We introduce new families of pure quantum states that are constructed on top of the well-known Gilmore-Perelomov group-theoretic coherent states. We do this by constructing unitaries as the exponential of operators quadratic in Cartan subalgebra elements and by applying these unitaries to regular group-theoretic coherent states. This enables us to generate entanglement not found in the coherent states themselves, while retaining many of their desirable properties. Most importantly, we explain how the expectation values of physical observables can be evaluated efficiently. Examples include generalized spin-coherent states and generalized Gaussian states, but our construction can be applied to any Lie group represented on the Hilbert space of a quantum system. We comment on their applicability as variational families in condensed matter physics and quantum information.
We propose a variational approach to the dynamics of the Bose-Hubbard model beyond the mean field approximation. To develop a numerical scheme, we use a discrete overcomplete set of Glauber coherent states and its connection to the generalized coherent states studied in depth by Perelomov [A. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag (Berlin, 1986)]. The variational equations of motion of the generalized coherent state parameters as well as of the coefficients in an expansion of the wavefunction in terms of those states are derived and solved for many-parti…
Spectral properties of the Hamiltonian function which characterizes a trapped ion are investigated. In order to study semiclassical dynamics of trapped ions, coherent state orbits are introduced as sub-manifolds of the quantum state space, with the Kähler structure induced by the transition probability. The time dependent variational principle (TDVP) is applied on coherent states' orbits. The Hamilton equations of motion on Kähler manifolds of the type of classical phase spaces naturally arise. The associated classical Hamiltonian is obtained from the expected values on symplectic coherent sta…
Quantum coherence inherently affects the dynamics and the performances of a quantum machine. Coherent control can, at least in principle, enhance the work extraction and boost the velocity of evolution in an open quantum system. Using advanced tools from the calculus of variations and reformulating the control problem in the instantaneous Hamiltonian eigenframe, we develop a general technique for minimizing a wide class of cost functionals when the external control has access to full rotations of the system Hamiltonian. The method is then applied both to time and heat loss minimization problem…
This paper has a dual purpose. One aim is to study the evolution of coherent states in ordinary quantum mechanics. This is done by means of a Hamiltonian approach to the evolution of the parameters that define the state. The stability of the solutions is studied. The second aim is to apply these techniques to the study of the stability of minisuperspace solutions in field theory. For a $λφ^4$ theory we show, both by means of perturbation theory and rigorously by means of theorems of the K.A.M. type, that the homogeneous minisuperspace sector is indeed stable for positive values of the paramete…
We investigate the energy transfer dynamics in a donor-acceptor model by developing a time-local master equation technique based on a variational transformation of the underlying Hamiltonian. The variational transformation allows a minimisation of the Hamiltonian perturbation term dependent on the system parameters, and consequently results in a versatile master equation valid over a range of system-bath coupling strengths, temperatures, and environmental spectral densities. While our formalism reduces to the well-known Redfield, Foerster and polaron forms in the appropriate limits, in general…
Standard Quantum Limit (SQL) of a classical mechanical force detection results from quantum back action perturbing evolution of a mechanical system. In this paper we show that usage of a multidimensional optical transducer may enable a broadband quantum back action evading measurement. We study a corresponding technique for measurement of a resonant signal force acting on a linear mechanical oscillator coupled to an optical system with three optical modes with separation nearly equal to the mechanical frequency. The measurement is performed by optical pumping of the central optical mode and me…
Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davydov D$_1$ trial states, also known as the multi-D$_1$ Ansatz. Accuracy in dynamics simulations is improved significantly over the single D$_1$ Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-D$_1$ Ansatz for various coupling strengths and initial conditions are also systematically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossov…
From coherence to mixedness: a driver of barren plateaus in variational quantum algorithms
School of Mathematics and Computational Science, Shangrao Normal College（上饶师范学院数学与计算科学学院）；School of Physical Science and Intelligent Education, Jiangxi Province Key Laboratory of Applied Optical Technology（江西应用光学技术重点实验室物理科学与智能教育学院）
We propose coherent (`Schrödinger catlike') states adapted to the parity symmetry providing a remarkable variational description of the ground and first excited states of vibron models for finite-($N$)-size molecules. Vibron models undergo a quantum shape phase transition (from linear to bent) at a critical value $ξ_c$ of a control parameter. These trial cat states reveal a sudden increase of vibration-rotation entanglement linear ($L$) and von Neumann ($S$) entropies from zero to $L^{(N)}_{\rm cat}(ξ)\simeq 1-{2}/{\sqrt{πN}}$ [to be compared with $L^{(N)}_{\rm max.}(ξ)=1-{1}/{(N+1)}$] and $S^…
This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.
