Searcharxiv⌕ Search

arXiv subjects

Tien D. Kieu

Publications and source records attributed to Tien D. Kieu.

At least 19 recordsLinked to original sources
arXiv2022

Quantum Central Limit Theorems, Emergence of Classicality and Time-dependent Differential Entropy

We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained hermitean operators. Thanks to the hermicity constraints, we obtain positive-definite distribution for the expectation values of observables. These probability distributions open some pathway for an emergence of classical behaviours in the limit of infinitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probability distributions so derived also enable us to evaluate the nontrivial time-dependence of certain differential entropies.

quant-ph↗
arXiv2019

Principle of Unattainability of Absolute Zero Temperature, the Third Law of Thermodynamics, and Projective Quantum Measurements

The Principle of Unattainability rules out the attainment of absolute zero temperature by any finite physical means, no matter how idealised they could be. Nevertheless, we clarify that the Third Law of Thermodynamics, as defined by Nernst's heat theorem statement, is distinct from the Principle of Unattainability in the sense that the Third Law is mathematically equivalent only to the unattainability of absolute zero temperature by {\em quasi-static adiabatic} processes. This, on the one hand, leaves open the possibility of attainability of absolute zero by non-adiabatic means, without violating the Third Law. On the other hand, we point out some apparent incompatibility between the Postulate of Projective Measurement in quantum mechanics and the Principle of Unattainability in that projective measurements of energy could result in zero temperature.

quant-ph↗
arXiv2019

A Class of Time-Energy Uncertainty Relations for Time-dependent Hamiltonians

A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave functions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for time-independent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasise the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation in general.

quant-ph↗
arXiv2019

The Travelling Salesman Problem and Adiabatic Quantum Computation: An Algorithm

An explicit algorithm for the travelling salesman problem is constructed in the framework of adiabatic quantum computation, AQC. The initial Hamiltonian for the AQC process admits canonical coherent states as the ground state, and the target Hamiltonian has the shortest tour as the desirable ground state. Some estimates/bounds are also given for the computational complexity of the algorithm with particular emphasis on the required energy resources, besides the space and time complexity, for the physical process of (quantum) computation in general.

quant-ph↗
arXiv2019

A Factorisation Algorithm in Adiabatic Quantum Computation

The problem of factorising positive integer $N$ into two integer factors $x$ and $y$ is first reformulated as an optimisation problem over the positive integer domain of either of the Diophantine polynomials $Q_N(x,y)=N^2(N-xy)^2 + x(x-y)^2$ or $R_N(x,y) = N^2(N-xy)^2 + (x-y)^2 + x$, of each of which the optimal solution is unique with $x\le \sqrt{N} \le y$, and $x=1$ if and only if $N$ is prime. An algorithm in the context of Adiabatic Quantum Computation is then proposed for the general factorisation problem.

quant-ph↗
arXiv2010

Finite temperature correlations in the Bose-Hubbard model: application of the Gauge $P$ representation

We study ultracold Bose gases in periodic potentials as described by the Bose-Hubbard model. In 1D and at finite temperature, we simulate ultracold Bose gases in imaginary time with the gauge $P$ representation. We study various quantities including the Luttinger parameter $K$, which is important for locating the boundaries of the Mott insulator lobes, and find a simple relation for the kinetic energy part of the Bose-Hubbard Hamiltonian. We show that for J=0, the stepwise pattern of the average number of particles per lattice site versus the chemical potential vanishes at temperatures above $T \approx 0.1 U$. Also, at chemical potential $μ=0.5 U$ and temperature $T=0.5 U$ by increasing $J$, the relative value of the number fluctuation decreases and approaches that of a coherent state.

cond-mat.quant-gas↗
arXiv2006

Quantum Adiabatic Computation and the Travelling Salesman Problem

The NP-complete problem of the travelling salesman (TSP) is considered in the framework of quantum adiabatic computation (QAC). We first derive a remarkable lower bound for the computation time for adiabatic algorithms in general as a function of the energy involved in the computation. Energy, and not just time and space, must thus be considered in the evaluation of algorithm complexity, in perfect accordance with the understanding that all computation is physical. We then propose, with oracular Hamiltonians, new quantum adiabatic algorithms of which not only the lower bound in time but also the energy requirement do not increase exponentially in the size of the input. Such an improvement in both time and energy complexity, as compared to all other existing algorithms for TSP, is apparently due to quantum entanglement. We also appeal to the general theory of Diophantine equations in a speculation on physical implementation of those oracular Hamiltonians.

quant-ph↗
arXiv2006

Reply to Andrew Hodges

We separate the criticisms of Hodges \cite{Hodges2005} and others into those against the algorithm itself and those against its physical implementation. We then point out that {\em all} those against the algorithm are either misleading or misunderstanding, and that the algorithm is self consistent. The only central argument against physical implementations of the algorithm, on the other hand, is based on an assumption that its Hamiltonians cannot be effectively constructed due to a lack of infinite precision. However, so far there is no known physical principle dictating why that cannot be done. To show that the criticism may not be a forgone conclusion, we point out the virtually unknown fact that, on the contrary, simple instances of Diophantine equations with apparently {\em infinitely precisely} integer coefficients have {\em already} been realised in experiments for certain quantum phase transitions. We also speculate on how central limit theorem of statistics might be of some help in the effective implementation of the required Hamiltonians.

