Analysis of Generalized Grover's Quantum Search Algorithms Using Recursion Equations

The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [PRA 60, 2742 (1999)] is generalized. It is applied to the large class of Grover's type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution is expressed using first order linear difference equations. These equations are solved exactly. The solution provides the number of iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, P_max. Both T and P_max are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments.Comment: 8 pages, no figures. To appear in Phys. Rev.

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 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

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

Potentially Diagnostic Electron Paramagnetic Resonance Spectra Elucidate the Underlying Mechanism of Mitochondrial Dysfunction in the Deoxyguanosine Kinase Deficient Rat Model of a Genetic Mitochondrial DNA Depletion Syndrome

A novel rat model for a well-characterized human mitochondrial disease, mitochondrial DNA depletion syndrome with associated deoxyguanosine kinase (DGUOK) deficiency, is described. The rat model recapitulates the pathologic and biochemical signatures of the human disease. The application of electron paramagnetic (spin) resonance (EPR) spectroscopy to the identification and characterization of respiratory chain abnormalities in the mitochondria from freshly frozen tissue of the mitochondrial disease model rat is introduced. EPR is shown to be a sensitive technique for detecting mitochondrial functional abnormalities in situ and, here, is particularly useful in characterizing the redox state changes and oxidative stress that can result from depressed expression and/or diminished specific activity of the distinct respiratory chain complexes. As EPR requires no sample preparation or non-physiological reagents, it provides information on the status of the mitochondrion as it was in the functioning state. On its own, this information is of use in identifying respiratory chain dysfunction; in conjunction with other techniques, the information from EPR shows how the respiratory chain is affected at the molecular level by the dysfunction. It is proposed that EPR has a role in mechanistic pathophysiological studies of mitochondrial disease and could be used to study the impact of new treatment modalities or as an additional diagnostic tool

Measuring Energy, Estimating Hamiltonians, and the Time-Energy Uncertainty Relation

Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine what is the Hamiltonian. We show that in general this requires a time $\Delta t$ which obeys the uncertainty relation $\Delta t \Delta H \gtrsim 1$ where $\Delta H$ is a measure of how accurately the unknown Hamiltonian must be estimated. We then apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can in principle be measured in an arbitrarily short time. On the other hand we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time $\Delta t$ which obeys the uncertainty relation $\Delta t \Delta E \gtrsim 1$ where $\Delta E$ is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail.Comment: 12pages, revised version with small correction

Quantum key distribution without alternative measurements

Entanglement swapping between Einstein-Podolsky-Rosen (EPR) pairs can be used to generate the same sequence of random bits in two remote places. A quantum key distribution protocol based on this idea is described. The scheme exhibits the following features. (a) It does not require that Alice and Bob choose between alternative measurements, therefore improving the rate of generated bits by transmitted qubit. (b) It allows Alice and Bob to generate a key of arbitrary length using a single quantum system (three EPR pairs), instead of a long sequence of them. (c) Detecting Eve requires the comparison of fewer bits. (d) Entanglement is an essential ingredient. The scheme assumes reliable measurements of the Bell operator.Comment: REVTeX, 5 pages, 2 figures. Published version with some comment

An entanglement monotone derived from Grover's algorithm

This paper demonstrates that how well a state performs as an input to Grover's search algorithm depends critically upon the entanglement present in that state; the more entanglement, the less well the algorithm performs. More precisely, suppose we take a pure state input, and prior to running the algorithm apply local unitary operations to each qubit in order to maximize the probability P_max that the search algorithm succeeds. We prove that, for pure states, P_max is an entanglement monotone, in the sense that P_max can never be decreased by local operations and classical communication.Comment: 7 page

Effects of Noisy Oracle on Search Algorithm Complexity

Grover's algorithm provides a quadratic speed-up over classical algorithms for unstructured database or library searches. This paper examines the robustness of Grover's search algorithm to a random phase error in the oracle and analyzes the complexity of the search process as a function of the scaling of the oracle error with database or library size. Both the discrete- and continuous-time implementations of the search algorithm are investigated. It is shown that unless the oracle phase error scales as O(N^(-1/4)), neither the discrete- nor the continuous-time implementation of Grover's algorithm is scalably robust to this error in the absence of error correction.Comment: 16 pages, 4 figures, submitted to Phys. Rev.

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