Thermally assisted adiabatic quantum computation

We study the effect of a thermal environment on adiabatic quantum computation using the Bloch-Redfield formalism. We show that in certain cases the environment can enhance the performance in two different ways: (i) by introducing a time scale for thermal mixing near the anticrossing that is smaller than the adiabatic time scale, and (ii) by relaxation after the anticrossing. The former can enhance the scaling of computation when the environment is super-Ohmic, while the latter can only provide a prefactor enhancement. We apply our method to the case of adiabatic Grover search and show that performance better than classical is possible with a super-Ohmic environment, with no a priori knowledge of the energy spectrum.     

Transitionless driving on local adiabatic quantum search algorithm

We apply the transitionless driving on the local adiabatic quantum search algorithm to speed up the adiabatic process.By studying quantum dynamics of the adiabatic search algorithm with the equivalent two-level system, we derive the transitionless driving Hamiltonian for the local adiabatic quantum search algorithm. We found that when adding a transitionless quantum driving term H_D(t) on the local adiabatic quantum search algorithm, the success rate is 1 exactly with arbitrary evolution time by solving the time-dependent Schr odinger equation in eigen-picture. Moreover, we show the reason for the drastic decrease of the evolution time is that the driving Hamiltonian increases the lowest eigenvalues to a maximum of ON~(1/2).     

Partial evolution based local adiabatic quantum search

Recently, Zhang and Lu provided a quantum search algorithm based on partial adiabatic evolution, which beats the time bound of local adiabatic search when the number of marked items in the unsorted database is larger than one. Later, they found that the above two adiabatic search algorithms had the same time complexity when there is only one marked item in the database. In the present paper, following the idea of Roland and Cerf [Roland J and Cerf N J 2002 Phys. Rev. A 65 042308], if within the small symmetric evolution interval defined by Zhang et al., a local adiabatic evolution is performed instead of the original “global” one, this “new” algorithm exhibits slightly better performance, although they are progressively equivalent with M increasing. In addition, the proof of the optimality for this partial evolution based local adiabatic search when M=1 is also presented. Two other special cases of the adiabatic algorithm obtained by appropriately tuning the evolution interval of partial adiabatic evolution based quantum search, which are found to have the same phenomenon above, are also discussed.     

Nonlinear evolution of quantum states in the adiabatic regime

We investigate adiabatic evolution of quantum states as governed by the nonlinear Schr?dinger equation and provide examples of applications with a nonlinear tunneling model for Bose-Einstein condensates. Our analysis not only spells out conditions for adiabatic evolution of eigenstates but also characterizes the motion of noneigenstates which cannot be obtained from the former in the absence of the superposition principle. We find that Aharonov-Anandan phases play the role of classical canonical actions and are conserved in the adiabatic evolution of noneigenstates.     

A modified quantum adiabatic evolution for the Deutsch–Jozsa problem

Deutsch–Jozsa algorithm has been implemented via a quantum adiabatic evolution by S. Das et al. [S. Das, R. Kobes, G. Kunstatter, Phys. Rev. A 65 (2002) 062310]. This adiabatic algorithm gives rise to a quadratic speed up over classical algorithms. We show that a modified version of the adiabatic evolution in that paper can improve the performance to constant time.     

Experimental implementation of local adiabatic evolution algorithms by an NMR quantum information processor

Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, using the local adiabatic evolution, the algorithms given by J. Roland and N.J. Cerf for Grover's search [J. Roland, N.J. Cerf, Quantum search by local adiabatic evolution, Phys. Rev. A 65 (2002) 042308] and by Saurya Das, Randy Kobes, and Gabor Kunstatter for the Deutsch-Jozsa algorithm [S. Das, R. Kobes, G. Kunstatter, Adiabatic quantum computation and Deutsh's algorithm, Phys. Rev. A 65 (2002) 062301], yield a complexity of order N (where N=2(n) and n is the number of qubits). In this paper, we report the experimental implementation of these local adiabatic evolution algorithms on a 2-qubit quantum information processor, by Nuclear Magnetic Resonance.     

On an evolution operator connecting Lagrangian and Hamiltonian formalisms

The unambiguous evolution operator K was recently introduced in the theory of constrained systems. By viewing K as a vector field over the Legendre transformation, we give an intrinsic characterization of it through simple and intuitive properties. Some immediate consequences are explored.     

A quantum search algorithm based on partial adiabatic evolution

This paper presents and implements a specified partial adiabatic search algorithm on a quantum circuit. It studies the minimum energy gap between the first excited state and the ground state of the system Hamiltonian and it finds that,in the case of M = 1,the algorithm has the same performance as the local adiabatic algorithm. However,the algorithm evolves globally only within a small interval,which implies that it keeps the advantages of global adiabatic algorithms without losing the speedup of the local adiabatic search algorithm.     

