barak et al. (STOC 2012) and our results yield a certain improvement for that
problem.
3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time
algorithm with substantially improved guarantees for recovering a planted
-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n
nonzero coordinates, we can recover it with high probability whenever , improving for prior methods which
intrinsically required
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