Related papers: Faster quantum and classical SDP approximations fo…
The advent of quantum algorithms has initiated a discourse on the potential for quantum speedups for optimization problems. However, several factors still hinder a practical realization of the potential benefits. These include the lack of…
Despite the numerous uses of semidefinite programming (SDP) and its universal solvability via interior point methods (IPMs), it is rarely applied to practical large-scale problems. This mainly owes to the computational cost of IPMs that…
Spectral methods are a leading approach for tensor PCA with a ``spiked" Gaussian tensor. The methods use the spectrum of a linear operator in a vector space with exponentially high dimension and in Ref. 1 it was shown that quantum…
We present a new generic approach to the condensed-matter ground-state problem which is complementary to variational techniques and works directly in the thermodynamic limit. Relaxing the ground-state problem, we obtain semidefinite…
Branch-and-bound is a widely used technique for solving combinatorial optimisation problems where one has access to two procedures: a branching procedure that splits a set of potential solutions into subsets, and a cost procedure that…
Many practically important NP-hard optimization problems are inherently higher-order polynomial optimizations, which are typically addressed using approximation algorithms. Classical relaxations express polynomial objectives over a…
It is well-known in physics that the limit of large quantum spin $S$ should be understood as a semiclassical limit. This raises the question of whether such emergent classicality facilitates the approximation of computationally hard quantum…
These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms. So far, we have only discovered a few techniques which can produce speed up versus classical…
Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to…
This paper initiates the study of quantum algorithms for matroid property problems. It is shown that quadratic quantum speedup is possible for the calculation problem of finding the girth or the number of circuits (bases, flats,…
Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a…
Vector set orthogonal normalization and matrix QR decomposition are fundamental problems in matrix analysis with important applications in many fields. We know that Gram-Schmidt process is a widely used method to solve these two problems.…
A major obstacle for quantum optimizers is the reformulation of constraints as a quadratic unconstrained binary optimization (QUBO). Current QUBO translators exaggerate the weight $M$ of the penalty terms. Classically known as the "Big-$M$"…
Quantum approximate optimization is one of the promising candidates for useful quantum computation, particularly in the context of finding approximate solutions to Quadratic Unconstrained Binary Optimization (QUBO) problems. However, the…
We present a quantum algorithmic routine that extends the realm of Grover-based heuristics for tackling combinatorial optimization problems with arbitrary efficiently computable objective and constraint functions. Building on previously…
There has been considerable progress in the design and construction of quantum annealing devices. However, a conclusive detection of quantum speedup over traditional silicon-based machines remains elusive, despite multiple careful studies.…
We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the…
In this paper, we extend a previously presented Grover-based heuristic to tackle general combinatorial optimization problems with linear constraints. We further describe the introduced method as a framework that enables performance…
In the past decade quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests…
Mixed-integer linear programming problems are extensively used in industry for a wide range of optimization tasks. However, as they get larger, they present computational challenges for classical solvers within practical time limits.…