211 research outputs found
Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games
We study quantum algorithms on search trees of unknown structure, in a model
where the tree can be discovered by local exploration. That is, we are given
the root of the tree and access to a black box which, given a vertex ,
outputs the children of .
We construct a quantum algorithm which, given such access to a search tree of
depth at most , estimates the size of the tree within a factor of in steps. More generally, the same algorithm can
be used to estimate size of directed acyclic graphs (DAGs) in a similar model.
We then show two applications of this result:
a) We show how to transform a classical backtracking search algorithm which
examines nodes of a search tree into an time
quantum algorithm, improving over an earlier quantum backtracking algorithm of
Montanaro (arXiv:1509.02374).
b) We give a quantum algorithm for evaluating AND-OR formulas in a model
where the formula can be discovered by local exploration (modeling position
trees in 2-player games). We show that, in this setting, formulas of size
and depth can be evaluated in quantum time . Thus,
the quantum speedup is essentially the same as in the case when the formula is
known in advance.Comment: Fixed some typo
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
Universal Quantum Speedup for Branch-and-Bound, Branch-and-Cut, and Tree-Search Algorithms
Mixed Integer Programs (MIPs) model many optimization problems of interest in
Computer Science, Operations Research, and Financial Engineering. Solving MIPs
is NP-Hard in general, but several solvers have found success in obtaining
near-optimal solutions for problems of intermediate size. Branch-and-Cut
algorithms, which combine Branch-and-Bound logic with cutting-plane routines,
are at the core of modern MIP solvers. Montanaro proposed a quantum algorithm
with a near-quadratic speedup compared to classical Branch-and-Bound algorithms
in the worst case, when every optimal solution is desired. In practice,
however, a near-optimal solution is satisfactory, and by leveraging tree-search
heuristics to search only a portion of the solution tree, classical algorithms
can perform much better than the worst-case guarantee. In this paper, we
propose a quantum algorithm, Incremental-Quantum-Branch-and-Bound, with
universal near-quadratic speedup over classical Branch-and-Bound algorithms for
every input, i.e., if classical Branch-and-Bound has complexity on an
instance that leads to solution depth , Incremental-Quantum-Branch-and-Bound
offers the same guarantees with a complexity of . Our
results are valid for a wide variety of search heuristics, including
depth-based, cost-based, and heuristics. Universal speedups are also
obtained for Branch-and-Cut as well as heuristic tree search. Our algorithms
are directly comparable to commercial MIP solvers, and guarantee near quadratic
speedup whenever . We use numerical simulation to verify that for typical instances of the Sherrington-Kirkpatrick model, Maximum
Independent Set, and Portfolio Optimization; as well as to extrapolate the
dependence of on input size parameters. This allows us to project the
typical performance of our quantum algorithms for these important problems.Comment: 25 pages, 5 figure
Quantum-accelerated constraint programming
Constraint programming (CP) is a paradigm used to model and solve constraint
satisfaction and combinatorial optimization problems. In CP, problems are
modeled with constraints that describe acceptable solutions and solved with
backtracking tree search augmented with logical inference. In this paper, we
show how quantum algorithms can accelerate CP, at both the levels of inference
and search. Leveraging existing quantum algorithms, we introduce a
quantum-accelerated filtering algorithm for the global
constraint and discuss its applicability to a broader family of global
constraints with similar structure. We propose frameworks for the integration
of quantum filtering algorithms within both classical and quantum backtracking
search schemes, including a novel hybrid classical-quantum backtracking search
method. This work suggests that CP is a promising candidate application for
early fault-tolerant quantum computers and beyond.Comment: published in Quantu
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
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