Optimal Hamiltonian Research Articles

Constructions of optimal-speed quantum evolutions: a comparative study

We present a comparative analysis of two different constructions of optimal-speed quantum Hamiltonian evolutions on the Bloch sphere. In the first approach (Mostafazadeh’s approach), the evolution is specified by a traceless stationary Hermitian Hamiltonian and occurs between two arbitrary qubit states by maximizing the energy uncertainty. In the second approach (Bender’s approach), instead, the evolution is characterized by a stationary Hermitian Hamiltonian which is not traceless and occurs between an initial qubit state on the north pole and an arbitrary final qubit state. In this second approach, the evolution occurs by minimizing the evolution time subject to the constraint that the difference between the largest and the smallest eigenvalues of the Hamiltonian is kept fixed. For both approaches we calculate explicitly the optimal Hamiltonian, the optimal unitary evolution operator and, finally, the optimal magnetic field configuration. Furthermore, we show in a clear way that Mostafazadeh’s and Bender’s approaches are equivalent when we extend Mostafazadeh’s approach to Hamiltonians with nonzero trace and, at the same time, focus on an initial quantum state placed on the north pole of the Bloch sphere. Finally, we demonstrate in both scenarios that the optimal unitary evolution operator is a rotation about an axis that is orthogonal to the unit Bloch vectors that correspond to the initial and final qubit states.

DFT-Based Tight-binding model of vdW bilayer χ3 and β12 borophene

In this work, a tight-binding (TB) model with the Slater–Koster (SK) approximation is presented to describe the electronic properties of vdW bilayer χ3 and β12 borophene with AA stack. This model is created by using four orbitals of S, Px, Py, and Pz for each boron atom. By fitting the results of the TB model and density functional theory (DFT) calculations using the least-squares method and the Levenberg-Marquardt nonlinear fitting algorithm, the optimal Hamiltonian and the overlap matrix are calculated. This TB model well describes energy bands around high-symmetry k-points, by considering three nearest neighbors in each layer and a non-orthogonal basis set.

Inverse Hamiltonian design by automatic differentiation

An ultimate goal of materials science is to deliver materials with desired properties at will. Solving the inverse problem to obtain an appropriate Hamiltonian directly from the desired properties has the potential to reach qualitatively new principles, but most research to date has been limited to quantitative determination of parameters within known models. Here, we develop a general framework that can automatically design a Hamiltonian with desired physical properties by using automatic differentiation. In the application to the quantum anomalous Hall effect, our framework can not only construct the Haldane model automatically but also generate Hamiltonians that exhibit a six-times larger anomalous Hall effect. In addition, the application to the photovoltaic effect gives an optimal Hamiltonian for electrons moving on a noncoplanar spin texture, which can generate ~ 700 Am−2 under solar radiation. This framework would accelerate materials exploration by automatic construction of models and principles.

Effect of configuration mixing on quadrupole and octupole collective states of transitional nuclei

A model is presented that simultaneously describes shape coexistence and quadrupole and octupole collective excitations within a theoretical framework based on the nuclear density functional theory and the interacting boson model. An optimal interacting-boson Hamiltonian that incorporates the configuration mixing between normal and intruder states, as well as the octupole degrees of freedom, is identified by means of self-consistent mean-field calculations using a universal energy density functional and a pairing interaction, with constraints on the triaxial quadrupole and the axially-symmetric quadrupole and octupole shape degrees of freedom. An illustrative application to the transitional nuclei $^{72}$Ge, $^{74}$Se, $^{74}$Kr, and $^{76}$Kr shows that the inclusion of the intruder states and the configuration mixing significantly lower the energy levels of the excited $0^+$ states, and that the predicted low-lying positive-parity states are characterized by the strong admixture of nearly spherical, weakly deformed oblate, and strongly deformed prolate shapes. The low-lying negative-parity states are shown to be dominated by the deformed intruder configurations.