quant-ph↗
arXiv2006

On the identification of the ground state based on occupation probabilities: An investigation of Smith's apparent counterexamples

We study a set of truncated matrices, given by Smith~\cite{Smith2005}, in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert's tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also argue that it is only an artefact of the truncation of the underlying Hilbert spaces, through showing its sensitivity to different boundary conditions available for such a truncation. It is maintained that the criterion, in general, should be applicable provided certain conditions are satisfied. We also point out that, apart from this one, other criteria serving the same identification purpose may also be available.

quant-ph↗
arXiv2006

A mathematical proof for a ground-state identification criterion

We give a mathematical proof for an identification criterion by a probability measure for the ground state among an infinite number of available states, or a finitely truncated number with appropriate boundary conditions, in a quantum adiabatic algorithm for Hilbert's tenth problem.

quant-ph↗
arXiv2005

Permanent magnetic lattices for ultracold atoms and quantum degenerate gases

We propose the use of periodic arrays of permanent magnetic films for producing magnetic lattices of microtraps for confining, manipulating and controlling small clouds of ultracold atoms and quantum degenerate gases. Using analytical expressions and numerical calculations we show that periodic arrays of magnetic films can produce one-dimensional (1D) and two-dimensional (2D) magnetic lattices with non-zero potential minima, allowing ultracold atoms to be trapped without losses due to spin flips. In particular, we show that two crossed layers of periodic arrays of parallel rectangular magnets plus bias fields, or a single layer of periodic arrays of square-shaped magnets with three different thicknesses plus bias fields, can produce 2D magnetic lattices of microtraps having non-zero potential minima and controllable trap depth. For arrays with micron-scale periodicity, the magnetic microtraps can have very large trap depths ($\sim$0.5 mK for the realistic parameters chosen for the 2D lattice) and very tight confinement.

cond-mat.other↗
arXiv2005

Quantum Heat Engines, the Second Law and Maxwell's Daemon

We introduce a class of quantum heat engines which consists of two-energy-eigenstate systems, the simplest of quantum mechanical systems, undergoing quantum adiabatic processes and energy exchanges with heat baths, respectively, at different stages of a cycle. Armed with this class of heat engines and some interpretation of heat transferred and work performed at the quantum level, we are able to clarify some important aspects of the second law of thermodynamics. In particular, it is not sufficient to have the heat source hotter than the sink, but there must be a minimum temperature difference between the hotter source and the cooler sink before any work can be extracted through the engines. The size of this minimum temperature difference is dictated by that of the energy gaps of the quantum engines involved. Our new quantum heat engines also offer a practical way, as an alternative to Szilard's engine, to physically realise Maxwell's daemon. Inspired and motivated by the Rabi oscillations, we further introduce some modifications to the quantum heat engines with single-mode cavities in order to, while respecting the second law, extract more work from the heat baths than is otherwise possible in thermal equilibria. Some of the results above are also generalisable to quantum heat engines of an infinite number of energy levels including 1-D simple harmonic oscillators and 1-D infinite square wells.

quant-ph↗
arXiv2005

The second law, Maxwell's daemon and work derivable from quantum heat engines

With a class of quantum heat engines which consists of two-energy-eigenstate systems undergoing, respectively, quantum adiabatic processes and energy exchanges with heat baths at different stages of a cycle, we are able to clarify some important aspects of the second law of thermodynamics. The quantum heat engines also offer a practical way, as an alternative to Szilard's engine, to physically realise Maxwell's daemon. While respecting the second law on the average, they are also capable of extracting more work from the heat baths than is otherwise possible in thermal equilibrium.

quant-ph↗
arXiv2005

Mathematical computability questions for some classes of linear and non-linear differential equations originated from Hilbert's tenth problem

Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schrödinger equations with some appropriate time-dependent Hamiltonians. We then raise the questions whether these two classes of differential equations are computable or not in some computation models of computable analysis. These are non-trivial and important questions given that: (i) not all computation models of computable analysis are equivalent, unlike the case with classical recursion theory; (ii) and not all models necessarily and inevitably reduce computability of real functions to discrete computations on Turing machines. However unlikely the positive answers to our computability questions, their existence should deserve special attention and be satisfactorily settled since such positive answers may also have interesting logical consequence back in the classical recursion theory for the Church-Turing thesis.

math.GM↗
arXiv2005

Hypercomputability of quantum adiabatic processes: Fact versus Prejudices

We give an overview of a quantum adiabatic algorithm for Hilbert's tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementation.

quant-ph↗
arXiv2004

Finiteness of the universe and computation beyond Turing computability

We clarify the confusion, misunderstanding and misconception that the physical finiteness of the universe, if the universe is indeed finite, would rule out all hypercomputation, the kind of computation that exceeds the Turing computability, while maintaining and defending the validity of Turing computation and the Church-Turing thesis.

quant-ph↗
arXiv2004

Using biased coins as oracles

While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set $X$ may be coded as a probability $p_{X}$ such that if a Turing machine is given a coin which lands heads with probability $p_{X}$ it can compute any function recursive in $X$ with arbitrarily high probability. We also show how the assumption of a non-recursive bias can be weakened by using a sequence of increasingly accurate recursive biases or by choosing the bias at random from a distribution with a non-recursive mean. We conclude by briefly mentioning some implications regarding the physical realisability of such methods.

cs.OH↗