We present a simple derivation of the formula for the Hamiltonian operator(s) that achieve the fastest possible unitary evolution between given initial and final states. We discuss how this formula is modified in pseudo-Hermitian quantum mechanics and provide an explicit expression for the most general optimal-speed quasi-Hermitian Hamiltonian. Our approach allows for an explicit description of the metric- (inner product-) dependence of the lower bound on the travel time and the universality (metric-independence) of the upper bound on the speed of unitary evolutions.
The evolution speed in projective Hilbert space is considered for Hermitian Hamiltonians and for non-Hermitian (NH) ones. Based on the Hilbert-Schmidt norm and the spectral norm of a Hamiltonian, resource-related upper bounds on the evolution speed are constructed. These bounds are valid also for NH Hamiltonians and they are illustrated for an optical NH Hamiltonian and for a non-Hermitian $\mathcal{PT}-$symmetric matrix Hamiltonian. Furthermore, the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective Hilbert space…
We report on a time scaling technique to enhance the performances of quantum protocols in non-Hermitian systems. The considered time scaling involves no extra-couplings and yields a significant enhancement of the quantum fidelity for a comparable amount of resources. We discuss the application of this technique to quantum state transfers in 2 and 3-level open quantum systems. We derive the quantum speed limit in a system governed by a non-Hermitian Hamiltonian. Interestingly, we show that, with an appropriate driving, the time-scaling technique preserves the optimality of the quantum speed wit…
From the unitary operator used for implementing two-state discrete-time quantum walk on one-, two- and three- dimensional lattice we obtain a two-component Dirac-like Hamiltonian. In particular, using different pairs of Pauli basis as position translation states we obtain three different form of Hamiltonians for evolution on one-dimensional lattice. We extend this to two- and three-dimensional lattices using different Pauli basis states as position translation states for each dimension and show that the external coin operation, which is necessary for one-dimensional walk is not a necessary req…
We consider finite-dimensional Markovian open quantum systems, and characterize the extent to which time-independent Hamiltonian control may allow to stabilize a target quantum state or subspace and optimize the resulting convergence speed. For a generic Lindblad master equation, we introduce a dissipation-induced decomposition of the associated Hilbert space, and show how it serves both as a tool to analyze global stability properties for given control resources and as the starting point to synthesize controls that ensure rapid convergence. The resulting design principles are illustrated in r…
Inequalities of Mandelstam-Tamm and Margolus-Levitin type provide lower bounds on the time it takes for a quantum system to evolve from one state into another. Knowledge of such bounds, called quantum speed limits, is of utmost importance in virtually all areas of physics, where determination of the minimum time required for a quantum process is of interest. Most Mandelstam-Tamm and Margolus-Levitin inequalities found in the literature have been derived from growth estimates for the Bures length, which is a statistical distance measure. In this paper we derive such inequalities by differential…
We theoretically investigate the effect of multi-mode dynamics on the creation of macroscopic superposition states (spin-cat states) in Bose-Einstein condensates via one-axis twisting. A two-component Bose-Einstein condensate naturally realises an effective one-axis twisting interaction, under which an initially separable state will evolve toward a spin-cat state. However, the large evolution times necessary to realise these states is beyond the scope of current experiments. This evolution time is proportional to the degree of asymmetry in the relative scattering lengths of the system, which r…
We propose a scheme to realize fast generation of three-dimensional entanglement between two atoms via superadiabatic-based shortcuts in an atom-cavity-fiber system. The scheme is experimentally feasible because of the same form of the counterdiabatic Hamiltonian as that of the effective Hamiltonian. Besides, numerical simulations are given to prove that the scheme is strongly robust against variations in various parameters and decoherence.
We propose a scheme for generating squeezed states in solid state circuits consisting of a nanomechanical resonator (NMR), a superconducting Cooper-pair box (CPB) and a superconducting transmission line resonator (STLR). The nonlinear interaction between the NMR and the STLR can be implemented by setting the external biased flux of the CPB at certain values. The interaction Hamiltonian between the NMR and the STLR is derived by performing Fr$\rm\ddot o$hlich transformation on the total Hamiltonian of the combined system. Just by adiabatically keeping the CPB at the ground state, we get the sta…
The NOON states are valuable quantum resources, which have a wide range of applications in quantum communication, quantum metrology, and quantum information processing. Here we propose a fast, concise and reliable protocol for deterministically generating the NOON states of two resonators coupled to a single $\triangle$-type superconducting qutrit. In particular, we derive the effective Hamiltonians at the multi-photon resonances by virtue of the strong counter-rotating interaction between the resonator modes and the qutrit. Based on these crucial effective Hamiltonians, our protocol simplifie…
We present a general approach to speed up the adiabatic process without adding the traditional counterdiabatic driving (CD) Hamiltonian. The strategy is to design an easy-to-get intermediate Hamiltonian to connect the original Hamiltonian and final transitionless Hamiltonian. With final transitionless Hamiltonian, the same target can be achieved as in the adiabatic process governed by the original Hamiltonian, but in a shorter time. We apply the present approach to a three-level system, and the result shows that the final transitionless Hamiltonian usually has the same structure as the origina…
We present two 2-body Hamiltonians that approximate the exact PH-Pfaffian wavefunction with their ground states for all the system sizes where this wavefunction has been numerically constructed to date. The approximate wavefunctions have high overlap with the original and reproduce well the low-lying entanglement spectrum and structure factor. The approximate generating Hamiltonians are obtained by an optimisation procedure where three to four pseudopotentials are varied in the neighbourhood of second Landau level Coulomb interaction or of a non-interacting model. They belong to a finite regio…
The Eigenvalue Moment Method (EMM), Handy (2001), Handy and Wang (2001)) is applied to the $H_α\equiv P^2 + iX^3 + iαX$ Hamiltonian, enabling the algebraic/numerical generation of converging bounds to the complex energies of the $L^2$ states, as argued (through asymptotic methods) by Delabaere and Trinh (J. Phys. A: Math. Gen. {\bf 33} 8771 (2000)).
