Derivation of the Quantum Probability Rule without the Frequency Operator

We present an alternative frequencists' proof of the quantum probability rule which does not make use of the frequency operator, with expectation that this can circumvent the recent criticism against the previous proofs which use it. We also argue that avoiding the frequency operator is not only for technical merits for doing so but is closely related to what quantum mechanics is all about from the viewpoint of many-world interpretation.Comment: 12 page

Sufficiency Criterion for the Validity of the Adiabatic Approximation

We examine the quantitative condition which has been widely used as a criterion for the adiabatic approximation but was recently found insufficient. Our results indicate that the usual quantitative condition is sufficient for a special class of quantum mechanical systems. For general systems, it may not be sufficient, but it along with additional conditions is sufficient. The usual quantitative condition and the additional conditions constitute a general criterion for the validity of the adiabatic approximation, which is applicable to all $N-$dimensional quantum systems. Moreover, we illustrate the use of the general quantitative criterion in some physical models.Comment: 9 pages, no figure,appearing in PRL98(2007)15040

Quantitative conditions do not guarantee the validity of the adiabatic approximation

In this letter, we point out that the widely used quantitative conditions in the adiabatic theorem are insufficient in that they do not guarantee the validity of the adiabatic approximation. We also reexamine the inconsistency issue raised by Marzlin and Sanders (Phys. Rev. Lett. 93, 160408, 2004) and elucidate the underlying cause.Comment: corrected typos. Eq. (32) is corrected. No other change

Effective Hamiltonian approach to adiabatic approximation in open systems

The adiabatic approximation in open systems is formulated through the effective Hamiltonian approach. By introducing an ancilla, we embed the open system dynamics into a non-Hermitian quantum dynamics of a composite system, the adiabatic evolution of the open system is then defined as the adiabatic dynamics of the composite system. Validity and invalidity conditions for this approximation are established and discussed. A High-order adiabatic approximation for open systems is introduced. As an example, the adiabatic condition for an open spin-$\frac 1 2$ particle in time-dependent magnetic fields is analyzed.Comment: 6 pages, 2 figure

Energy and Efficiency of Adiabatic Quantum Search Algorithms

We present the results of a detailed analysis of a general, unstructured adiabatic quantum search of a data base of $N$ items. In particular we examine the effects on the computation time of adding energy to the system. We find that by increasing the lowest eigenvalue of the time dependent Hamiltonian {\it temporarily} to a maximum of $\propto \sqrt{N}$, it is possible to do the calculation in constant time. This leads us to derive the general theorem which provides the adiabatic analogue of the $\sqrt{N}$ bound of conventional quantum searches. The result suggests that the action associated with the oracle term in the time dependent Hamiltonian is a direct measure of the resources required by the adiabatic quantum search.Comment: 6 pages, Revtex, 1 figure. Theorem modified, references and comments added, sections introduced, typos corrected. Version to appear in J. Phys.

Improved Bounds on Quantum Learning Algorithms

In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is $O(\frac{\log |C|}{\sqrt{{\hat{\gamma}}^{C}}})$, where $\hat{\gamma}^{C}$ is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using $O(\frac{\log |C| \log \log |C|}{\sqrt{{\hat{\gamma}}^{C}}})$ quantum black-box queries. We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other. Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for $(\epsilon,\delta)$-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain $\{0,1\}^n$ from $\Omega(\frac{d}{n})$ to $\Omega(\frac{1}{\epsilon}\log \frac{1}{\delta}+d+\frac{\sqrt{d}}{\epsilon})$. This new lower bound comes closer to matching known upper bounds for classical PAC learning.Comment: Minor corrections. 18 pages. To appear in Quantum Information Processing. Requires: algorithm.sty, algorithmic.sty to buil

Time-Dependent Open String Solutions in 2+1 Dimensional Gravity

We find general, time-dependent solutions produced by open string sources carrying no momentum flow in 2+1 dimensional gravity. The local Poincar\'e group elements associated with these solutions and the coordinate transformations that transform these solutions into Minkowski metric are obtained. We also find the relation between these solutions and the planar wall solutions in 3+1 dimensions.Comment: CU-TP-619, 18 pages. (minor changes

Effect of magnetic field on the strange star

We study the effect of a magnetic field on the strage quark matter and apply to strange star. We found that the strange star becomes more compact in presence of strong magnetic field.Comment: 10 pages (LaTex) and 3 postscript figures available on reques

A Quantum Approach to Classical Statistical Mechanics

We present a new approach to study the thermodynamic properties of $d$-dimensional classical systems by reducing the problem to the computation of ground state properties of a $d$-dimensional quantum model. This classical-to-quantum mapping allows us to deal with standard optimization methods, such as simulated and quantum annealing, on an equal basis. Consequently, we extend the quantum annealing method to simulate classical systems at finite temperatures. Using the adiabatic theorem of quantum mechanics, we derive the rates to assure convergence to the optimal thermodynamic state. For simulated and quantum annealing, we obtain the asymptotic rates of $T(t) \approx (p N) /(k_B \log t)$ and $\gamma(t) \approx (Nt)^{-\bar{c}/N}$, for the temperature and magnetic field, respectively. Other annealing strategies, as well as their potential speed-up, are also discussed.Comment: 4 pages, no figure

Criticality, the area law, and the computational power of PEPS

The projected entangled pair state (PEPS) representation of quantum states on two-dimensional lattices induces an entanglement based hierarchy in state space. We show that the lowest levels of this hierarchy exhibit an enormously rich structure including states with critical and topological properties as well as resonating valence bond states. We prove, in particular, that coherent versions of thermal states of any local 2D classical spin model correspond to such PEPS, which are in turn ground states of local 2D quantum Hamiltonians. This correspondence maps thermal onto quantum fluctuations, and it allows us to analytically construct critical quantum models exhibiting a strict area law scaling of the entanglement entropy in the face of power law decaying correlations. Moreover, it enables us to show that there exist PEPS within the same class as the cluster state, which can serve as computational resources for the solution of NP-hard problems