88,952 research outputs found
Robust quantum spatial search
Quantum spatial search has been widely studied with most of the study
focusing on quantum walk algorithms. We show that quantum walk algorithms are
extremely sensitive to systematic errors. We present a recursive algorithm
which offers significant robustness to certain systematic errors. To search N
items, our recursive algorithm can tolerate errors of size O(1/\sqrt{\ln N})
which is exponentially better than quantum walk algorithms for which tolerable
error size is only O(\ln N/\sqrt{N}). Also, our algorithm does not need any
ancilla qubit. Thus our algorithm is much easier to implement experimentally
compared to quantum walk algorithms
Quantum search algorithm tailored to clause satisfaction problems
Many important computer science problems can be reduced to clause
satisfaction problem. We are given Boolean variables and
clauses where each clause is a function of values of some of the
variables. We want to find an assignment of variables for which all
clauses are satisfied. Let be a binary function which is if
clause is satisfied by the assignment else .
Then the solution is for which , where is the AND
function of all . In quantum computing, Grover`s algorithm can be
used to find . A crucial component of this algorithm is the selective phase
inversion of the solution state encoding . is implemented by
computing for all in superposition which requires computing AND of
all binary functions . Hence there must be coupling between the
computation circuits for each . In this paper, we present an
alternative quantum search algorithm which relaxes the requirement of such
couplings. Hence it offers implementation advantages for clause satisfaction
problems
Postprocessing can speed up general quantum search algorithms
A general quantum search algorithm aims to evolve a quantum system from a
known source state to an unknown target state . It uses
a diffusion operator having source state as one of its eigenstates and
, where denotes the selective phase inversion of
state. It evolves to a particular state ,
call it w-state, in time steps where is and is a characteristic of the diffusion operator. Measuring
the w-state gives the target state with the success probability of
and applications of the algorithm can boost it from to
, making the total time complexity . In the special case
of Grover's algorithm, is and is very close to . A more
efficient way to boost the success probability is quantum amplitude
amplification provided we can efficiently implement . Such an efficient
implementation is not known so far. In this paper, we present an efficient
algorithm to approximate selective phase inversions of the unknown eigenstates
of an operator using phase estimation algorithm. This algorithm is used to
efficiently approximate which reduces the time complexity of general
algorithm to . Though algorithms are known to exist,
our algorithm offers physical implementation advantages.Comment: Accepted for publication in Physical Review A. arXiv admin note:
substantial text overlap with arXiv:1210.464
Adiabatic Quantum Computation with a 1D projector Hamiltonian
Adiabatic quantum computation is based on the adiabatic evolution of quantum
systems. We analyse a particular class of qauntum adiabatic evolutions where
either the initial or final Hamiltonian is a one-dimensional projector
Hamiltonian on the corresponding ground state. The minimum energy gap which
governs the time required for a successful evolution is shown to be
proportional to the overlap of the ground states of the initial and final
Hamiltonians. We show that such evolutions exhibit a rapid crossover as the
ground state changes abruptly near the transition point where the energy gap is
minimum. Furthermore, a faster evolution can be obtained by performing a
partial adiabatic evolution within a narrow interval around the transition
point. These results generalize and quantify earlier works.Comment: revised versio
Quantum computers can search rapidly by using almost any selective transformations
The search problem is to find a state satisfying certain properties out of a
given set. Grover's algorithm drives a quantum computer from a prepared initial
state to the target state and solves the problem quadratically faster than a
classical computer. The algorithm uses selective transformations to distinguish
the initial state and target state from other states. It does not succeed
unless the selective transformations are very close to phase-inversions. Here
we show a way to go beyond this limitation. An important application lies in
quantum error-correction, where the errors can cause the selective
transformations to deviate from phase-inversions. The algorithms presented here
are robust to errors as long as the errors are reproducible and reversible.
This particular class of systematic errors arise often from imperfections in
apparatus setup. Hence our algorithms offer a significant flexibility in the
physical implementation of quantum search.Comment: 8 pages, Accepted for publication in PR
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