In this paper, we propose an experimentally viable scheme to enhance the sensitivity of force detection in a hybrid optomechanical setup assisted by squeezed vacuum injection, beyond the standard quantum limit (SQL). The scheme is based on a combination of the coherent quantum noise cancellation (CQNC) strategy with a variational homodyne detection of the cavity output spectrum in which the phase of the local oscillator is optimized. In CQNC, realizing a negative-mass oscillator in the system leads to exact cancellation of the backaction noise from the mechanics due to destructive quantum inte…
We present applications of variational -- wavelet approach to three different models of nonlinear beam motions with underlying collective behaviour: Vlasov-Maxwell-Poisson systems, envelope dynamics, beam-beam model. We have the representation for dynamical variables as a multiresolution (multiscales) expansion via high-localized nonlinear eigenmodes in the base of compactly supported wavelet bases. Numerical modelling demonstrates formation of coherent structures and stable patterns.
Hamiltonians of a wide-spread class of $G_{inv}$-invariant nonlinear quantum models, including multiboson and frequency conversion ones, are expressed as non-linear functions of $sl(2)$ generators. It enables us to use standard variational schemes, based on $sl(2)$ generalized coherent states as trial functions, for solving both spectral and evolution tasks. In such a manner a new analytical expression is found for energy spectra in a mean-field approximation which is beyond quasi-equidistant ones obtained earlier.
We investigate bright solitons in the one-dimensional Schrödinger equation in the framework of an extended variational approach. We apply the latter to the stationary ground state of the system as well as to coherent collisions between two or more solitons. Using coupled Gaussian trial wave functions, we demonstrate that the variational approach is a powerful method to calculate the soliton dynamics. This method has the advantage that it is computationally faster compared to numerically exact grid calculations. In addition, it goes far beyond the capability of analytical ground state solutions…
We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: Kähler and non-Kähler. Traditional variational methods typically require the variational family to be a Kähler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in…
We use spin-coherent states as a time-dependent variational ansatz for a semiclassical description of a large family of Heisenberg models. In addition to common approaches we also evaluate the square variance of the Hamiltonian in terms of coherent states. This quantity turns out to have a natural interpretation with respect to time-dependent solutions of the equations of motion and allows for an estimate of quantum fluctuations in a semiclassical regime. The general results are applied to solitons, instantons and vortices in several one- and two-dimensional models.
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the dimension of the problem space grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm can efficiently find the eigenvalue of a given eigenvector but requires fully coherent evolution. We present an alternative approach that greatly reduces the requirements for coherent evolution and we combine this method with a new approach to state…
Quasiclassical dynamics of trapped ions is characterized by applying the time dependent variational principle (TDVP) on coherent state orbits, in case of quadrupole and octupole combined (Paul and Penning) and radiofrequency (RF) traps. A dequantization algorithm is proposed, by which the classical Hamilton (energy) function associated to the system results as the expectation value of the quantum Hamiltonian on squeezed coherent states. We develop such method and particularize the quantum Hamiltonian for a combined and for a RF trap, with axial symmetry and a RF anharmonic electric potential. …
Coherent states, known as displaced vacuum states, play an important role in quantum information processing, quantum machine learning,and quantum optics. In this article, two ways to digitally prepare coherent states in quantum circuits are introduced. First, we construct the displacement operator by decomposing it into Pauli matrices via ladder operators, i.e., creation and annihilation operators. The high fidelity of the digitally generated coherent states is verified compared with the Poissonian distribution in Fock space. Secondly, by using Variational Quantum Algorithms, we choose differe…
A new proof for the completeness of the coherent states $D(α)\mid f>$ for the Heisenberg Weyl group and the groups $SU(2)$ and $SU(1,1)$ is presented. Generalizations of these results and their consequences are disussed.
We use coherent states as a time-dependent variational ansatz for a semiclassical treatment of the dynamics of anharmonic quantum oscillators. In this approach the square variance of the Hamiltonian within coherent states is of particular interest. This quantity turns out to have natural interpretation with respect to time-dependent solutions of the semiclassical equations of motion. Moreover, our approach allows for an estimate of the decoherence time of a classical object due to quantum fluctuations. We illustrate our findings at the example of the Toda chain.