A new method for generating analytical expression of quantum Hamiltonian from non-linear differential equation with stationary energy level has been formulated.Further calculation of energy levels have been carried out analytically using and numerically using matrix diagonalisation method.
In this paper, we propose a concept to use a quantum speed limit (QSL) as a measure of robustness of states, defining that a state with bigger QSL is more robust. In this perspective, it is important to have an explicitly-computable QSL, because then we can formulate an engineering problem of Hamiltonian that makes a target state robust against decoherence. Hence we derive a new explicitly-computable QSL that is applicable to general Markovian open quantum systems. This QSL is tighter than another explicitly-computable QSL, in an important setup such that decoherence is small. Also the Hamilto…
We observe that, within the effective generating function formalism for the implementation of canonical transformations within wave mechanics, non-trivial canonical transformations which leave invariant the form of the Hamilton function of the classical analogue of a quantum system manifest themselves in an integral equation for its stationary state eigenfunctions. We restrict ourselves to that subclass of these dynamical symmetries for which the corresponding effective generating functions are necessaarily free of quantum corrections. We demonstrate that infinite families of such transformati…
We propose and discuss a theoretical scheme to speed up Zeno dynamic passage by an external acceleration Hamiltonian. This scheme is a flexible and experimentally feasible acceleration because the acceleration Hamiltonian does not adhere rigidly to an invariant relationship, whereas it can be a more general form $\sum u_{j}(t)H_{cj}$. Here $H_{cj}$ can be arbitrarily selected without any limitation, and therefore one can always construct an acceleration Hamiltonian by only using realizable $H_{cj}$. Applying our scheme, we finally design an experimentally feasible Hamiltonian as an example to …
We derive generalized quantum speed limit inequalities that represent limitations on the time evolution of quantum states. They are extensions of the original inequality and are applied to the overlap between the time-evolved state and an arbitrary state. We can discuss the lower limit of the overlap, in addition to the upper limit as in the original inequality, which allows us to estimate the minimum time for the evolution toward a target state. The inequalities are written by using an arbitrary reference state and are flexibly used to obtain a tight bound. We demonstrate these properties by …
We consider a pure 2m-qubit initial state to evolve under a particular quantum me- chanical spin Hamiltonian, which can be written in terms of the adjacency matrix of the Johnson network J(2m;m). Then, by using some techniques such as spectral dis- tribution and stratification associated with the graphs, employed in [1, 2], a maximally entangled GHZ state is generated between the antipodes of the network. In fact, an explicit formula is given for the suitable coupling strengths of the hamiltonian, so that a maximally entangled state can be generated between antipodes of the network. By using s…
The transitionless tracking (TT) algorithm enables the exact tracking of quantum adiabatic dynamics in an arbitrary short time by adding a counterdiabatic Hamiltonian to the original adiabatic Hamiltonian. By applying Husimi's method originally developed for a quantum parametric oscillator (QPO) to the transitionless QPO achieved using the TT algorithm, we obtain the transition probability generating function with a time-dependent parameter constituted with solutions of the corresponding classical parametric oscillator (CPO). By obtaining the explicit solutions of this CPO using the phase-ampl…
We formulate an optimization problem of Hamiltonian design based on the variational principle. Given a variational ansatz for a Hamiltonian we construct a loss function to be minimised as a weighted sum of relevant Hamiltonian properties specifying thereby the search query. Using fractional quantum Hall effect as a test system we illustrate how the framework can be used to determine a generating Hamiltonian of a finite-size model wavefunction (Moore-Read Pfaffian and Read-Rezayi states), find optimal conditions for an experiment or "extrapolate" given wavefunctions in a certain universality cl…
Generating ground states of any local Hamiltonians seems to be impossible in quantum polynomial time. In this paper, we give evidence for the impossibility by applying an argument used in the quantum-computational-supremacy approach. More precisely, we show that if ground states of any $3$-local Hamiltonians can be approximately generated in quantum polynomial time with postselection, then ${\sf PP}={\sf PSPACE}$. Our result is superior to the existing findings in the sense that we reduce the impossibility to an unlikely relation between classical complexity classes. We also discuss what makes…
We discuss a spectrum generating algebra in the supersymmetric quantum mechanical system which is defined as a series of solutions to a specific differential equation. All Hamiltonians have equally spaced eigenvalues, and we realize both positive and negative mode generators of a subalgebra of $W_{1+\infty}$ without use of negative power of raising/lowering operators of the system. All features in the supersymmetric case are generalized to the parasupersymmetric systems of order 2.
