Quantum mechanical Hamiltonian models of discrete processes that erase their own histories: Application to Turing machines

Work done before on the construction of quantum mechanical Hamiltonian models of Turing machines and general discrete processes is extended here to include processes which erase their own histories. The models consist of three phases: the forward process phase in which a mapT is iterated and a history of iterations is generated, a copy phase, which is activated if and only ifT reaches a fix point, and an erase phase, which erases the iteration history, undoes the iterations ofT, and recovers the initial state except for the copy system. A ballast system is used to stop the evolution at the desired state. The general model so constructed is applied to Turing machines. The main changes are that the system undergoing the evolution corresponding toT iterations becomes three systems corresponding to the internal machine, the computation tape, and computation head. Also the copy phase becomes more complex since it is desired that this correspond also to a copying Turing machine.     

Thermodynamics under Dynamic Forcing

Beginning with a system that is governed by an arbitrary time-dependent Hamiltonian, we exhibit an existence proof for a unitary generator that has an arbitrary initial value and yet contact transforms the representation to one governed by any given kinematically equivalent Hamiltonian. By choosing the initial value of the unitary operator to be unity, we are able to compare the behaviour of the same system under two different Hamiltonians and the same initial state vector. We thus are able to establish that the eventual physical states evolving from two distinct initial quantum state vectors will become practically indistinguishable under one of the two Hamiltonians if and only if they do so under the other. For the restricted class of systems for which one of the two Hamiltonians is a time-independent energy operator, and also generates equilibrium thermodynamics, then the condition for merging under the time-dependent Hamiltonian is the same as under the time-independent one. The two states must have the same initial energy. As a special case of the above, we choose the time-independent Hamiltonian to be the relativistic energy measuring operator for the time-dependent Hamiltonian, as associated with the chosen initial time. If the system under the time-dependent Hamiltonian is such that its relativistic energy measuring operator for any fixed time generates equilibrium thermodynamics, then we are led rigorously to the conclusion that the instantaneous relativistic energy for the system under the time-dependent Hamiltonian is simply a well-defined function of time and depends only on the initial energy and not on any other initial conditions. For a composite system that is of the above type, and in addition consists of one very small system in contact with a very large one, which is called a generalized reservoir, we consider a specific initial physical state for the large system, and various states for the small one. The eventual dynamic state of the composite system is essentially independent of the initial state of the small system which has almost no influence on the total composite energy. Hence the eventual dynamic state of the small system is shown rigorously to be independent of its initial state. For a forced system with a time-dependent Hamiltonian, we discuss the assignment of equilibrium thermodynamic potentials to a representation with a time-independent Hamiltonian. We discuss the concept of a process under a time-dependent Hamiltonian. Such a process is a natural generalization of the static and quasi-static processes. Also, we verify all of the theory with both general and specific examples of electromagnetic interactions.     

First order in the plaquette expansion

The plaquette expansion, a general non-perturbative method for calculating the properties of lattice Hamiltonian systems, is analytically investigated at the first non-trivial order for an arbitrary system. At this level the approximation describes systems with either a bounded or an unbounded spectrum, depending on simple inequalities formed from the first four cumulants. The analysis yields analytic forms for the ground state energy, mass gaps and density of states in the thermodynamic limit. An exact form for the resolvent oeprator is given for a finite number of sites, as well as the asymptotic form in the thermodynamic limit.     

Boundary Hamiltonian Theory for Gapped Topological Orders

We report our systematic construction of the lattice Hamiltonian model of topological orders on open surfaces,with explicit boundary terms. We do this mainly for the Levin-Wen string-net model. The full Hamiltonian in our approach yields a topologically protected, gapped energy spectrum, with the corresponding wave functions robust under topology-preserving transformations of the lattice of the system. We explicitly present the wavefunctions of the ground states and boundary elementary excitations. The creation and hopping operators of boundary quasi-particles are constructed. It is found that given a bulk topological order, the gapped boundary conditions are classified by Frobenius algebras in its input data. Emergent topological properties of the ground states and boundary excitations are characterized by(bi-) modules over Frobenius algebras.     

Quantum simulation of classical thermal states

We establish a connection between ground states of local quantum Hamiltonians and thermal states of classical spin systems. For any discrete classical statistical mechanical model in any spatial dimension, we find an associated quantum state such that the reduced density operator behaves as the thermal state of the classical system. We show that all these quantum states are unique ground states of a universal 5-body local quantum Hamiltonian acting on a (polynomially enlarged) qubit system on a 2D lattice. The only free parameters of the quantum Hamiltonian are coupling strengths of two-body interactions, which allow one to choose the type and dimension of the classical model as well as the interaction strength and temperature. This opens the possibility to study and simulate classical spin models in arbitrary dimension using a 2D quantum system.     

