Quantum computing of molecular magnet Mn12_{12}

Quantum computation in molecular magnets is studied by solving the time-dependent Schr\"{o}dinger equation numerically. Following Leuenberger and Loss (Nature (London) 410, 789(2001)), an external oscillating magnetic field is applied to populate and manipulate the spin coherent states in molecular magnet Mn$_{12}$. The conditions to realize parallel recording and reading data bases of Grover algorithsm in molecular magnets are discussed in details. It is found that an accurate duration time of magnetic pulse as well as the amplitudes are required to design the device of quantum computing.Comment: 3 pages, 1 figur

Schemes of implementation in NMR of quantum processors and Deutsch-Jozsa algorithm by using virtual spin representation

Schemes of experimental realization of the main two qubit processors for quantum computers and Deutsch-Jozsa algorithm are derived in virtual spin representation. The results are applicable for every four quantum states allowing the required properties for quantum processor implementation if for qubit encoding virtual spin representation is used. Four dimensional Hilbert space of nuclear spin 3/2 is considered in details for this aimComment: 15 pages, 3 figure

Quantum Search with Two-atom Collisions in Cavity QED

We propose a scheme to implement two-qubit Grover's quantum search algorithm using Cavity Quantum Electrodynamics. Circular Rydberg atoms are used as quantum bits (qubits). They interact with the electromagnetic field of a non-resonant cavity . The quantum gate dynamics is provided by a cavity-assisted collision, robust against decoherence processes. We present the detailed procedure and analyze the experimental feasibility.Comment: 4 pages, 2 figure

Entangling capacity of global phases and implications for Deutsch-Jozsa algorithm

We investigate the creation of entanglement by the application of phases whose value depends on the state of a collection of qubits. First we give the necessary and sufficient conditions for a given set of phases to result in the creation of entanglement in a state comprising of an arbitrary number of qubits. Then we analyze the creation of entanglement between any two qubits in three qubit pure and mixed states. We use our result to prove that entanglement is necessary for Deutsch-Jozsa algorithm to have an exponential advantage over its classical counterpart.Comment: All 8 figures at the en

Quantum Probabilistic Subroutines and Problems in Number Theory

We present a quantum version of the classical probabilistic algorithms $\grave{a}$ la Rabin. The quantum algorithm is based on the essential use of Grover's operator for the quantum search of a database and of Shor's Fourier transform for extracting the periodicity of a function, and their combined use in the counting algorithm originally introduced by Brassard et al. One of the main features of our quantum probabilistic algorithm is its full unitarity and reversibility, which would make its use possible as part of larger and more complicated networks in quantum computers. As an example of this we describe polynomial time algorithms for studying some important problems in number theory, such as the test of the primality of an integer, the so called 'prime number theorem' and Hardy and Littlewood's conjecture about the asymptotic number of representations of an even integer as a sum of two primes.Comment: 9 pages, RevTex, revised version, accepted for publication on PRA: improvement in use of memory space for quantum primality test algorithm further clarified and typos in the notation correcte

Multi-valued Logic Gates for Quantum Computation

We develop a multi-valued logic for quantum computing for use in multi-level quantum systems, and discuss the practical advantages of this approach for scaling up a quantum computer. Generalizing the methods of binary quantum logic, we establish that arbitrary unitary operations on any number of d-level systems (d > 2) can be decomposed into logic gates that operate on only two systems at a time. We show that such multi-valued logic gates are experimentally feasible in the context of the linear ion trap scheme for quantum computing. By using d levels in each ion in this scheme, we reduce the number of ions needed for a computation by a factor of log d.Comment: Revised version; 8 pages, 3 figures; to appear in Physical Review

Use of Quadrupolar Nuclei for Quantum Information processing by Nuclear Magnetic Resonance: Implementation of a Quantum Algorithm

Physical implementation of Quantum Information Processing (QIP) by liquid-state Nuclear Magnetic Resonance (NMR), using weakly coupled spin-1/2 nuclei of a molecule, is well established. Nuclei with spin$>$1/2 oriented in liquid crystalline matrices is another possibility. Such systems have multiple qubits per nuclei and large quadrupolar couplings resulting in well separated lines in the spectrum. So far, creation of pseudopure states and logic gates have been demonstrated in such systems using transition selective radio-frequency pulses. In this paper we report two novel developments. First, we implement a quantum algorithm which needs coherent superposition of states. Second, we use evolution under quadrupolar coupling to implement multi qubit gates. We implement Deutsch-Jozsa algorithm on a spin-3/2 (2 qubit) system. The controlled-not operation needed to implement this algorithm has been implemented here by evolution under the quadrupolar Hamiltonian. This method has been implemented for the first time in quadrupolar systems. Since the quadrupolar coupling is several orders of magnitude greater than the coupling in weakly coupled spin-1/2 nuclei, the gate time decreases, increasing the clock speed of the quantum computer.Comment: 16 pages, 3 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

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

Interaction-free generation of entanglement

In this paper, we study how to generate entanglement by interaction-free measurement. Using Kwiat et al.'s interferometer, we construct a two-qubit quantum gate that changes a particle's trajectory according to the other particle's trajectory. We propose methods for generating the Bell state from an electron and a positron and from a pair of photons by this gate. We also show that using this gate, we can carry out the Bell measurement with the probability of 3/4 at the maximum and execute a controlled-NOT operation by the method proposed by Gottesman and Chuang with the probability of 9/16 at the maximum. We estimate the success probability for generating the Bell state by our procedure under imperfect interaction.Comment: 18 pages, Latex2e, 11 eps figures, v2: minor corrections and one reference added, v3: a minor correctio