Optimal parent Hamiltonians for time-dependent states

Given a generic time-dependent many-body quantum state, we determine the associated parent Hamiltonian. This procedure may require, in general, interactions of any sort. Enforcing the requirement of a fixed set of engineerable Hamiltonians, we find the optimal Hamiltonian once a set of realistic elementary interactions is defined. We provide three examples of this approach. We first apply the optimization protocol to the ground states of the one-dimensional Ising model and a ferromagnetic $p$-spin model but with time-dependent coefficients. We also consider a time-dependent state that interpolates between a product state and the ground state of a $p$-spin model. We determine the time-dependent optimal parent Hamiltonian for these states and analyze the capability of this Hamiltonian of generating the state evolution. Finally, we discuss the connections of our approach to shortcuts to adiabaticity.

Towards Necessary and Sufficient Stability Conditions for NMPC

Abstract The analysis of convergence and stability conditions for Non-linear Model Predictive Control (NMPC) schemes has seen significant progress during the last decades. Numerous results in the literature state sufficient conditions for convergence and/or stability of NMPC. Yet, necessary and sufficient conditions are—to the best of the author’s knowledge—not known and, given the variety of choices for MPC design, seemingly appear to be difficult. This paper analyzes the convergence of NMPC of continuous-time systems in time-invariant settings. We introduce the receding-horizon Hamiltonian—i.e., the value of the optimal control Hamiltonian along the sequence of OCPs—as a novel tool for stability/convergence analysis. Based on the assumption of strict dissipativity, we prove necessary and sufficient conditions for asymptotic convergence of the closed loop to the optimal steady state. Numerical results obtained for the Van de Vusse reactor illustrate the proposed conditions.

Activating Hidden Metrological Usefulness.

We consider bipartite entangled states that cannot outperform separable states in any linear interferometer. Then, we show that these states can still be more useful metrologically than separable states if several copies of the state are provided or an ancilla is added to the quantum system. We present a general method to find the local Hamiltonian for which a given quantum state performs the best compared to separable states. We obtain analytically the optimal Hamiltonian for some quantum states with a high symmetry. We show that all bipartite entangled pure states outperform separable states in metrology. Some potential applications of the results are also suggested.

Optimal control for multi-parameter quantum estimation with time-dependent Hamiltonians

We investigate simultaneous estimation of multi-parameter with time-dependent Hamiltonians. We analytically obtain the maximum quantum Fisher information matrix for two-parameter encoded in time-dependent three-level systems. The optimal coherent control scheme is proposed to increase the simultaneous multi-parameter estimation precisions. In an example of a spin-1 particle in a uniformly rotating magnetic field, the optimal coherent Hamiltonians for different parameters can be chosen to be completely same. However, in general, the optimal coherent Hamiltonians for different parameters are incompatible. In this situation, we suggest a variance method to obtain the optimal coherent Hamiltonian for simultaneously estimating multiple parameters, and obtain the optimal simultaneous estimation precision of two-parameter in a three-level Landau-Zener Hamiltonian.

Sufficient and Necessary Conditions for an Edge in the Optimal Hamiltonian Cycle Based on Frequency Quadrilaterals

The symmetric traveling salesman problem is studied according to frequency graphs computed with frequency quadrilaterals. Here, we provide the sufficient and necessary conditions for an optimal Hamiltonian cycle edge based on frequency quadrilaterals. If the probability that an edge has frequency 5 in a frequency quadrilateral is 1, it belongs to the optimal Hamiltonian cycle. For an optimal Hamiltonian cycle edge, the probability that it has frequency 5 in a frequency quadrilateral tends to 1 as the scale of traveling salesman problem is sufficiently large.