We present a physically motivated variational wave function for the ground state of the asymmetric quantum Rabi model (AQRM). The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states, and relies only on three variational parameters in the regimes of interest where the squeezing effect becomes negligible. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. We use the variational result to calculate various relevant physical observables of the ground state…
Variational quantum algorithms are promising applications of noisy intermediate-scale quantum (NISQ) computers. These algorithms consist of a number of separate prepare-and-measure experiments that estimate terms in a Hamiltonian. The number of terms can become overwhelmingly large for problems at the scale of NISQ hardware that may soon be available. We approach this problem from the perspective of contextuality, and use unitary partitioning (developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190 (2020)]) to define variational quantum eigensolver procedures in which addit…
We use coherent states as trial states for a variational approach to study a system of a finite number of three-level atoms interacting in a dipolar approximation with a one-mode electromagnetic field. The atoms are treated as semi-distinguishable using different cooperation numbers and representations of SU(3). We focus our analysis on the quantum phases of the system as well as the behavior of the most relevant observables near the phase transitions. The results are computed for all three possible configurations ($Ξ$, $Λ$ and $V$) of the three-level atoms.
Université Paris Cité（巴黎西岱大学）；CNRS, Matériaux et Phénomènes Quantiques（法国国家科学研究中心，物质与现象量子研究所）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）
For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of $SU(2)$ and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of $SU(1,1)$. We obtain the evolution operator in both of the Lie algebras. The number-type eigenstates and the coherent states are constructed group-theoretically for both the time-independent and the time-dependent harmonic oscillators with the singular term. It is shown that the squeeze operator transforms unitarily the time-dependent basis…
Georgia Institute of Technology（佐治亚理工学院）；IBM Research（IBM研究院）；Hon Hai (Foxconn) Research Institute（鸿海（富士康）研究院）
The noisy intermediate-scale quantum (NISQ) devices enable the implementation of the variational quantum circuit (VQC) for quantum neural networks (QNN). Although the VQC-based QNN has succeeded in many machine learning tasks, the representation and generalization powers of VQC still require further investigation, particularly when the dimensionality of classical inputs is concerned. In this work, we first put forth an end-to-end quantum neural network, TTN-VQC, which consists of a quantum tensor network based on a tensor-train network (TTN) for dimensionality reduction and a VQC for functiona…
A system of $N_a$ atoms of $n$-levels interacting dipolarly with $\ell$ modes of electromagnetic field is considered. The energy surface of the system is constructed from the direct product of the coherent states of U$(n)$ in the totally symmetric representation for the matter times the $\ell$ coherent states of the electromagnetic field. A variational analysis shows that the collective region is divided into $\ell$ zones, inside each of which only one mode of the electromagnetic field contributes to the ground state. In consequence, the polychromatic phase diagram for the ground state natural…
Theoretical Guarantees of Variational Quantum Algorithm with Guiding States
Centre for Quantum Software and Information, School of Computer Science, Faculty of Engineering and Information Technology, University of Technology Sydney（悉尼科技大学工程与信息科技学院计算机科学学院量子软件与信息中心）；Centre for Quantum Computation and Communication Technology, University of Technology Sydney（悉尼科技大学量子计算与通信技术研究中心）
Effective low-energy theories represent powerful theoretical tools to reduce the complexity in modeling interacting quantum many-particle systems. However, common theoretical methods rely on perturbation theory, which limits their applicability to weak interactions. Here we introduce the Variational Adiabatic Gauge Transformation (VAGT), a non-perturbative hybrid quantum algorithm that can use nowadays quantum computers to learn the variational parameters of the unitary circuit that brings the Hamiltonian to either its block-diagonal or full-diagonal form. If a Hamiltonian can be diagonalized …
We present an ansatz for the ground states of the Quantum Sherrington-Kirkpatrick model, a paradigmatic model for quantum spin glasses. Our ansatz, based on the concept of generalized coherent states, very well captures the fundamental aspects of the model, including the ground state energy and the position of the spin glass phase transition. It further enables us to study some previously unexplored features, such as the non-vanishing longitudinal field regime and the entanglement structure of the ground states. We find that the ground state entanglement can be captured by a simple ensemble of…
