587 research outputs found
A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II : Supmech and Quantum Systems
Supmech, which is noncommutative Hamiltonian mechanics \linebreak (NHM)
(developed in paper I) with two extra ingredients : positive observable valued
measures (PObVMs) [which serve to connect state-induced expectation values and
classical probabilities] and the `CC condition' [which stipulates that the sets
of observables and pure states be mutually separating] is proposed as a
universal mechanics potentially covering all physical phenomena. It facilitates
development of an autonomous formalism for quantum mechanics. Quantum systems,
defined algebraically as supmech Hamiltonian systems with non-supercommutative
system algebras, are shown to inevitably have Hilbert space based realizations
(so as to accommodate rigged Hilbert space based Dirac bra-ket formalism),
generally admitting commutative superselection rules. Traditional features of
quantum mechanics of finite particle systems appear naturally. A treatment of
localizability much simpler and more general than the traditional one is given.
Treating massive particles as localizable elementary quantum systems, the
Schrdinger wave functions with traditional Born interpretation appear
as natural objects for the description of their pure states and the
Schrdinger equation for them is obtained without ever using a
classical Hamiltonian or Lagrangian. A provisional set of axioms for the
supmech program is given.Comment: 55 pages; some modifications in text; improved treatment of
topological aspects and of Noether invariants; results unchange
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
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
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
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