An underactuated drift-free left-invariant control system on the Lie group ISO(3,1)

The purpose of our paper is to study a class of left-invariant, drift-free optimal control problem on the Lie group ISO(3,1). The left-invariant, drift-free optimal control problems involves finding a trajectory-control pair on ISO(3,1), which minimize a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*G using the optimal Hamiltonian on G*, where the maximum principle yields the optimal control. We use energy methods (Arnold’s method, in this case) to give sufficient conditions fornonlinear stability of the equilibrium states. Around this equilibrium states we might be able to find the periodical orbits using Moser's theorem, as future work. For the some unstable equilibrium states, a quadratic control is considered in order to stabilize the dynamics.

Experimental implementation of generalized transitionless quantum driving

It is known that high intensity fields are usually required to implement shortcuts to adiabaticity via transitionless quantum driving (TQD). Here, we show that this requirement can be relaxed by exploiting the gauge freedom of generalized TQD, which is expressed in terms of an arbitrary phase when mimicking the adiabatic evolution. We experimentally investigate the performance of generalized TQD in comparison to both traditional TQD and adiabatic dynamics. By using a Yb+171 trapped ion hyperfine qubit, we implement a Landau-Zener adiabatic Hamiltonian and its (traditional and generalized) TQD counterparts. We show that the generalized theory provides energy-optimal Hamiltonians for TQD, with no additional fields required. In addition, the optimal TQD Hamiltonian for the Landau-Zener model is investigated under dephasing. Even using less intense fields, optimal TQD exhibits fidelities that are more robust against a decohering environment, with performance superior to that provided by the adiabatic dynamics.

Quantum-limited biochemical magnetometers designed using the Fisher information and quantum reaction control

Radical-ion pairs and their reactions have triggered the study of quantum effects in biological systems. This is because they exhibit a number of effects best understood within quantum information science, and at the same time are central in understanding the avian magnetic compass and the spin transport dynamics in photosynthetic reaction centers. Here we address radical-pair reactions from the perspective of quantum metrology. Since the coherent spin motion of radical-pairs is effected by an external magnetic field, these spin-dependent reactions essentially realize a biochemical magnetometer. Using the quantum Fisher information, we find the fundamental quantum limits to the magnetic sensitivity of radical-pair magnetometers. We then explore how well the usual measurement scheme considered in radical-pair reactions, the measurement of reaction yields, approaches the fundamental limits. In doing so, we find the optimal hyperfine interaction Hamiltonian that leads to the best magnetic sensitivity as obtained from reaction yields. This is still an order of magnitude smaller than the absolute quantum limit. Finally, we demonstrate that with a realistic quantum reaction control reminding of Ramsey interferometry, here presented as a quantum circuit involving the spin-exchange interaction and a recently proposed molecular switch, we can approach the fundamental quantum limit within a factor of 2. This work opens the application of well-advanced quantum metrology methods to biological systems.

Optimal Hamiltonian Simulation by Quantum Signal Processing

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i)transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii)transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii)projecting this ancilla with near unity success probability.

A Binomial Distribution Model for the Traveling Salesman Problem Based on Frequency Quadrilaterals

We study the symmetric traveling salesman problem via frequency graphs. One computes the frequency of edges by computing how many times an edge occurs in an optimal path involving four vertices. The edges that are in the Optimal Hamiltonian Cycle ($OHC$) have a higher frequency than most edges that are not in the $OHC$ and thus edges with a low frequency can safely be ignored when searching for the optimal solution. A binomial distribution model is introduced for the symmetric traveling salesman problem based on frequency quadrilaterals. When the frequency of each edge is computed with $N$ frequency quadrilaterals, our model suggests that the minimum frequency of an edge in the $OHC$ is $F_\mathrm{min} =(\epsilon_\mathrm{min}+1)N$ where $\frac{4}{3}+\frac{4}{3(n-2)} < \epsilon_\mathrm{min} < 4$. This suggests a heuristic to reduce the number of edges that need to be considered in the search for the $OHC$ which is to keep only those edges whose frequencies are $\geq F_\mathrm{min}$. We explore this heuristic in several real-world examples. An erratum for this paper has been published on August 2017 (see link on the right)

Optimal Control and Hamiltonian System

In this paper, an optimal control for Hamiltonian control systems with external variables will be formulated and analysed. Necessary and sufficient conditions which lead to Pantryagin’s principle are stated and elaborated. Finally it is shown how the Pontryagin’s principle fits very well to the theory of Hamiltonian systems. The case of Potryagin’s maximum principle will be considered in detail since it is capable of dealing with both unbounded continuous controls and bounded controls which are possibly discontinuous.