Product-State Approximations to Quantum States

We show that for any many-body quantum state there exists an unentangled quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded approximation error. If we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our approximations allow an extensive error in energy, which is the scale considered by the quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states.     

On the possibility of electric transport mediated by long living intrinsic localized solectron modes

We consider a polaron Hamiltonian in which not only the lattice and the electron-lattice interactions, but also the electron hopping term is affected by anharmonicity. We find that the one-electron ground states of this system are localized in a wide range of the parameter space. Furthermore, low energy excited states, generated either by additional momenta in the lattice sites or by appropriate initial electron conditions, lead to states constituted by a localized electron density and an associated lattice distortion, which move together through the system, at subsonic or supersonic velocities. Thus we investigate here the localized states above the ground state which correspond to moving electrons. We show that besides the stationary localized electron states (proper polaron states) there exist moving localized solectron states which can be easily excited. The evolution of these localized states suggests their potential as new carriers for fast electric charge transport.     

Robust bulk states

We predict topologically robust zero energy bulk states in a disordered tight binding lattice. We explore a new kind of order and discuss that zero energy states exist in a system iff its Hamiltonian is noninvertible. We show that they are robust against any kind of disorder as long as the disordered Hamiltonian is noninvertible, too.     

Adiabatic evolution under quantum control

One difficulty with adiabatic quantum computation is the limit on the computation time. Here we propose two schemes to speed-up the adiabatic evolution. To apply this controlled adiabatic evolution to adiabatic quantum computation, we design one of the schemes without any explicit knowledge of the instantaneous eigenstates of the final Hamiltonian. Whereas in another scheme, we assume that the ground state of the Hamiltonian is known, and this information can be used to design the control. By these techniques, a linear speed-up proportional to the nonlinearity can be predicted. As an illustration, we study a two-level system driven by a time-dependent magnetic field under the control. The problem of finding an item in an unsorted database by adiabatic evolution is also examined. The physics behind the control scheme is interpreted.     

Construction of coherent states for Morse potential: A su(2)-like approach

We propose a generalized su(2) algebra that perfectly describes the discrete energy part of the Morse potential. Then, we examine particular examples and the approach can be applied to any Morse oscillator and to practically any physical system whose spectrum is finite. Further, we construct the Klauder coherent state for Morse potential satisfying the resolution of identity with a positive measure, obtained through the solution of truncated Stieltjes moment problem. The time evolution of the uncertainty relation of the constructed coherent states is analyzed. The uncertainty relation is more localized for small values of radius of convergence.     

Theoretical evidence for equivalence between the ground states of the strong coupling BCS Hamiltonian and the antiferromagnetic Heisenberg model

By explicitly computing wave function overlap via exact diagonalization in finite systems, we provide evidence indicating that, in the limit of strong coupling, i.e., Delta/t--> infinity the ground state of the Gutzwiller-projected BCS Hamiltonian (accompanied by proper particle-number projection) is identical to the exact ground state of the 2D antiferromagnetic Heisenberg model on the square lattice. This identity is adiabatically connected to a very high overlap between the ground states of the projected BCS Hamiltonian and the t-J model at moderate doping.     

Energy evolution in the time-dependent harmonic oscillator with arbitrary external forcing

The classical Hamiltonian system of time-dependent harmonic oscillator driven by an arbi- trary external time-dependent force is considered. Exact analytical solution of the corresponding equations of motion is constructed in the framework of the technique based on WKB approach. Energy evolution for the ensemble of uniformly distributed w.r.t, the canonical angle initial conditions on the initial invariant torus is studied. Exact expressions for the energy moments of arbitrary order taken at arbitrary time moment are analytically derived. Corresponding character- istic function is analytically constructed in the form of infinite series and numerically evaluated for certain values of the system parameters. Energy distribution function is numerically obtained in some particular cases. In the limit of small initial ensemble's energy the relevant formula for the energy distribution function is analytically derived. Keywords: classical dynamics, Hamiltonian systems, linear oscillator, energy distribution func- tion, WKB approach.     

