Eigenvalues Of Hamiltonian Research Articles

Two-sided Lieb–Thirring bounds

We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb–Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of (−Δ + V + M)uM = 1 in Rd; here M∈R is chosen so that the operator is positive. We further prove that the infimum of (uM−1−M) is a lower bound for the ground state energy E0 and derive a simple iteration scheme converging to E0.

Sampling Error Analysis in Quantum Krylov Subspace Diagonalization

Quantum Krylov subspace diagonalization (QKSD) is an emerging method used in place of quantum phase estimation in the early fault-tolerant era, where limited quantum circuit depth is available. In contrast to the classical Krylov subspace diagonalization (KSD) or the Lanczos method, QKSD exploits the quantum computer to efficiently estimate the eigenvalues of large-size Hamiltonians through a faster Krylov projection. However, unlike classical KSD, which is solely concerned with machine precision, QKSD is inherently accompanied by errors originating from a finite number of samples. Moreover, due to difficulty establishing an artificial orthogonal basis, ill-conditioning problems are often encountered, rendering the solution vulnerable to noise. In this work, we present a nonasymptotic theoretical framework to assess the relationship between sampling noise and its effects on eigenvalues. We also propose an optimal solution to cope with large condition numbers by eliminating the ill-conditioned bases. Numerical simulations of the one-dimensional Hubbard model demonstrate that the error bound of finite samplings accurately predicts the experimental errors in well-conditioned regions.

Error-mitigated variational algorithm on a photonic processor.

Our study demonstrates successful error mitigation of indistinguishably-related noise in a quantum photonic processor through the application of the zero-noise extrapolation (ZNE) technique. By measuring observable values at different error levels, we were able to extrapolate toward a noise-free regime. We examined the impact of partial distinguishability of photons in a two-qubit processor implementing the variational quantum eigensolver for a Schwinger Hamiltonian. Our findings highlight the effectiveness of the extrapolation technique in mitigating indistinguishably-related noise and improving the accuracy of the Hamiltonian eigenvalue estimation.

Weakly q-deformed Heisenberg algebra and non-Hermitian Hamiltonians: Application in statistical physics

We have investigated a weakly q-deformed algebra, which leads to non-Hermitian operators. It has been explicitly shown that the harmonic oscillator Hamiltonian is quasi-Hermitian, and that the corresponding physical Hilbert-space metric Θ differs from that obtained in Ref. [1], by hermitizing the position operator. The reality of the Hamiltonian eigenvalues has been illustrated by analytically computing the first order correction to the energy spectrum of the harmonic oscillator (HO). Furthermore, we have shown that this q-deformation leads to a GUP with minimal uncertainties in both position and momentum measurements. Moreover, the physical implications of this q-deformation, have been investigated by studying the thermostatistics of a system of HOs and an ideal gas. For both systems, we computed the partition function, and then we derived some thermodynamic functions. In addition, for the ideal gas, a modified equation of state and a generalized Mayer's equation, consistent with the real behavior of gases, have been established. The obtained results show that the impact of this model would be significant at high temperatures. However, unlike earlier research, we observe that this q-deformation induced the similar corrections regardless of the system under study.

Effective potential between static sources in quenched light-front Yukawa theory

We compute a nonperturbative effective potential between two static fermions in light-front Yukawa theory as a Hamiltonian eigenvalue problem. Fermion pair production is suppressed, to make possible an exact analytic solution in the form of a coherent state of bosons that form clouds around the sources. The effective potential is essentially an interference term between individual clouds. The model is regulated with Pauli-Villars bosons and fermions, to achieve consistent quantization and renormalization of masses and couplings. This extends earlier work on scalar Yukawa theory where Pauli-Villars regularization did not play a central role. The key result is that the nonperturbative solution restores rotational symmetry even though the light-front formulation of Yukawa theory, with its preferred axis, appears antithetical to such a symmetry. Published by the American Physical Society 2024

KMS Dirichlet forms, coercivity and superbounded Markovian semigroups on von Neumann algebras

We introduce a construction of Dirichlet forms on von Neumann algebras M associated to any eigenvalue of the Araki modular Hamiltonian of a faithful normal non-tracial state, providing also conditions by which the associated Markovian semigroups are GNS symmetric. The structure of these Dirichlet forms is described in terms of spatial derivations. Coercivity bounds are proved and the spectral growth is derived. We introduce a regularizing property of positivity preserving semigroups (superboundedness) stronger than hypercontractivity, in terms of the symmetric embedding of M into its standard space L2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(M)$$\\end{document} and the associated noncommutative Lp(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(M)$$\\end{document} spaces. We prove superboundedness for a special class of positivity preserving semigroups and that some of them are dominated by the Markovian semigroups associated to the Dirichlet forms introduced above, for type I factors M. These tools are applied to a general construction of the quantum Ornstein–Uhlembeck semigroups of the Canonical Commutation Relations CCR and some of their non-perturbative deformations.