An optimum Hamiltonian for non-Hermitian quantum evolution and the complex Bloch sphere

For a quantum system governed by a non-Hermitian Hamiltonian, we studied the problem of obtaining an optimum Hamiltonian that generates nonunitary transformations of a given initial state into a certain final state in the smallest time τ. The analysis is based on the relationship between the states of the two-dimensional subspace of the Hilbert space spanned by the initial and final states and the points of the two-dimensional complex Bloch sphere.     

On the construction of supersymmetric quantum mechanical Hamiltonian

Starting from an arbitrary supersymmetric quantum mechanical Hamilonian (SSQM) (with unbroken spersymmetry) we construct a sequence of SSQM Hamiltonians, each one having the same energy spectrum as the original one with different eigenvectors. The alogorithm developed here is verified for a transcendental type superpotential.     

On global solutions for non-linear Hamiltonian evolution equations

It is shown that partial differential equations of Hamiltonian type admit global solutions in time if (a) the initial data is near equilibrium (or the coupling constant is small) (b) the linear terms have positive energy and (c) the non-linear terms are smooth functions in the topology of the linearized energy norm. The non-linear terms need not have positive energy. The result is applied to non-linear wave equations in which the interaction energy is not necessarily positive.Partially supported by NSF Grant GP 15735.     

Experimental implementation of an adiabatic quantum optimization algorithm

We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well-suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.     

Collisions for the quantum Coulomb Hamiltonian

We study the propagation of phase space singularities for the time dependent Schrödinger equation with potential having Coulomb-type singularities in space dimension equal tothree. We prove that the singularities (frequency set) of the solution are reflected by a Coulomb center exactly as in the classical problem, i.e. the frequency set follows theregularized trajectories of Classical Mechanics after a collision.     

Genetic code evolution as an initial driving force for molecular evolution

There is an intrinsic relationship between the molecular evolution in primordial period and genomic or proteomic properties of contemporary species. The genomic data may help us understand the driving force of evolution of life at a molecular level. In the absence of evidence, numerous problems in molecular evolution had to fall into a twilight zone of speculation and controversy in the past. Here we show that delicate variation patterns of genomic base compositions and amino acid frequencies resulted from the genetic code evolution, which underlies the molecular evolution. The theoretical results agree with the experimental observations very well, not only in the evolutionary trends of amino acid frequencies and genomic base compositions but also in many detailed characters. Inversely, the genomic data of contemporary species can help us unravel the genetic code chronology and amino acid chronology. Our results may shed light on the intrinsic mechanism of molecular evolution and the genetic code evolution.     

A sufficient condition for the Hamiltonian evolution

We give a sufficient condition for the time evolution of a finite quantum system to be a Hamiltonian one. We use the semigroup formalism recently developed by Kossakowski, Gorini, Sudarshan et al.     

On the time evolution automorphisms of the CCR-algebra for quantum mechanics

In ordinary quantum mechanics for finite systems, the time evolution induced by Hamiltonians of the form is studied from the point of view of *-automorphisms of the CCRC*-algebra (see Ref. [1, 2]). It is proved that those Hamiltonians do not induce *-automorphisms of this algebra in the cases: a) and b)V L (,dx) L ¹ (,dx), except when the potential is trivial.     

On the discrete spectrum of the N-body quantum mechanical Hamiltonian: II

Using the Weinberg-van Winter equations we prove finiteness of the discrete spectrum of the N-body quantum mechanical Hamiltonian with pair potentials satisfying |V(x)| C(1 + |x|²)^– , > 1 in case the threshold of the continuous spectrum is negative and determined exclusively by eigenvalues of two-cluster Hamiltonians.     

On an adiabatic theory of multielectron systems

An adiabatic approximation is constructed for the outer atomic electron using its weak bond as compared to other electrons. The adiabatic approach is well-grounded for a helium atom and is extended to other atoms and crystals of rare gases, a functional dependence being established between electron properties of crystals and free atom constants.     

On an equivalent Hermitian Hamiltonian for the wrong-sign quartic potential

In this paper I present the results of a calculation with J. Mateo, in which, by a judicious choice of the contour on which the Schrödinger equation for the potential ?gz ⁴ is posed, we were able to give an explicit construction of an equivalent Hermitian Hamiltonian with the same spectrum. I also discuss the functional-integral approach to constructing equivalent Hamiltonians. In many cases this gives the simplest derivation. However, in this particular case it only gives the classical Hamiltonian, without a linear term, which is in fact an anomaly that can only be obtained by a careful discretization of the functional integral.