11 research outputs found
Controlled quantum search on structured databases
We present quantum algorithms to search for marked vertices in structured
databases with low connectivity. Adopting a multi-stage search process, we
achieve a success probability close to on Cayley trees with large
branching factors. We find that the number of stages required is given by the
height of the Cayley tree. At each stage, the jumping rate should be chosen as
different values. The dominant term of the runtime in the search process is
proportional to for the Cayley tree of height with
vertices. We further find that one can control the number of stages by
adjusting the weight of the edges in the graphs. The multi-stage search process
can be merged into a single stage, and then an optimal runtime proportional to
is achieved, yielding a substantial speedup. The search process is
quite robust under various kinds of small perturbations.Comment: 10 pages, 18 figure
Hamiltonian and measuring time for analog quantum search
We derive in this study a Hamiltonian to solve with certainty the analog
quantum search problem analogue to the Grover algorithm. The general form of
the initial state is considered. Since the evaluation of the measuring time for
finding the marked state by probability of unity is crucially important in the
problem, especially when the Bohr frequency is high, we then give the exact
formula as a function of all given parameters for the measuring time.Comment: 5 page
Quantum Dueling: an Efficient Solution for Combinatorial Optimization
In this paper, we present a new algorithm for generic combinatorial
optimization, which we term quantum dueling. Traditionally, potential solutions
to the given optimization problems were encoded in a ``register'' of qubits.
Various techniques are used to increase the probability of finding the best
solution upon measurement. Quantum dueling innovates by integrating an
additional qubit register, effectively creating a ``dueling'' scenario where
two sets of solutions compete. This dual-register setup allows for a dynamic
amplification process: in each iteration, one register is designated as the
'opponent', against which the other register's more favorable solutions are
enhanced through a controlled quantum search. This iterative process gradually
steers the quantum state within both registers toward the optimal solution.
With a quantitative contraction for the evolution of the state vector,
classical simulation under a broad range of scenarios and hyper-parameter
selection schemes shows that a quadratic speedup is achieved, which is further
tested in more real-world situations. In addition, quantum dueling can be
generalized to incorporate arbitrary quantum search techniques and as a quantum
subroutine within a higher-level algorithm. Our work demonstrates that
increasing the number of qubits allows the development of previously
unthought-of algorithms, paving the way for advancement of efficient quantum
algorithm design.Comment: 18 pages, 10 figure
General framework for quantum search algorithms
Grover's quantum search algorithm drives a quantum computer from a prepared
initial state to a desired final state by using selective transformations of
these states. Here, we analyze a framework when one of the selective
trasformations is replaced by a more general unitary transformation. Our
framework encapsulates several previous generalizations of the Grover's
algorithm. We show that the general quantum search algorithm can be improved by
controlling the transformations through an ancilla qubit. As a special case of
this improvement, we get a faster quantum algorithm for the two-dimensional
spatial search.Comment: revised versio
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