Fast Quantum Search Algorithms in Protein Sequence Comparison - Quantum Biocomputing

Quantum search algorithms are considered in the context of protein sequence comparison in biocomputing. Given a sample protein sequence of length m (i.e m residues), the problem considered is to find an optimal match in a large database containing N residues. Initially, Grover's quantum search algorithm is applied to a simple illustrative case - namely where the database forms a complete set of states over the 2^m basis states of a m qubit register, and thus is known to contain the exact sequence of interest. This example demonstrates explicitly the typical O(sqrt{N}) speedup on the classical O(N) requirements. An algorithm is then presented for the (more realistic) case where the database may contain repeat sequences, and may not necessarily contain an exact match to the sample sequence. In terms of minimizing the Hamming distance between the sample sequence and the database subsequences the algorithm finds an optimal alignment, in O(sqrt{N}) steps, by employing an extension of Grover's algorithm, due to Boyer, Brassard, Hoyer and Tapp for the case when the number of matches is not a priori known.Comment: LaTeX, 5 page

Quantum computing with four-particle decoherence-free states in ion trap

Quantum computing gates are proposed to apply on trapped ions in decoherence-free states. As phase changes due to time evolution of components with different eigenenergies of quantum superposition are completely frozen, quantum computing based on this model would be perfect. Possible application of our scheme in future ion-trap quantum computer is discussed.Comment: 10 pages, no figures. Comments are welcom

Quantum search by measurement

We propose a quantum algorithm for solving combinatorial search problems that uses only a sequence of measurements. The algorithm is similar in spirit to quantum computation by adiabatic evolution, in that the goal is to remain in the ground state of a time-varying Hamiltonian. Indeed, we show that the running times of the two algorithms are closely related. We also show how to achieve the quadratic speedup for Grover's unstructured search problem with only two measurements. Finally, we discuss some similarities and differences between the adiabatic and measurement algorithms.Comment: 8 pages, 2 figure

Grover search with pairs of trapped ions

The desired interference required for quantum computing may be modified by the wave function oscillations for the implementation of quantum algorithms[Phys.Rev.Lett.84(2000)1615]. To diminish such detrimental effect, we propose a scheme with trapped ion-pairs being qubits and apply the scheme to the Grover search. It can be found that our scheme can not only carry out a full Grover search, but also meet the requirement for the scalable hot-ion quantum computing. Moreover, the ion-pair qubits in our scheme are more robust against the decoherence and the dissipation caused by the environment than single-particle qubits proposed before.Comment: RevTe

Geometric Strategy for the Optimal Quantum Search

We explore quantum search from the geometric viewpoint of a complex projective space $CP$, a space of rays. First, we show that the optimal quantum search can be geometrically identified with the shortest path along the geodesic joining a target state, an element of the computational basis, and such an initial state as overlaps equally, up to phases, with all the elements of the computational basis. Second, we calculate the entanglement through the algorithm for any number of qubits $n$ as the minimum Fubini-Study distance to the submanifold formed by separable states in Segre embedding, and find that entanglement is used almost maximally for large $n$. The computational time seems to be optimized by the dynamics as the geodesic, running across entangled states away from the submanifold of separable states, rather than the amount of entanglement itself.Comment: revtex, 10 pages, 7 eps figures, uses psfrag packag

Implementation of a Deutsch-like quantum algorithm utilizing entanglement at the two-qubit level, on an NMR quantum information processor

We describe the experimental implementation of a recently proposed quantum algorithm involving quantum entanglement at the level of two qubits using NMR. The algorithm solves a generalisation of the Deutsch problem and distinguishes between even and odd functions using fewer function calls than is possible classically. The manipulation of entangled states of the two qubits is essential here, unlike the Deutsch-Jozsa algorithm and the Grover's search algorithm for two bits.Comment: 4 pages, two eps figure

Quantum Computing of Quantum Chaos in the Kicked Rotator Model

We investigate a quantum algorithm which simulates efficiently the quantum kicked rotator model, a system which displays rich physical properties, and enables to study problems of quantum chaos, atomic physics and localization of electrons in solids. The effects of errors in gate operations are tested on this algorithm in numerical simulations with up to 20 qubits. In this way various physical quantities are investigated. Some of them, such as second moment of probability distribution and tunneling transitions through invariant curves are shown to be particularly sensitive to errors. However, investigations of the fidelity and Wigner and Husimi distributions show that these physical quantities are robust in presence of imperfections. This implies that the algorithm can simulate the dynamics of quantum chaos in presence of a moderate amount of noise.Comment: research at Quantware MIPS Center http://www.quantware.ups-tlse.fr, revtex 11 pages, 13 figs, 2 figs and discussion adde

A Self Assembled Nanoelectronic Quantum Computer Based on the Rashba Effect in Quantum Dots

Quantum computers promise vastly enhanced computational power and an uncanny ability to solve classically intractable problems. However, few proposals exist for robust, solid state implementation of such computers where the quantum gates are sufficiently miniaturized to have nanometer-scale dimensions. Here I present a new approach whereby a complete computer with nanoscale gates might be self-assembled using chemical synthesis. Specifically, I demonstrate how to self-assemble the fundamental unit of this quantum computer - a 2-qubit universal quantum controlled-NOT gate - based on two exchange coupled multilayered quantum dots. Then I show how these gates can be wired using thiolated conjugated molecules as electrical connectors. A qubit is encoded in the ground state of a quantum dot spin-split by the Rashba interaction. Arbitrary qubit rotations are effected by bringing the spin splitting energy in a target quantum dot in resonance with a global ac magnetic field by applying a potential pulse of appropriate amplitude and duration to the dot. The controlled dynamics of the 2-qubit controlled-NOT operation (XOR) can be realized by exploiting the exchange coupling with the nearest neighboring dot. A complete prescription for initialization of the computer and data input/output operations is presented.Comment: 22 pages, 4 figure

Basic concepts in quantum computation

Section headings: 1 Qubits, gates and networks 2 Quantum arithmetic and function evaluations 3 Algorithms and their complexity 4 From interferometers to computers 5 The first quantum algorithms 6 Quantum search 7 Optimal phase estimation 8 Periodicity and quantum factoring 9 Cryptography 10 Conditional quantum dynamics 11 Decoherence and recoherence 12 Concluding remarksComment: 37 pages, lectures given at les Houches Summer School on "Coherent Matter Waves", July-August 199