We consider quantum systems with causal dynamics in discrete spacetimes, also known as quantum cellular automata (QCA). Due to time-discreteness this type of dynamics is not characterized by a Hamiltonian but by a one-time-step unitary. This can be written as the exponential of a Hamiltonian but in a highly non-unique way. We ask if any of the Hamiltonians generating a QCA unitary is local in some sense, and we obtain two very different answers. On one hand, we present an example of QCA for which all generating Hamiltonians are fully non-local, in the sense that interactions do not decay with …
Supersymmetry allows one to build a hierarchy of Hamiltonians that share the same spectral properties and which are pairwise connected through common superpotentials. The iso-spectral properties of these Hamiltonians imply that the dynamics and therefore control of different eigenstates are connected through supersymmetric intertwining relations. In this work we explore how this enables one to study general dynamics, shortcuts to adiabaticity (STA) and quantum speed limits for distinct states of different supersymmetric partner potentials by using the infinite box as an example.
We consider the problem of preparing topologically ordered states using unitary and non-unitary circuits, as well as local time-dependent Hamiltonian and Liouvillian evolutions. We prove that for any topological code in $D$ dimensions, the time required to encode logical information into the ground space is at least $Ω(d^{1/(D-1)})$, where $d$ is the code distance. This result is tight for the toric code, giving a scaling with the linear system size. More generally, we show that the linear scaling is necessary even when dropping the requirement of encoding: preparing any state close to the gro…
The speed of information propagation is finite in quantum systems with local interactions. In many such systems, local operators spread ballistically in time and can be characterized by a "butterfly velocity", which can be measured via out-of-time-ordered correlation functions. In general, the butterfly velocity can depend asymmetrically on the direction of information propagation. In this work, we construct a family of simple 2-local Hamiltonians for understanding the asymmetric hydrodynamics of operator spreading. Our models live on a one dimensional lattice and exhibit asymmetric butterfly …
We report on the generation of single-photon pulse trains at a repetition rate of up to 1 GHz. We achieve this high speed by modulating the external voltage applied on an electrically contacted quantum dot microlens, which is optically excited by a continuous-wave laser. By modulating the photoluminescence of the quantum dot microlens using a square-wave voltage, single-photon emission is triggered with a response time as short as 270 ps being 6.5 times faster than the radiative lifetime of 1.75 ns. This large reduction in the characteristic emission time is enabled by a rapid capacitive gatin…
The family F of all potentials V(x) for which the Hamiltonian H in one space dimension possesses a high order Lie symmetry is determined. A sub-family F', which contains a class of potentials allowing a realization of so(2,1) as spectrum generating algebra of H through differential operators of finite order, is identified. Furthermore and surprisingly, the families F and F' are shown to be related to the stationary KdV hierarchy. Hence, the "harmless" Hamiltonian H connects different mathematical objects, high order Lie symmetry, realization of so(2,1)-spectrum generating algebra and families …
One often needs to estimate how fast an evolving state of a quantum system can depart from some target state or target subspace of a Hilbert space. Such estimates are known as quantum speed limits. We derive a quantum speed limit for a general time-dependent target subspace. When the target subspace is an instantaneous invariant subspace of a time-dependent Hamiltonian, the obtained quantum speed limit bounds the adiabatic fidelity, which is a figure of merit of quantum adiabaticity. We also compare two states evolving under two different Hamiltonians and derive a bound on the Loschmidt echo.
The effects of physical interactions are usually incorporated into the quantum theory by including the corresponding terms in the Hamiltonian. Here we consider the effects of including the gravitational potential energy of massive particles in the Hamiltonian of quantum electrodynamics. This results in a predicted correction to the speed of light that is proportional to the fine structure constant. The correction to the speed of light obtained in this way depends on the gravitational potential and not the gravitational field, which is not gauge invariant and presumably nonphysical. Nevertheles…
Information Processing in Quantum Thermodynamic Systems: an Autonomous Hamiltonian Approach
University of Massachusetts, Boston（马萨诸塞大学波士顿分校）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；University of Maryland, Baltimore County（马里兰大学巴尔的摩分校）；Quantum Science Institute, University of Maryland, Baltimore County（马里兰大学巴尔的摩分校量子科学研究所）
We analyze the dynamical equations obeyed by a classical system with position-dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the Hamiltonian for this system and find the modifications required in the Euler-Lagrange and Hamilton's equations to reproduce the appropriate Newton's dynamical law. Since the Hamiltonian is not time invariant, we get a constant of motion suited to write the dynamical equations in the form of the Hamilton's ones. The time-dependent first integrals of motion ar…
We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum spe…
When discussing consequences of symmetries of dynamical systems based on Noether's first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether's first theorem. Rather a much less restrictive statement applies,…
In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra sl(2, C). This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass $\wp$ type, one of them being r…
