ON THE ADIABATIC EVOLUTION OF ONE-DIMENSIONAL PROJECTOR HAMILTONIANS

2012 ◽  
Vol 10 (04) ◽  
pp. 1250046 ◽  
Author(s):  
JIE SUN ◽  
SONG-FENG LU
Keyword(s):  
Ground State ◽  
Time Complexity ◽  
The Other ◽  
Constant Time ◽  
Simple Analysis ◽  
Infinite Time ◽  
Adiabatic Evolution ◽  
One Dimensional ◽  
Original Time

In this paper, we discuss the adiabatic evolution of one-dimensional projector Hamiltonians. Three kinds of adiabatic algorithms for this problem are shown, in which two of them provide a quadratic speedup over the other one. But when the ground state of the initial Hamiltonian and that of the final Hamiltonian have a zero overlap, the algorithms above all fail, in the sense of infinite time complexity. A corresponding revised method for this phenomenon through adding a driving Hamiltonian is also shown, from which a constant time complexity can be gained. But by a simple analysis, we find that the original time complexity for the adiabatic evolution is shifted to implement the driving Hamiltonian, which is supported by several early works in the literature.

On the General Class of Models of Adiabatic Evolution

2016 ◽  
Vol 23 (03) ◽  
pp. 1650016 ◽  
Author(s):  
Jie Sun ◽  
Songfeng Lu ◽  
Fang Liu
Keyword(s):  
Ground State ◽  
General Class ◽  
Time Complexity ◽  
State Energy ◽  
Search Problem ◽  
Quantum Search ◽  
Final States ◽  
Adiabatic Evolution ◽  
Speed Up ◽  
Effective Use

The general class of models of adiabatic evolution was proposed to speed up the usual adiabatic computation in the case of quantum search problem. It was shown [8] that, by temporarily increasing the ground state energy of a time-dependent Hamiltonian to a suitable quantity, the quantum computation can perform the calculation in time complexity O(1). But it is also known that if the overlap between the initial and final states of the system is zero, then the computation based on the generalized models of adiabatic evolution can break down completely. In this paper, we find another severe limitation for this class of adiabatic evolution-based algorithms, which should be taken into account in applications. That is, it is still possible that this kind of evolution designed to deal with the quantum search problem fails completely if the interpolating paths in the system Hamiltonian are chosen inappropriately, while the usual adiabatic evolutions can do the same job relatively effectively. This implies that it is not always recommendable to use nonlinear paths in adiabatic computation. On the contrary, the usual simple adiabatic evolution may be sufficient for effective use.

ENTANGLEMENT PROPERTIES OF ADIABATIC QUANTUM ALGORITHMS

2009 ◽  
Vol 07 (08) ◽  
pp. 1531-1539 ◽  
Author(s):  
JIAYAN WEN ◽  
YI HUANG ◽  
DAOWEN QIU
Keyword(s):  
Quantum System ◽  
Search Algorithm ◽  
Quantum Algorithms ◽  
The Other ◽  
Constant Time ◽  
Quantum Search ◽  
Adiabatic Evolution ◽  
Algorithm Performance ◽  
Quantum Search Algorithm ◽  
Speed Up

In this paper, by constructing a more entangled quantum system, we shorten the adiabatic quantum search algorithm to constant time. On the other hand, we show that the speed-up of adiabatic quantum algorithms by selecting particular adiabatic evolution paths or injecting energy into the quantum system can be explained as a form of entanglement enlargement. These findings suggest that entanglement plays a fundamental role for the efficiency of algorithm performance.

scholarly journals Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

2021 ◽  
pp. 1-19
Author(s):  
Augusto Modanese
Keyword(s):  
Cellular Automata ◽  
Parallel Computation ◽  
Time Complexity ◽  
Regular Languages ◽  
Constant Time ◽  
Language Recognition ◽  
One Dimensional ◽  
Acceptance Condition

After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (one-dimensional) cellular automata: sublinear-time computation. The model considered is that of ACAs, which are language acceptors whose acceptance condition depends on the states of all cells in the automaton. We prove a time hierarchy theorem for sublinear-time ACA classes, analyze their intersection with the regular languages, and, finally, establish strict inclusions in the parallel computation classes [Formula: see text] and (uniform) [Formula: see text]. As an addendum, we introduce and investigate the concept of a decider ACA (DACA) as a candidate for a decider counterpart to (acceptor) ACAs. We show the class of languages decidable in constant time by DACAs equals the locally testable languages, and we also determine [Formula: see text] as the (tight) time complexity threshold for DACAs up to which no advantage compared to constant time is possible.

