NONLINEAR QUANTUM MECHANICS IMPLIES POLYNOMIAL-TIME SOLUTION FOR NP-COMPLETE AND P PROBLEMS
If quantum states exhibit small nonlinearities during time evolution, then quantum computers can be used to solve NP-complete problems in polynomial time. We provide algorithms that solve NP-complete and #P oracle problems by exploiting nonlinear quantum logic gates. It is argued that virtually any deterministic nonlinear quantum theory will include such gates, and the method is explicitly demonstrated using the Weinberg model of nonlinear quantum mechanics.

title: Computational capacity of the universe.
abstract: All physical systems register and process information. The laws of physics determine the amount of information that a physical system can register (number of bits) and the number of elementary logic operations that a system can perform (number of ops). The Universe is a physical system. The amount of information that the Universe can register and the number of elementary operations that it can have performed over its history are calculated. The Universe can have performed 10(120) ops on 10(90) bits ( 10(120) bits including gravitational degrees of freedom).

title: ANALOG QUANTUM ERROR CORRECTION
abstract: Quantum error-correction routines are developed for continuous quantum variables such as position and momentum. The result of such analog quantum error correction is the construction of composite continuous quantum variables that are largely immune to the effects of noise and decoherence.

title: Quantum Computation over Continuous Variables
abstract: This paper provides necessary and sufficient conditions for constructing a universal quantum computer over continuous variables. As an example, it is shown how a universal quantum computer for the amplitudes of the electromagnetic field might be constructed using simple linear devices such as beam splitters and phase shifters, together with squeezers and nonlinear devices such as Kerr-effect fibers and atoms in optical cavities. Such a device could in principle perform “floating point” computations. Problems of noise, finite precision, and error correction are discussed.

title: Estimation of the local density of states on a quantum computer
abstract: We report an efficient quantum algorithm for estimating the local density of states (LDOS) on a quantum computer. The LDOS describes the redistribution of energy levels of a quantum system under the influence of a perturbation. Sometimes known as the 'strength function' from nuclear spectroscopy experiments, the shape of the LDOS is directly related to the survivial probability of unperturbed eigenstates, and has recently been related to the fidelity decay (or 'Loschmidt echo') under imperfect motion reversal. For quantum systems that can be simulated efficiently on a quantum computer, the LDOS estimation algorithm enables an exponential speedup over direct classical computation.

title: Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors
abstract: We describe a new polynomial time quantum algorithm that uses the quantum fast Fourier transform to find eigenvalues and eigenvectors of a local Hamiltonian, and that can be applied in cases (commonly found in ab initio physics and chemistry problems) for which all known classical algorithms require exponential time. Applications of the algorithm to specific problems are considered, and we find that classically intractable and interesting problems from atomic physics may be solved with between 50 and 100 quantum bits.

title: Comparisons of fully differential exact results for O(α) virtual corrections to single hard bremsstrahlung in e+e− annihilation at high energies
abstract: We present comparisons of the fully differential exact virtual correction to the important single hard bremsstrahlung process in e + e − annihilation at high energies, which is essential for precision studies of the Standard Model from 1 GeV to 1 TeV, as calculated by two completely independent methods and groups. We show that the two sets of results are in excellent agreement. Phenomenological implications are discussed.