Learning adiabatic quantum algorithms over optimization problems

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Davide Pastorello ◽  
Enrico Blanzieri ◽  
Valter Cavecchia
Keyword(s):  
Optimization Problems ◽  
Quantum Algorithms

scholarly journals Applying Quantum Optimization Algorithms for Linear Programming

2017 ◽  
Author(s):  
Mert Side ◽  
Volkan Erol
Keyword(s):  
Resource Allocation ◽  
Linear Programming ◽  
Production Scheduling ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Quantum Optimization ◽  
Special Case

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.

scholarly journals Efficient Quantum Walk Circuits for Metropolis-Hastings Algorithm

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 287 ◽  
Author(s):  
Jessica Lemieux ◽  
Bettina Heim ◽  
David Poulin ◽  
Krysta Svore ◽  
Matthias Troyer
Keyword(s):  
Discrete Optimization ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Quantum Walk ◽  
Numerical Results ◽  
Circuit Implementation ◽  
Arithmetic Operations ◽  
Detailed Circuit ◽  
Discrete Optimization Problems ◽  
Direct Implementation

We present a detailed circuit implementation of Szegedy's quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly arithmetic operations. We thus reformulate the quantum walk, circumventing its implementation altogether by closely following the classical Metropolis-Hastings walk. We also present heuristic quantum algorithms that use the quantum walk in the context of discrete optimization problems and numerically study their performances. Our numerical results indicate polynomial quantum speedups in heuristic settings.

scholarly journals Warm-starting quantum optimization

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 479
Author(s):  
Daniel J. Egger ◽  
Jakub Mareček ◽  
Stefan Woerner
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Randomized Rounding ◽  
Initial State ◽  
Warm Start ◽  
Performance Guarantees ◽  
Unique Games ◽  
Quantum Optimization

There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

scholarly journals Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 502
Author(s):  
Benjamin Zanger ◽  
Christian B. Mendl ◽  
Martin Schulz ◽  
Martin Schreiber
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Optimization Problems ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
High Order ◽  
Quantum Computers ◽  
Integration Methods ◽  
Linear Ordinary Differential Equations ◽  
Legendre Collocation Method

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

scholarly journals Applying Quantum Optimization Algorithms for Linear Programming

2017 ◽  
Author(s):  
Mert Side ◽  
Volkan Erol
Keyword(s):  
Resource Allocation ◽  
Linear Programming ◽  
Production Scheduling ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Quantum Optimization ◽  
Special Case

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.

scholarly journals Detecting Quantum Speedup of HHL Algorithm for Linear Programming Scenarios

2017 ◽  
Author(s):  
Volkan Erol ◽  
Mert Side
Keyword(s):  
Linear Programming ◽  
Production Scheduling ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Equation System ◽  
Linear Equation System ◽  
The Matrix ◽  
Speedup Ratio ◽  
Special Case ◽  
Quantum Speedup

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems, we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we look at quantum speedup ratios of HHL Algorithm for different scenarios of linear programming. In a first scenario we look quantum speedup ratio (S(N)) as a function of phase transition and the ratio (κ) between the greatest and smallest eigenvalues of the matrix in linear equation system. As a second scenario, we investigate the changes in S(N) as a function of κ and s, which is the coefficient for defining the matrix as s-sparse.

scholarly journals On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing

2021 ◽  
Vol 2 (4) ◽  
pp. 1-21
Author(s):  
Stuart Hadfield
Keyword(s):  
Quantum Computing ◽  
Boolean Functions ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Building Blocks ◽  
Constraint Satisfaction Problems ◽  
Boolean Formula ◽  
Combinatorial Optimization Problems ◽  
Compute Function ◽  
Real Functions

Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli Z operators (Ising spin operators) with the terms of the sum corresponding to the function’s Fourier expansion. For many classes of Boolean functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation, i.e., as hard as computing its number of satisfying assignments. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses each acting on a fixed number of bits as is common in constraint satisfaction problems. We show composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks, which are particularly suitable for direct implementation as classical software. We further apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results to quantum algorithms for optimization. A goal of this work is to provide a design toolkit for quantum optimization which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to provide a unified framework for the various constructions appearing in the literature.

QUANTUM ALGORITHMS FOR OPTIMIZATION USING ASYMPTOTIC QUANTUM SEARCH

2008 ◽  
Vol 06 (04) ◽  
pp. 935-944
Author(s):  
RUBENS V. RAMOS ◽  
PAULO B. M. SOUSA
Keyword(s):  
Optimization Problems ◽  
Quantum Algorithms ◽  
Main Property ◽  
Quantum Circuit ◽  
Quantum Search ◽  
Minimal Distance ◽  
Unitary Operation ◽  
And Mathematics ◽  
Engineering Physics ◽  
Asymptotic Quantum

Problems of optimization are very common in several areas like engineering, physics, economics and mathematics and usually, they are very difficult to solve. Basically, one has to find the minimum or maximum of an objective function. In this work, we chose two optimization problems and we present quantum algorithms to solve them. The problems are: (1) to find the best decomposition of a unitary operation achievable by a programmable quantum circuit; (2) to find the minimal distance of a linear code. The quantum algorithms used are asymptotic quantum search and their main property is the fact that only one measurement is required.

Solving a class of continuous global optimization problems using quantum algorithms

2002 ◽  
Vol 296 (1) ◽  
pp. 9-14 ◽  
Author(s):  
V. Protopopescu ◽  
J. Barhen
Keyword(s):  
Global Optimization ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Continuous Global Optimization

scholarly journals Convex optimization using quantum oracles

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 220 ◽  
Author(s):  
Joran van Apeldoorn ◽  
András Gilyén ◽  
Sander Gribling ◽  
Ronald de Wolf
Keyword(s):  
Convex Optimization ◽  
Convex Set ◽  
Optimization Problems ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Classical Literature ◽  
Membership Queries ◽  
Convex Optimization Problems ◽  
Classical Work ◽  
Speed Up

We study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using O~(1) quantum queries to a membership oracle, which is an exponential quantum speed-up over the Ω(n) membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with a simplification of recent classical work of Lee, Sidford, and Vempala gives our efficient separation oracle. This in turn implies, via a known algorithm, that O~(n) quantum queries to a membership oracle suffice to implement an optimization oracle (the best known classical upper bound on the number of membership queries is quadratic). We also prove several lower bounds: Ω(n) quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and Ω(n) quantum separation queries are needed if it does not.

Sign in / Sign up

Export Citation Format

Share Document