Class Problem!: Why the Inconsistent Application of Rule 23\u27s Class Certification Requirements During Overbreadth Analysis is a Threat to Litigant Certainty

Rule 23 of the Federal Rules of Civil Procedure is home to the class action device. It is well-documented that this rule significantly impacts our legal system. As a result, the need for its effective utilization has been apparent since its introduction. Despite this, federal courts have inconsistently applied the rule during their analyses of overbroad class definitions at the class certification stage. Consequently, parties involved in such litigation have been exposed to unnecessary costs and the potential for forum shopping. Nonetheless, this judicial inconsistency has gone largely unrecognized because it does not implicate the results of class certification. Hence, courts here must first recognize the general need for uniformity before a precise standard for overbreadth analysis may be chosen. Only then, this Note argues, may the aforementioned detrimental consequences be avoided

Influence of qubit displacements on quantum logic operations in a silicon-based quantum computer with constant interaction

The errors caused by qubit displacements from their prescribed locations in an ensemble of spin chains are estimated analytically and calculated numerically for a quantum computer based on phosphorus donors in silicon. We show that it is possible to polarize (initialize) the nuclear spins even with displaced qubits by using Controlled NOT gates between the electron and nuclear spins of the same phosphorus atom. However, a Controlled NOT gate between the displaced electron spins is implemented with large error because of the exponential dependence of exchange interaction constant on the distance between the qubits. If quantum computation is implemented on an ensemble of many spin chains, the errors can be small if the number of chains with displaced qubits is small

Decoherence and Spin Echo in Biological Systems

The spin echo approach is extended to include bio-complexes for which the interaction with dynamical noise is strong. Significant restoration of the free induction decay signal due to homogeneous (decoherence) and inhomogeneous (dephasing) broadening is demonstrated analytically and numerically, for both an individual dimer of interacting chlorophylls and for an ensemble of dimers. This approach is based on an exact and closed system of ordinary differential equations that can be easily solved for a wide range of parameters that are relevant for bio-applications.Comment: 5 pages, 5 figure

An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-K\"uhn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms. As an application, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions

Creation of entanglement in a scalable spin quantum computer with long-range dipole-dipole interaction between qubits

Creation of entanglement is considered theoretically and numerically in an ensemble of spin chains with dipole-dipole interaction between the spins. The unwanted effect of the long-range dipole interaction is compensated by the optimal choice of the parameters of radio-frequency pulses implementing the protocol. The errors caused by (i) the influence of the environment,(ii) non-selective excitations, (iii) influence of different spin chains on each other, (iv) displacements of qubits from their perfect locations, and (v) fluctuations of the external magnetic field are estimated analytically and calculated numerically. For the perfectly entangled state the z component, M, of the magnetization of the whole system is equal to zero. The errors lead to a finite value of M. If the number of qubits in the system is large, M can be detected experimentally. Using the fact that M depends differently on the parameters of the system for each kind of error, varying these parameters would allow one to experimentally determine the most significant source of errors and to optimize correspondingly the quantum computer design in order to decrease the errors and M. Using our approach one can benchmark the quantum computer, decrease the errors, and prepare the quantum computer for implementation of more complex quantum algorithms.Comment: 31 page

Stability of the Ground State of a Harmonic Oscillator in a Monochromatic Wave

Classical and quantum dynamics of a harmonic oscillator in a monochromatic wave is studied in the exact resonance and near resonance cases. This model describes, in particular, a dynamics of a cold ion trapped in a linear ion trap and interacting with two lasers fields with close frequencies. Analytically and numerically a stability of the ``classical ground state'' (CGS) -- the vicinity of the point ($x=0, p=0$) -- is analyzed. In the quantum case, the method for studying a stability of the quantum ground state (QGS) is suggested, based on the quasienergy representation. The dynamics depends on four parameters: the detuning from the resonance, $\delta=\ell-\Omega/\omega$, where $\Omega$ and $\omega$ are, respectively, the wave and the oscillator's frequencies; the positive integer (resonance) number, $\ell$; the dimensionless Planck constant, $h$, and the dimensionless wave amplitude, $\epsilon$. For $\delta=0$, the CGS and the QGS are unstable for resonance numbers $\ell=1, 2$. For small $\epsilon$, the QGS becomes more stable with increasing $\delta$ and decreasing $h$. When $\epsilon$ increases, the influence of chaos on the stability of the QGS is analyzed for different parameters of the model, $\ell$, $\delta$ and $h$.Comment: RevTeX, 38 pages, 24 figure

Perturbation Theory for Quantum Computation with Large Number of Qubits

We describe a new and consistent perturbation theory for solid-state quantum computation with many qubits. The errors in the implementation of simple quantum logic operations caused by non-resonant transitions are estimated. We verify our perturbation approach using exact numerical solution for relatively small (L=10) number of qubits. A preferred range of parameters is found in which the errors in processing quantum information are small. Our results are needed for experimental testing of scalable solid-state quantum computers.Comment: 8 pages RevTex including 2 figure