The fastest known classical algorithm deciding the @math -colorability of @math -vertex graph requires running time @math for @math . In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time @math using quantum random access memory (QRAM). Our approach is based on 's quantum dynamic programming with applications of Grover's search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph @math -coloring problem with running time @math . In the polynomial-space quantum algorithm, we essentially show @math -time classical algorithms that can be improved quadratically by Grover's search.
There is almost no previous theoretical work on quantum algorithms for the graph coloring problems. F "urer mentioned that Grover's algorithm can be applied to branching algorithms so that Beigel and Eppstein's algorithm for the graph 3-coloring problem can be improved to running time @math @cite_2 . The quantum algorithms for Theorem are basically obtained by applying Grover's search to generalized Byskov's reduction algorithms on the basis of F "urer's observation.
@cite_2
2040927484 1727475799 1568345435 2796145042
Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of N items laid out in d spatial dimensions can be searched in time of order radical (N) for d>2, and in time of order radical (N) poly(log N) for d=2. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that radical (N) speedup can also be achieved on the hypercube. We show that full radical (N) speedup can be achieved on a d-dimensional periodic lattice for d>4. In d=4, the quantum walk search algorithm takes time of order radical (N) poly(log N), and in d<4, the algorithm does not provide substantial speedup. In his seminal paper, Grover points out the prospect of faster solutions for an NP-complete problem like SAT. If there are n variables, then an obvious classical deterministic algorithm checks out all 2n truth assignments in about 2n steps, while his quantum search algorithm can find a satisfying truth assignment in about 2n 2 steps. For several NP-complete problems, many sophisticated classical algorithms have been designed. They are still exponential, but much faster than the brute force algorithms. The question arises whether their running time can still be decreased from T(n) to O (√T(n)) by using a quantum computer. Isolated positive examples are known, and some speed-up has been obtained for wider classes. Here, we present a simple method to obtain the full T(n) to O(√T(n)) speed-up for most of the many nontrivial exponential time algorithms for NP-hard problems. The method works whenever the widely used technique of recursive decomposition is employed. This included all currently known algorithms for which such a speedup has not yet been known. We introduce a quantum algorithm that produces approximate solutions for combinatorial optimization problems. The algorithm depends on a positive integer p and the quality of the approximation improves as p is increased. The quantum circuit that implements the algorithm consists of unitary gates whose locality is at most the locality of the objective function whose optimum is sought. The depth of the circuit grows linearly with p times (at worst) the number of constraints. If p is fixed, that is, independent of the input size, the algorithm makes use of efficient classical preprocessing. If p grows with the input size a different strategy is proposed. We study the algorithm as applied to MaxCut on regular graphs and analyze its performance on 2-regular and 3-regular graphs for fixed p. For p = 1, on 3-regular graphs the quantum algorithm always finds a cut that is at least 0.6924 times the size of the optimal cut. We present a new quantum algorithm that either finds a triangle (a copy of @math ) in an undirected graph @math on @math nodes, or it outputs reject'' if @math is triangle free. The algorithm uses @math queries, and it is based on a new design concept of Ambainis that incorporates the benefits of quantum walks into Grover search. The algorithm both improves on, and is simpler than a recent algorithm of Szegedy which has @math query complexity. The Triangle Problem was first treated in [BDH+01], where an algorithm with @math query complexity was presented (here @math is the number of edges of @math ).