Depiction of Hamiltonian PT-symmetry

The theory of PT-symmetry describes the non-hermitian Hamiltonian with real energy levels, which means that the Hamiltonian <i>H</i> is invariant neither under parity operator <i>P</i>, nor under time reversal operator <i>T</i>, <i>PTH</i> = <i>H</i>. Whether the Hamiltonian is real and symmetric is not a necessary condition for ensuring the fundamental axioms of quantum mechanics: real energy levels and unitary time evolution. The theory of PT-symmetry plays a significant role in studying quantum physics and quantum information science, Researchers have paid much attention to how to describe PT-symmetry of Hamiltonian. In the paper, we define operator <i>F</i> according to the PT-symmetry theory and the normalized eigenfunction of Hamiltonian. Then we first describe the PT-symmetry of Hamiltonian in dimensionless cases after finding the features of commutator and anti-commutator of operator <i>CPT</i> and operator <i>F</i>. Furthermore, we find that this method can also quantify the PT-symmetry of Hamiltonian in dimensionless case. <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> represents the part of PT-symmetry broken, and <i>J</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> represents the part of PT-symmetry. If <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> = 0, Hamiltonian <i>H</i> is globally PT-symmetric. Once <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> ≠ 0, Hamiltonian <i>H</i> is PT-symmetrically broken. In addition, we propose another method to describe PT-symmetry of Hamiltonian based on real and imaginary parts of eigenvalues of Hamiltonian, to judge whether the Hamiltonian is PT symmetric. Re<i>F</i> = 1/4||(<i>CPTF</i>+<i>F</i>)||CPT represents the sum of squares of real part of the eigenvalue <i>E<sub>n</sub></i> of Hamiltonian <i>H</i>, Im<i>F</i> = 1/4||(<i>CPTF</i>–<i>F</i>)||CPT is the sum of imaginary part of the eigenvalue <i>E<sub>n</sub></i> of a Hamiltonian <i>H</i>. If Im<i>F</i> = 0, Hamiltonian <i>H</i> is globally PT-symmetric. Once Im<i>F</i> ≠ 0, Hamiltonian <i>H</i> is PT-symmetrically broken. Re<i>F</i> = 0 implies that Hamiltonian <i>H</i> is PT-asymmetric, but it is a sufficient condition, not necessary condition. The later is easier to realize in the experiment, but the studying conditions are tighter, and it further requires that <i>CPT</i> <inline-formula><tex-math id="Z-20240108115351">\begin{document}$\phi_n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115351.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115351.png"/></alternatives></inline-formula>(<i>x</i>) = <inline-formula><tex-math id="Z-20240108115401">\begin{document}$\phi_n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115401.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115401.png"/></alternatives></inline-formula>(<i>x</i>). If we only pay attention to whether PT-symmetry is broken, it is simpler to use the latter method. The former method is perhaps better to quantify the PT-symmetrically broken part and the part of local PT-symmetry.

Simultaneous estimation of multiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers

We introduce a multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) method to simultaneously estimate multiple eigenvalues of a quantum Hamiltonian on early fault-tolerant quantum computers. Our theoretical analysis demonstrates that the algorithm exhibits Heisenberg-limited scaling in terms of circuit depth and total cost. Notably, the proposed quantum circuit utilizes just one ancilla qubit, and with appropriate initial state conditions, it achieves significantly shorter circuit depths compared to circuits based on quantum phase estimation (QPE). Numerical results suggest that compared to QPE, the circuit depth can be reduced by around two orders of magnitude under several settings for estimating ground-state and excited-state energies of certain quantum systems.

Degenerate Perturbation Theory for Models of Quantum Field Theory with Symmetries

We consider Hamiltonians of models describing non-relativistic quantum mechanical matter coupled to a relativistic field of bosons. If the free Hamiltonian has an eigenvalue, we show that this eigenvalue persists also for nonzero coupling. The eigenvalue of the free Hamiltonian may be degenerate provided there exists a symmetry group acting irreducibly on the eigenspace. Furthermore, if the Hamiltonian depends analytically on external parameters then so does the eigenvalue and eigenvector. Our result applies to the ground state as well as resonance states. For our results, we assume a mild infrared condition. The proof is based on operator theoretic renormalization. It generalizes the method used in Griesemer and Hasler (Ann Henri Poincaré 10(3):577–621, 2009) to non-degenerate situations, where the degeneracy is protected by a symmetry group, and utilizes Schur’s lemma from representation theory.

