We present a general quantum algorithm for solving finite-horizon dynamic programming problems. Up to polylogarithmic factors, our algorithm provides a quadratic quantum advantage in terms of the number of states of a given dynamic programming problem. This speedup comes at the expense of the appearance of other polynomial factors representative of the number of actions of the dynamic programming problem, the maximum value of the instantaneous reward, and the time horizon of the problem. Our algorithm can be applied to combinatorial optimization problems solved classically using dynamic programming techniques. As one application, we show that the travelling salesperson problem can be solved in $O^*(\lceil c \rceil^4 \sqrt{2^n})$ on a quantum computer, where $n$ is the number of vertices of the underlying graph and $\lceil c \rceil$ is its maximum edge-weight. As another example, we show that the minimum set-cover problem can be solved in $O(\sqrt{2^n} \operatorname{poly}(m, n))$ , where $m$ is the number of sets used to cover a universe of size $n$ . Finally, we prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for solving dynamic programming problems, and show that no greater-than-quadratic speedup in either the number of states or number of actions can be achieved for solving dynamic programming problems using quantum algorithms.We present a general quantum algorithm for solving finite-horizon dynamic programming problems. Up to polylogarithmic factors, our algorithm provides a quadratic quantum advantage in terms of the number of states of a given dynamic programming problem. This speedup comes at the expense of the appearance of other polynomial factors representative of the number of actions of the dynamic programming problem, the maximum value of the instantaneous reward, and the time horizon of the problem. Our algorithm can be applied to combinatorial optimization problems solved classically using dynamic programming techniques. As one application, we show that the travelling salesperson problem can be solved in $O^*(\lceil c \rceil^4 \sqrt{2^n})$ on a quantum computer, where $n$ is the number of vertices of the underlying graph and $\lceil c \rceil$ is its maximum edge-weight. As another example, we show that the minimum set-cover problem can be solved in $O(\sqrt{2^n} \operatorname{poly}(m, n))$, where $m$ is the number of sets used to cover a universe of size $n$. Finally, we prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for solving dynamic programming problems, and show that no greater-than-quadratic speedup in either the number of states or number of actions can be achieved for solving dynamic programming problems using quantum algorithms.

Jun 5, 201913 citations

Quantum Speedups for Exponential-Time Dynamic Programming Algorithms

Andris Ambainis, Kaspars Balodis, Jānis Iraids +3 more

In this paper we study quantum algorithms for NP-complete problems whose best classical algorithm is an exponential time application of dynamic programming. We introduce the path in the hypercube problem that models many of these dynamic programming algorithms. In this problem we are asked whether there is a path from $0^n$ to $1^n$ in a given subgraph of the Boolean hypercube, where the edges are all directed from smaller to larger Hamming weight. We give a quantum algorithm that solves path in the hypercube in time $O^*(1.817^n)$ . The technique combines Grover's search with computing a partial dynamic programming table. We use this approach to solve a variety of vertex ordering problems on graphs in the same time $O^*(1.817^n)$ , and graph bandwidth in time $O^*(2.946^n)$ . Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time $O^*(1.728^n)$ .In this paper we study quantum algorithms for NP-complete problems whose best classical algorithm is an exponential time application of dynamic programming. We introduce the path in the hypercube problem that models many of these dynamic programming algorithms. In this problem we are asked whether there is a path from $0^n$ to $1^n$ in a given subgraph of the Boolean hypercube, where the edges are all directed from smaller to larger Hamming weight. We give a quantum algorithm that solves path in the hypercube in time $O^*(1.817^n)$. The technique combines Grover's search with computing a partial dynamic programming table. We use this approach to solve a variety of vertex ordering problems on graphs in the same time $O^*(1.817^n)$, and graph bandwidth in time $O^*(2.946^n)$. Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time $O^*(1.728^n)$.

arXiv:1807.05209Jul 13, 20181 citation

Quantum Algorithms for Optimization Problems

Dilnoz Muhamediyeva, Rustam Raximov, Shavkat Urokov

Lecture notes in computer scienceJul 29, 20250 citations

Quantum Algorithms and Applications

Melinda Andrews, Joseph Stephen Dulny, Andrew N. Glaudell +6 more

Sep 1, 20200 citations

New quantum algorithm for studying NP-complete problems

Masanori Ohya, Igor Vasil'evich Volovich

Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm. keywords: Quantum Algorithm, NP-complete problem, Chaotic Dynamics 1Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm. keywords: Quantum Algorithm, NP-complete problem, Chaotic Dynamics 1

Jan 1, 200337 citations

O^*(\lceil c \rceil^4 \sqrt{2^n})

on a quantum computer, where

n

is the number of vertices of the underlying graph and

\lceil c \rceil

is its maximum edge-weight. As another example, we show that the minimum set-cover problem can be solved in

O(\sqrt{2^n} \operatorname{poly}(m, n))

, where

m

is the number of sets used to cover a universe of size

n

. Finally, we prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for solving dynamic programming problems, and show that no greater-than-quadratic speedup in either the number of states or number of actions can be achieved for solving dynamic programming problems using quantum algorithms.

0^n

1^n

in a given subgraph of the Boolean hypercube, where the edges are all directed from smaller to larger Hamming weight. We give a quantum algorithm that solves path in the hypercube in time

O^*(1.817^n)

. The technique combines Grover's search with computing a partial dynamic programming table. We use this approach to solve a variety of vertex ordering problems on graphs in the same time

O^*(1.817^n)

, and graph bandwidth in time

O^*(2.946^n)

. Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time

O^*(1.728^n)