11,026 research outputs found
Finding a marked node on any graph by continuous-time quantum walk
Spatial search by discrete-time quantum walk can find a marked node on any
ergodic, reversible Markov chain quadratically faster than its classical
counterpart, i.e.\ in a time that is in the square root of the hitting time of
. However, in the framework of continuous-time quantum walks, it was
previously unknown whether such general speed-up is possible. In fact, in this
framework, the widely used quantum algorithm by Childs and Goldstone fails to
achieve such a speedup. Furthermore, it is not clear how to apply this
algorithm for searching any Markov chain . In this article, we aim to
reconcile the apparent differences between the running times of spatial search
algorithms in these two frameworks. We first present a modified version of the
Childs and Goldstone algorithm which can search for a marked element for any
ergodic, reversible by performing a quantum walk on its edges. Although
this approach improves the algorithmic running time for several instances, it
cannot provide a generic quadratic speedup for any . Secondly, using the
framework of interpolated Markov chains, we provide a new spatial search
algorithm by continuous-time quantum walk which can find a marked node on any
in the square root of the classical hitting time. In the scenario where
multiple nodes are marked, the algorithmic running time scales as the square
root of a quantity known as the extended hitting time. Our results establish a
novel connection between discrete-time and continuous-time quantum walks and
can be used to develop a number of Markov chain-based quantum algorithms.Comment: This version deals only with new algorithms for spatial search by
continuous-time quantum walk (CTQW) on ergodic, reversible Markov chains.
Please see arXiv:2004.12686 for results on the necessary and sufficient
conditions for the optimality of the Childs and Goldstone algorithm for
spatial search by CTQ
Speed-up via Quantum Sampling
The Markov Chain Monte Carlo method is at the heart of efficient
approximation schemes for a wide range of problems in combinatorial enumeration
and statistical physics. It is therefore very natural and important to
determine whether quantum computers can speed-up classical mixing processes
based on Markov chains. To this end, we present a new quantum algorithm, making
it possible to prepare a quantum sample, i.e., a coherent version of the
stationary distribution of a reversible Markov chain. Our algorithm has a
significantly better running time than that of a previous algorithm based on
adiabatic state generation. We also show that our methods provide a speed-up
over a recently proposed method for obtaining ground states of (classical)
Hamiltonians.Comment: 8 pages, fixed some minor typo
Quantum speedup for active learning agents
Can quantum mechanics help us in building intelligent robots and agents? One
of the defining characteristics of intelligent behavior is the capacity to
learn from experience. However, a major bottleneck for agents to learn in any
real-life situation is the size and complexity of the corresponding task
environment. Owing to, e.g., a large space of possible strategies, learning is
typically slow. Even for a moderate task environment, it may simply take too
long to rationally respond to a given situation. If the environment is
impatient, allowing only a certain time for a response, an agent may then be
unable to cope with the situation and to learn at all. Here we show that
quantum physics can help and provide a significant speed-up for active learning
as a genuine problem of artificial intelligence. We introduce a large class of
quantum learning agents for which we show a quadratic boost in their active
learning efficiency over their classical analogues. This result will be
particularly relevant for applications involving complex task environments.Comment: Minor updates, 14 pages, 3 figure
Search via Quantum Walk
We propose a new method for designing quantum search algorithms for finding a
"marked" element in the state space of a classical Markov chain. The algorithm
is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of
the Markov chain. The main new idea is to apply quantum phase estimation to the
quantum walk in order to implement an approximate reflection operator. This
operator is then used in an amplitude amplification scheme. As a result we
considerably expand the scope of the previous approaches of Ambainis (2004) and
Szegedy (2004). Our algorithm combines the benefits of these approaches in
terms of being able to find marked elements, incurring the smaller cost of the
two, and being applicable to a larger class of Markov chains. In addition, it
is conceptually simple and avoids some technical difficulties in the previous
analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in
Section
Hitting Time of Quantum Walks with Perturbation
The hitting time is the required minimum time for a Markov chain-based walk
(classical or quantum) to reach a target state in the state space. We
investigate the effect of the perturbation on the hitting time of a quantum
walk. We obtain an upper bound for the perturbed quantum walk hitting time by
applying Szegedy's work and the perturbation bounds with Weyl's perturbation
theorem on classical matrix. Based on the definition of quantum hitting time
given in MNRS algorithm, we further compute the delayed perturbed hitting time
(DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the
upper bound for DPQHT is actually greater than the difference between the
square root of the upper bound for a perturbed random walk and the square root
of the lower bound for a random walk.Comment: 9 page
Efficient Circuits for Quantum Walks
We present an efficient general method for realizing a quantum walk operator
corresponding to an arbitrary sparse classical random walk. Our approach is
based on Grover and Rudolph's method for preparing coherent versions of
efficiently integrable probability distributions. This method is intended for
use in quantum walk algorithms with polynomial speedups, whose complexity is
usually measured in terms of how many times we have to apply a step of a
quantum walk, compared to the number of necessary classical Markov chain steps.