Zermelo navigation in the quantum brachistochrone

We analyse the optimal times for implementing unitary quantum gates in a constrained finite dimensional controlled quantum system. The family of constraints studied is that the permitted set of (time dependent) Hamiltonians is the unit ball of a norm induced by an inner product on . We also consider a generalization of this to arbitrary norms. We construct a Randers metric, by applying a theorem of Shen on Zermelo navigation, the geodesics of which are the time optimal trajectories compatible with the prescribed constraint. We determine all geodesics and the corresponding time optimal Hamiltonian for a specific constraint on the control i.e. for any given value of . Some of the results of Carlini et al are re-derived using alternative methods. A first order system of differential equations for the optimal Hamiltonian is obtained and shown to be of the form of the Euler–Poincaré equations. We illustrate that this method can form a methodology for determining which physical substrates are effective at supporting the implementation of fast quantum computation.

Variational truncated Wigner approximation.

In this paper we reconsider the notion of an optimal effective Hamiltonian for the semiclassical propagation of the Wigner distribution in phase space. An explicit expression for the optimal effective Hamiltonian is obtained in the short-time limit by minimizing the Hilbert-Schmidt distance between the semiclassical approximation and the real state of the system. The method is illustrated for the quartic oscillator.

How fast and robust is the quantum adiabatic passage?

We study the assisted adiabatic passage, and equivalently the transitionless quantum driving, as a quantum brachistochrone trajectory. The optimal Hamiltonian for given constraints is constructed from the quantum brachistochrone equation. We discuss how the adiabatic passage is realized as the solution of the equation. The formulation of the quantum brachistochrone is based on the principle of least action. We utilize it to discuss the stability of the adiabatic passage.

Path planning for simple wheeled robots: sub-Riemannian and elastic curves on SE(2)

SUMMARYThis paper presents a motion planning method for a simple wheeled robot in two cases: (i) where translational and rotational speeds are arbitrary, and (ii) where the robot is constrained to move forwards at unit speed. The motions are generated by formulating a constrained optimal control problem on the Special Euclidean group SE(2). An application of Pontryagin's maximum principle for arbitrary speeds yields an optimal Hamiltonian which is completely integrable in terms of Jacobi elliptic functions. In the unit speed case, the rotational velocity is described in terms of elliptic integrals, and the expression for the position is reduced to quadratures. Reachable sets are defined in the arbitrary speed case, and a numerical plot of the time-limited reachable sets is presented for the unit speed case. The resulting analytical functions for the position and orientation of the robot can be parametrically optimised to match prescribed target states within the reachable sets. The method is shown to be easily adapted to obstacle avoidance for static obstacles in a known environment.

Exploring the Cooling Limit of Quantum Mechanical Oscillators via Optimization

In this work, we have studied the optimal cooling limit when cooling a quantum mechanical oscillator by coupling it to an auxiliary. Since this limit is necessarily confined by the energy transfer rate of the control, i.e., the magnitude of the spectrum of the control Hamiltonian, we search for the optimal control Hamiltonian that can achieve the maximal cooling fidelity, given a bounded energy transfer rate. First we analyze and present a class of Hamiltonian which generates the perfect cooling result in the absence of decoherence. Then with perturbation analysis, we further show that such Hamiltonian is also optimal at least to the first order of the dynamical dissipative terms. With the help of numerical optimization, we have studied cases with different damping rates of the system and the auxiliary, and derived the plot of the optimal cooling fidelity curve versus time as well as the optimal time point to achieve the maximal fidelity.