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arXiv:1906.02229

The Problem of Dynamic Programming on a Quantum Computer

Pooya Ronagh

Published Jun 5, 2019• 5 citations
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Abstract

We discuss the problem of finite-horizon dynamic programming (DP) on a quantum computer. We introduce a query model for studying quantum and classical algorithms for solving DP problems, and provide example oracle constructions for the travelling salesperson problem, the minimum set-cover problem, and the edit distance problem. We formulate open questions regarding quadratic quantum speedups for DP and discuss their implications. We then prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for DP, and show that no greater-than-quadratic speedup can be achieved for solving DP problems.

DOI
10.48550/arxiv.1906.02229

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We discuss the problem of finite-horizon dynamic programming (DP) on a quantum computer. We introduce a query model for studying quantum and classical algorithms for solving DP problems, and provide example oracle constructions for the travelling salesperson problem, the minimum set-cover problem, and the edit distance problem. We formulate open questions regarding quadratic quantum speedups for DP and discuss their implications. We then prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for DP, and show that no greater-than-quadratic speedup can be achieved for solving DP problems.
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∗(⌈c⌉42n)O^*(\lceil c \rceil^4 \sqrt{2^n})O∗(⌈c⌉42n​) on a quantum computer, where nnn is the number of vertices of the underlying graph and ⌈c⌉\lceil c \rceil⌈c⌉ is its maximum edge-weight. As another example, we show that the minimum set-cover problem can be solved in O(2npoly⁡(m,n))O(\sqrt{2^n} \operatorname{poly}(m, n))O(2n​poly(m,n)), where mmm is the number of sets used to cover a universe of size nnn. 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.
Combinatorial optimization problems involve find-ing an optimal solution from a finite set of feasible solutions.Dynamic Programming attempts to solve a combinatorial prob-lem by performing an exhaustive search over the entire domain of the problem.In many such cases, this exhaustive search is not effective or tractable.Travelling salesperson problem is an important combinatorial optimization problem in which the salesperson is given N cities to travel with the condition that he has to travel each city exactly once and return to the origin.Among the finite set of feasible routes, the salesperson has to choose an optimal route with shortest total distance.The route can be optimal either by cost or by distance.The classical algorithm which attempts to find the shortest distance/cost to travel all the cities run in exponential time complexity.For small number of cities, the classical algorithm is effective and feasible but as the number of cities increase, the number of possible routes and the time required to compute them grows exponentially.However, addressing this problem by implementing quantum computing significantly reduces the time required to compute these routes.A quantum algorithm is required to run on a quantum computer to obtain meaningful results as quantum computers use Quantum Bits which differ from the bits used by a classical computer.A.
In this work, we design quantum algorithms that are more efficient than classical algorithms to solve time-dependent and finite-horizon Markov Decision Processes (MDPs) in two distinct settings: (1) In the exact dynamics setting, where the agent has full knowledge of the environment's dynamics (i.e., transition probabilities), we prove that our Quantum Value Iteration (QVI)\textbf{Quantum Value Iteration (QVI)}Quantum Value Iteration (QVI) algorithm QVI-1\textbf{QVI-1}QVI-1 achieves a quadratic speedup in the size of the action space (A)(A)(A) compared with the classical value iteration algorithm for computing the optimal policy (π∗π^{*}π∗) and the optimal V-value function (V0∗V_{0}^{*}V0∗​). Furthermore, our algorithm QVI-2\textbf{QVI-2}QVI-2 provides an additional speedup in the size of the state space (S)(S)(S) when obtaining near-optimal policies and V-value functions. Both QVI-1\textbf{QVI-1}QVI-1 and QVI-2\textbf{QVI-2}QVI-2 achieve quantum query complexities that provably improve upon classical lower bounds, particularly in their dependences on SSS and AAA. (2) In the generative model setting, where samples from the environment are accessible in quantum superposition, we prove that our algorithms QVI-3\textbf{QVI-3}QVI-3 and QVI-4\textbf{QVI-4}QVI-4 achieve improvements in sample complexity over the state-of-the-art (SOTA) classical algorithm in terms of AAA, estimation error (ε)(ε)(ε), and time horizon (H)(H)(H). More importantly, we prove quantum lower bounds to show that QVI-3\textbf{QVI-3}QVI-3 and QVI-4\textbf{QVI-4}QVI-4 are asymptotically optimal, up to logarithmic factors, assuming a constant time horizon.
We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P\mathcal PP and an instance III of P\mathcal PP, such a speedup can be expressed in terms of the average degree δδδ of the dependency digraph GP(I)G_{\mathcal{P}}(I)GP​(I) of III, determined by a recursive formulation of P\mathcal PP. The nodes of this graph are the subproblems of P\mathcal PP induced by III and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in O~(∣V(GP(I))∣δ)\tilde{O}(|V(G_{\mathcal{P}}(I))| \sqrtδ)O~(∣V(GP​(I))∣δ​) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted nnn-vertex digraph with mmm edges that runs in O~(nnm)\tilde{O}(n\sqrt{nm})O~(nnm​) time, which improves the best known classical upper bound when m∈Ω(n1.4)m \in Ω(n^{1.4})m∈Ω(n1.4).
We investigate the quantum algorithms for dynamic programming by Ambainis et al. (SODA'19). While giving provable complexity speedups and applicable to a variety of NP-hard problems, these algorithms have a notable drawback: they require a large amount of Quantum Random Access Memory (QRAM), which potentially could be very challenging to implement in a physical quantum computer. In this work, we study how we can improve the space complexity by trading it for time, while still retaining a speedup over the classical algorithms. We show novel quantum time-space tradeoffs by combining different classical approaches with quantum techniques. For instance, we show that the Travelling Salesman Problem can be solved quantumly in O~(1.859n)\widetilde O(1.859^n)O(1.859n) time and O~(1.315n)\widetilde O(1.315^n)O(1.315n) QRAM space.
We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P and an instance I of P, such a speedup can be expressed in terms of the average degree δ of the dependency digraph GP(I) of I, determined by a recursive formulation of P. The nodes of this graph are the subproblems of P induced by I and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in Õ(|V (GP(I))|√δ) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted n-vertex digraph with m edges that runs in Õ(n√nm) time, which improves the best known classical upper bound when m ∈ Ω(n1.4).