39 research outputs found
Quantum walk speedup of backtracking algorithms
We describe a general method to obtain quantum speedups of classical
algorithms which are based on the technique of backtracking, a standard
approach for solving constraint satisfaction problems (CSPs). Backtracking
algorithms explore a tree whose vertices are partial solutions to a CSP in an
attempt to find a complete solution. Assume there is a classical backtracking
algorithm which finds a solution to a CSP on n variables, or outputs that none
exists, and whose corresponding tree contains T vertices, each vertex
corresponding to a test of a partial solution. Then we show that there is a
bounded-error quantum algorithm which completes the same task using O(sqrt(T)
n^(3/2) log n) tests. In particular, this quantum algorithm can be used to
speed up the DPLL algorithm, which is the basis of many of the most efficient
SAT solvers used in practice. The quantum algorithm is based on the use of a
quantum walk algorithm of Belovs to search in the backtracking tree. We also
discuss how, for certain distributions on the inputs, the algorithm can lead to
an exponential reduction in expected runtime.Comment: 23 pages; v2: minor changes to presentatio
Quantum walks can unitarily match random walks on finite graphs
Quantum and random walks were proven to be equivalent on finite graphs by
demonstrating how to construct a time-dependent random walk sharing the exact
same evolution of vertex probability of any given discrete-time coined quantum
walk. Such equivalence stipulated a deep connection between the processes that
is far stronger than simply considering quantum walks as quantum analogues of
random walks. This article expands on the connection between quantum and random
walks by demonstrating a procedure that constructs a time-dependent quantum
walk matching the evolution of vertex probability of any given random walk in a
unitary way. It is a trivial fact that a quantum walk measured at all time
steps of its evolution degrades to a random walk. More interestingly, the
method presented describes a quantum walk that matches a random walk without
measurement operations, such that the unitary evolution of the quantum walk
captures the probability evolution of the random walk. The construction
procedure is general, covering both homogeneous and non-homogeneous random
walks. For the homogeneous random walk case, the properties of unitary
evolution imply that the quantum walk described is time-dependent since
homogeneous quantum walks do not converge for arbitrary initial conditionsComment: 9 pages, 1 figur
Practical implementation of a quantum backtracking algorithm
In previous work, Montanaro presented a method to obtain quantum speedups for
backtracking algorithms, a general meta-algorithm to solve constraint
satisfaction problems (CSPs). In this work, we derive a space efficient
implementation of this method. Assume that we want to solve a CSP with
constraints on variables and that the union of the domains in which these
variables take their value is of cardinality . Then, we show that the
implementation of Montanaro's backtracking algorithm can be done by using data qubits. We detail an implementation of the predicate associated
to the CSP with an additional register of qubits. We explicit our
implementation for graph coloring and SAT problems, and present simulation
results. Finally, we discuss the impact of the usage of static and dynamic
variable ordering heuristics in the quantum setting.Comment: 18 pages, 10 figure
A Unified Framework of Quantum Walk Search
Many quantum algorithms critically rely on quantum walk search, or the use of quantum walks to speed up search problems on graphs. However, the main results on quantum walk search are scattered over different, incomparable frameworks, such as the hitting time framework, the MNRS framework, and the electric network framework. As a consequence, a number of pieces are currently missing. For example, recent work by Ambainis et al. (STOC\u2720) shows how quantum walks starting from the stationary distribution can always find elements quadratically faster. In contrast, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements.
We present a new quantum walk search framework that unifies and strengthens these frameworks, leading to a number of new results. For example, the new framework effectively finds marked elements in the electric network setting. The new framework also allows to interpolate between the hitting time framework, minimizing the number of walk steps, and the MNRS framework, minimizing the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. In addition to quantum walks and phase estimation, our new algorithm makes use of quantum fast-forwarding, similar to the recent results by Ambainis et al. This perspective also enables us to derive more general complexity bounds on the quantum walk algorithms, e.g., based on Monte Carlo type bounds of the corresponding classical walk. As a final result, we show how in certain cases we can avoid the use of phase estimation and quantum fast-forwarding, answering an open question of Ambainis et al