An integrable model of a planar tri-atomic molecule

A model of a planar tri-atomic molecule is presented, which is integrable in the Born–Oppenheimer adiabatic approximation. The molecular Hamiltonian is the sum of a nuclear vibrational energy operator and an electronic Hamiltonian, where vibrations of nuclei are defined to be motions with vanishing total angular momentum in the center-of-mass system, and where the electronic Hamiltonian is assumed to be a traceless 2 × 2 Hermitian matrix defined on Ṙ3, the shape space of the planar three-body system. Once an eigenvalue of the electronic Hamiltonian is chosen, vibrational-electronic interaction is introduced through covariant differential operators acting on sections of the eigen-line bundle associated with the chosen eigenvalue. The Hamiltonian for nuclear motion coupled with electronic state is then described in terms of these covariant differential operators together with the chosen eigenvalue as a potential for nuclear motion. The eigenvalues of the nuclear Hamiltonian are evaluated for bound states. In the case that vibrational-electronic interaction is restricted to small vibrational-electronic one around a symmetric configuration of the nuclei, a remark is made on a relation to a well-known Hamiltonian describing the dynamic Jahn–Teller effect for a planar tri-atomic molecule X3.

A bulk-edge correspondence between the second Chern number and the spectral flow

A one-parameter family of semi-quantum Hamiltonians admitting the time-reversal symmetry is defined on the product S2(r1)×S2(r2) of two-spheres of respective radii r1 and r2, and a corresponding one-parameter family of quantum Hamiltonians is defined in terms of the angular momentum operators associated with respective spheres. The second Chern number of the eigen-vector bundle associated with a positive eigenvalue of the semi-quantum Hamiltonian is a piecewise constant function of the parameter with jumps at singular parameter values leading to the emergence of total degeneracy points for the eigenvalues. Correspondingly, the quantum Hamiltonian has edge-state eigenvalues which, as functions of the parameter, pass the zero energy level at the singular parameter values and the bulk-state eigenvalues prove to correspond to eigenvalues of the semi-quantum Hamiltonian. It is shown that the second-Chern number c2(T) at a non-singular parameter value T is in one-to-one correspondence with the spectral flow as a function of the end point of the interval (−∞,T). The bulk-edge correspondence is also shown to hold between the linearized semi-quantum and quantum systems, where the semi-quantum Hamiltonian is linearized at each of the degeneracy points of the eigenvalues and quantized to define linear quantum Hamiltonians. On this level, the spectral flow of each linear quantum Hamiltonian with a parameter μ proves to coincide with the index of the linear quantum Hamiltonian with μ=0.

A method for fast calculating the electronic states in 2D quantum structures based on AIIIBV nitrides

The paper presents a method for fast calculating the electronic states in two-dimensional quantum structures based on AIIIBV nitrides. The method is based on the representation of electronic states in the form of a linear combination of bulk wave functions of materials, from which quantum structures are made. The parameters and criteria for the selection of bulk wave functions that provides fast convergence of the numerical procedures for calculating the eigenvalues of the quantum Hamiltonian have been considered. The results of the calculations have been given both for one polar InGaN/GaN quantum well and for a system of several quantum wells. Being based on the full band structure of AIIIBV nitrides with a wurtzite-type crystal lattice, the proposed approach takes into account the states far from the center of the Brillouin zone, while preserving the computational efficiency of traditional methods of envelope function in approximating the effective mass.

Dynamical approach to shortcuts to adiabaticity for general two-level non-Hermitian systems

In this Letter, we apply a class of general two-level non-Hermitian Hamiltonians whose off-diagonal elements are neither equal nor conjugate of each other to study the dynamical approach to shortcut to adiabaticity which is based on engineering the Hamiltonian and dynamical properties of the system to remove any unwanted probability amplitude or implant the desired probability amplitude. Using the special characteristics that one of the eigenvalues of the general two-level non-Hermitian Hamiltonian is real and the other is complex, we can decay (amplify) the population exponentially in the undesired (desired) eigenstate, and maintain the conservation of probability amplitude of the other eigenstate. Applying this approach to general two-level non-Hermitian systems, we find that the efficient population transfer can be achieved by our method, which works even in almost instantaneous manner and has a lower cost compared with the common approach of shortcut to adiabaticity.

Exact and efficient Lanczos method on a quantum computer

We present an algorithm that uses block encoding on a quantum computer to exactly construct a Krylov space, which can be used as the basis for the Lanczos method to estimate extremal eigenvalues of Hamiltonians. While the classical Lanczos method has exponential cost in the system size to represent the Krylov states for quantum systems, our efficient quantum algorithm achieves this in polynomial time and memory. The construction presented is exact in the sense that the resulting Krylov space is identical to that of the Lanczos method, so the only approximation with respect to the exact method is due to finite sample noise. This is possible because, unlike previous quantum Krylov methods, our algorithm does not require simulating real or imaginary time evolution. We provide an explicit error bound for the resulting ground state energy estimate in the presence of noise. For our method to be successful efficiently, the only requirement on the input problem is that the overlap of the initial state with the true ground state must be Ω(1/poly(n)) for n qubits.