A honeycomb lattice model simulating the surface states of topological insulators

A honeycomb lattice model exhibiting the quantum spin-Hall effect is proposed, where the low-energy properties of the electrons are mainly determined by the energy spectrum in the vicinity of the Γ point, for suitable parameters. The nontrivial topology of the energy bands is revealed by calculating the Chern numbers, Berry curvature distribution, and edge state spectrum. We further show that in the continuum limit, the model Hamiltonian is equivalent to the effective model for the surface states in thin films of three-dimensional topological insulators. As a consequence, this lattice model provides a useful tool for numerical simulation of the physical properties of the surface states.     

Finite cell calculations on the periodic Anderson Hamiltonian

We report finite cell calculations on the one-dimensional periodic Anderson Hamiltonian. The ground state for two electrons per site is found to be an insulating non-magnetic singlet, which evolves continuously from the noninteracting U = 0 limit to the large U mixed valence and Kondo lattice regimes. The calculations for four sites given energy gaps which agree well with results for the infinite lattice in the few cases where they are known.     

Statistical properties of polariton states

Summary The unitary transformation that relates free polarization and photon states to polariton states is constructed. The time evolution of an arbitrary initial state in terms of the polariton Hamiltonian is presented. The many-photon components of polariton states as well as the transition probabilities to polariton states and the intrinsec and time-dependent polariton squeezing are discussed.     

Graphical Models in Reconstructability Analysis and Bayesian Networks

Reconstructability Analysis (RA) and Bayesian Networks (BN) are both probabilistic graphical modeling methodologies used in machine learning and artificial intelligence. There are RA models that are statistically equivalent to BN models and there are also models unique to RA and models unique to BN. The primary goal of this paper is to unify these two methodologies via a lattice of structures that offers an expanded set of models to represent complex systems more accurately or more simply. The conceptualization of this lattice also offers a framework for additional innovations beyond what is presented here. Specifically, this paper integrates RA and BN by developing and visualizing: (1) a BN neutral system lattice of general and specific graphs, (2) a joint RA-BN neutral system lattice of general and specific graphs, (3) an augmented RA directed system lattice of prediction graphs, and (4) a BN directed system lattice of prediction graphs. Additionally, it (5) extends RA notation to encompass BN graphs and (6) offers an algorithm to search the joint RA-BN neutral system lattice to find the best representation of system structure from underlying system variables. All lattices shown in this paper are for four variables, but the theory and methodology presented in this paper are general and apply to any number of variables. These methodological innovations are contributions to machine learning and artificial intelligence and more generally to complex systems analysis. The paper also reviews some relevant prior work of others so that the innovations offered here can be understood in a self-contained way within the context of this paper.     

Lattice Yang-Mills theory at nonzero temperature and the confinement problem

We discuss finite temperature lattice Yang-Mills theory with special attention to the confinement problem. The relationship between the confinement criteria of Wilson, Polyakov, and 't Hooft is clarified by establishing a string of inequalities between the corresponding string tensions. The close connection between finite temperature Yang-Mills models and spin models is exploited to obtain new and rather sharp upper bounds for the critical coupling constant above which there is confinement. This same analogy also allows us to establish infrared bounds for the gauge models that yield a lower bound for this critical coupling and thereby show the existence of a weak coupling regime without confinement at nonzero temperature in three or more space dimensions. Finally we discuss extension of our results to other forms of the lattice action, the Hamiltonian lattice models of Kogut and Susskind and 't Hooft'sN → ∞ limit.     

Free energy and the relative entropy

Under the assumption of an identity determining the free energy of a state of a statistical mechanical system relative to a given equilibrium state by means of the relative entropy, it is shown: first, that there is in any physically definable convex set of states a unique state of minimum free energy measured relative to a given equilibrium state; second, that if a state has finite free energy relative to an equilibrium state, then the set of its time translates is a weakly relatively compact set; and third, that a unique perturbed equilibrium state exists following a change in Hamiltonian that is bounded below.     

Generation of entanglement in finite spin systems via adiabatic quantum computation

For a finite XY chain and a finite two-dimensional Ising lattice, it is shown that the paramagnetic ground state is adiabatically transformed to the Greenberger-Horne-Zeilinger state in the ferromagnetic phase by changing slowly the external magnetic field. It is found that the fidelity between the Greenberger-Horne-Zeilinger state and an adiabatically evolved state depends on the interpolation schemes as well as the energy gap between the ground and exited states. A possibility whether quantum phase transitions can be simulated on adiabatic quantum computation is discussed.     

Energy landscape of 3D spin Hamiltonians with topological order

We explore the feasibility of a quantum self-correcting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local bounded-strength interactions which have a topologically ordered ground state but do not have stringlike logical operators. We prove that any sequence of local errors mapping a ground state of such a Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is tight up to a constant factor for one particular 3D spin Hamiltonian.