Entanglement for excited states of ultracold bosonic atoms in one-dimensional harmonic traps with contact interaction

2015 ◽  
Vol 29 (30) ◽  
pp. 1550189 ◽  
Author(s):  
Hsuan Tung Peng ◽  
Yew Kam Ho
Keyword(s):  
Quantum Entanglement ◽  
Excited States ◽  
Ground State ◽  
The Other ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
One Dimensional ◽  
Ultracold Bosonic Atoms ◽  
Bosonic Atoms ◽  
Atom System

We have investigated quantum entanglement for two interacting ultracold bosonic atoms in one-dimensional harmonic traps. The effective potential is modeled by delta interaction. For this two-atom system, we have investigated quantum entanglement properties, such as von Neumann entropy and linear entropy for its ground state and excited states. Using a computational scheme that is different from previously employed, a total of the lowest 16 states are studied. Here we show the dependencies of entanglement properties under various interacting strengths. Comparisons for the ground state entanglement are made with earlier results in the literature. New results for the other 15 excited states are reported here.

scholarly journals Ground state and stability of the fractional plateau phase in metallic Shastry–Sutherland system TmB4

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Matúš Orendáč ◽  
Slavomír Gabáni ◽  
Pavol Farkašovský ◽  
Emil Gažo ◽  
Jozef Kačmarčík ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Phase Diagram ◽  
Ground State ◽  
Constant Magnetic Field ◽  
Paramagnetic Phase ◽  
The Other ◽  
Zero Field ◽  
Plateau Phase ◽  
Low Field ◽  
Zero Field Cooling

AbstractWe present a study of the ground state and stability of the fractional plateau phase (FPP) with M/Msat = 1/8 in the metallic Shastry–Sutherland system TmB4. Magnetization (M) measurements show that the FPP states are thermodynamically stable when the sample is cooled in constant magnetic field from the paramagnetic phase to the ordered one at 2 K. On the other hand, after zero-field cooling and subsequent magnetization these states appear to be of dynamic origin. In this case the FPP states are closely associated with the half plateau phase (HPP, M/Msat = ½), mediate the HPP to the low-field antiferromagnetic (AF) phase and depend on the thermodynamic history. Thus, in the same place of the phase diagram both, the stable and the metastable (dynamic) fractional plateau (FP) states, can be observed, depending on the way they are reached. In case of metastable FP states thermodynamic paths are identified that lead to very flat fractional plateaus in the FPP. Moreover, with a further decrease of magnetic field also the low-field AF phase becomes influenced and exhibits a plateau of the order of 1/1000 Msat.

scholarly journals Pair-correlation ansatz for the ground state of interacting bosons in an arbitrary one-dimensional potential

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Przemysław Kościk ◽  
Arkadiusz Kuroś ◽  
Adam Pieprzycki ◽  
Tomasz Sowiński
Keyword(s):  
Boundary Conditions ◽  
Ground State ◽  
Pair Correlation ◽  
Contact Interactions ◽  
One Dimensional ◽  
Particle Correlations ◽  
Variational Scheme ◽  
Full Control ◽  
Arbitrary Shapes

AbstractWe derive and describe a very accurate variational scheme for the ground state of the system of a few ultra-cold bosons confined in one-dimensional traps of arbitrary shapes. It is based on assumption that all inter-particle correlations have two-body nature. By construction, the proposed ansatz is exact in the noninteracting limit, exactly encodes boundary conditions forced by contact interactions, and gives full control on accuracy in the limit of infinite repulsions. We show its efficiency in a whole range of intermediate interactions for different external potentials. Our results manifest that for generic non-parabolic potentials mutual correlations forced by interactions cannot be captured by distance-dependent functions.

scholarly journals Assessing a Linear Nanosystem's Limiting Reliability from its Components

2008 ◽  
Vol 45 (03) ◽  
pp. 879-887 ◽  
Author(s):  
Nader Ebrahimi
Keyword(s):  
Present State ◽  
Size Range ◽  
Two Dimensions ◽  
The Other ◽  
One Dimension ◽  
Present Moment ◽  
One Dimensional ◽  
The One ◽  
Fixed Moment ◽  
Nanosized Systems

Nanosystems are devices that are in the size range of a billionth of a meter (1 x 10-9) and therefore are built necessarily from individual atoms. The one-dimensional nanosystems or linear nanosystems cover all the nanosized systems which possess one dimension that exceeds the other two dimensions, i.e. extension over one dimension is predominant over the other two dimensions. Here only two of the dimensions have to be on the nanoscale (less than 100 nanometers). In this paper we consider the structural relationship between a linear nanosystem and its atoms acting as components of the nanosystem. Using such information, we then assess the nanosystem's limiting reliability which is, of course, probabilistic in nature. We consider the linear nanosystem at a fixed moment of time, say the present moment, and we assume that the present state of the linear nanosystem depends only on the present states of its atoms.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

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