We consider a finer notion of complexity including the number of elementary
gates it takes to implement each step of the quantum walk with some desired
accuracy. The difference in complexity for various implementation approaches is
that our method scales linearly in the sparsity parameter and
poly-logarithmically with the inverse of the desired precision. The best
previously known general methods either scale quadratically in the sparsity
parameter, or polynomially in the inverse precision. Our approach is especially
relevant for implementing quantum walks corresponding to classical random walks
like those used in the classical algorithms for approximating permanents and
sampling from binary contingency tables. In those algorithms, the sparsity
parameter grows with the problem size, while maintaining high precision is
required.Comment: Modified abstract, clarified conclusion, added application section in
appendix and updated reference
On the adiabatic condition and the quantum hitting time of Markov chains
We present an adiabatic quantum algorithm for the abstract problem of
searching marked vertices in a graph, or spatial search. Given a random walk
(or Markov chain) on a graph with a set of unknown marked vertices, one can
define a related absorbing walk where outgoing transitions from marked
vertices are replaced by self-loops. We build a Hamiltonian from the
interpolated Markov chain and use it in an adiabatic quantum
algorithm to drive an initial superposition over all vertices to a
superposition over marked vertices. The adiabatic condition implies that for
any reversible Markov chain and any set of marked vertices, the running time of
the adiabatic algorithm is given by the square root of the classical hitting
time. This algorithm therefore demonstrates a novel connection between the
adiabatic condition and the classical notion of hitting time of a random walk.
It also significantly extends the scope of previous quantum algorithms for this
problem, which could only obtain a full quadratic speed-up for state-transitive
reversible Markov chains with a unique marked vertex.Comment: 22 page
Faster quantum mixing for slowly evolving sequences of Markov chains
Markov chain methods are remarkably successful in computational physics,
machine learning, and combinatorial optimization. The cost of such methods
often reduces to the mixing time, i.e., the time required to reach the steady
state of the Markov chain, which scales as , the inverse of the
spectral gap. It has long been conjectured that quantum computers offer nearly
generic quadratic improvements for mixing problems. However, except in special
cases, quantum algorithms achieve a run-time of , which introduces a costly dependence on the Markov chain size
not present in the classical case. Here, we re-address the problem of mixing of
Markov chains when these form a slowly evolving sequence. This setting is akin
to the simulated annealing setting and is commonly encountered in physics,
material sciences and machine learning. We provide a quantum memory-efficient
algorithm with a run-time of ,
neglecting logarithmic terms, which is an important improvement for large state
spaces. Moreover, our algorithms output quantum encodings of distributions,
which has advantages over classical outputs. Finally, we discuss the run-time
bounds of mixing algorithms and show that, under certain assumptions, our
algorithms are optimal.Comment: 20 pages, 2 figure
Quantum walks can find a marked element on any graph
We solve an open problem by constructing quantum walks that not only detect
but also find marked vertices in a graph. In the case when the marked set
consists of a single vertex, the number of steps of the quantum walk is
quadratically smaller than the classical hitting time of any
reversible random walk on the graph. In the case of multiple marked
elements, the number of steps is given in terms of a related quantity
which we call extended hitting time.
Our approach is new, simpler and more general than previous ones. We
introduce a notion of interpolation between the random walk and the
absorbing walk , whose marked states are absorbing. Then our quantum walk
is simply the quantum analogue of this interpolation. Contrary to previous
approaches, our results remain valid when the random walk is not
state-transitive. We also provide algorithms in the cases when only
approximations or bounds on parameters (the probability of picking a
marked vertex from the stationary distribution) and are
known.Comment: 50 page
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