Topological insulators and geometry of vector bundles

For a long time, band theory of solids has focused on the energy spectrum, or Hamiltonian eigenvalues. Recently, it was realized that the collection of eigenvectors also contains important physical information. The local geometry of eigenspaces determines the electric polarization, while their global twisting gives rise to the metallic surface states in topological insulators. These phenomena are central topics of the present notes. The shape of eigenspaces is also responsible for many intriguing physical analogies, which have their roots in the theory of vector bundles. We give an informal introduction to the geometry and topology of vector bundles and describe various physical models from this mathematical perspective.

Fluorescence in quantum dynamics: Accurate spectra require post-mean-field approaches.

Real time modeling of fluorescence with vibronic resolution entails the representation of the light-matter interaction coupled to a quantum-mechanical description of the phonons and is therefore a challenging problem. In this work, taking advantage of the difference in timescales characterizing internal conversion and radiative relaxation-which allows us to decouple these two phenomena by sequentially modeling one after the other-we simulate the electron dynamics of fluorescence through a master equation derived from the Redfield formalism. Moreover, we explore the use of a recent semiclassical dissipative equation of motion [C. M. Bustamante et al., Phys. Rev. Lett. 126, 087401 (2021)], termed coherent electron electric-field dynamics (CEED), to describe the radiative stage. By comparing the results with those from the full quantum-electrodynamics treatment, we find that the semiclassical model does not reproduce the right amplitudes in the emission spectra when the radiative process involves the de-excitation to a manifold of closely lying states. We argue that this flaw is inherent to any mean-field approach and is the case with CEED. This effect is critical for the study of light-matter interaction, and this work is, to our knowledge, the first one to report this problem. We note that CEED reproduces the correct frequencies in agreement with quantum electrodynamics. This is a major asset of the semiclassical model, since the emission peak positions will be predicted correctly without any prior assumption about the nature of the molecular Hamiltonian. This is not so for the quantum electrodynamics approach, where access to the spectral information relies on knowledge of the Hamiltonian eigenvalues.

Conservation laws of optical Bloch equations for the tripod scheme - contribution of double dark-state

Results are presented for the conservation laws (CLs) of the optical Bloch equations (OBEs) for the 4-level tripod atomic scheme under the action of three laser fields. CLs are constructed for OBEs that include the spontaneous emission, collisional broadening and time of flight relaxation, and also without these effects, i.e. for the Liouville von Neumann equation (LNE). We used the direct method to construct CLs as the linear combinations of the density matrix elements and their products. These CLs were in turn used to obtain another set of CLs that explicitly depend on time and stand with relaxation due to the time of flight in OBEs. We also constructed CLs for OBEs representing special case of the tripod system that supports the coherent population trapping (CPT), the phenomenon characterized by the ‘dark-state’. Tripod scheme has two dark-states with the same Hamiltonian eigenvalue. As we explain, this peculiarity of tripod scheme contributes to additional CLs of both LNE and OBEs with all relaxation and decoherence effects included, when CPT is present.

Eigenstate thermalization hypothesis in two-dimensional XXZ model with or without SU(2) symmetry.

We investigate the eigenstate thermalization properties of the spin-1/2 XXZ model in two-dimensional rectangular lattices of size L_{1}×L_{2} under periodic boundary conditions. Exploiting the symmetry property, we can perform an exact diagonalization study of the energy eigenvalues up to system size 4×7 and of the energy eigenstates up to 4×6. Numerical analysis of the Hamiltonian eigenvalue spectrum and matrix elements of an observable in the Hamiltonian eigenstate basis supports that the two-dimensional XXZ model follows the eigenstate thermalization hypothesis. When the spin interaction is isotropic, the XXZ model Hamiltonian conserves the total spin and has SU(2) symmetry. We show that the eigenstate thermalization hypothesis is still valid within each subspace where the total spin is a good quantum number.

Trace minimization method via penalty for linear response eigenvalue problems

<p style='text-indent:20px;'>In various applications, such as the computation of energy excitation states of electrons and molecules, and the analysis of interstellar clouds, the linear response eigenvalue problem, which is a special type of the Hamiltonian eigenvalue problem, is frequently encountered. However, traditional eigensolvers may not be applicable to this problem owing to its inherently large scale. In fact, we are usually more interested in computing some of the smallest positive eigenvalues. To this end, a trace minimum principle optimization model with orthogonality constraint has been proposed. On this basis, we propose an unconstrained surrogate model called trace minimization via penalty, and we establish its equivalence with the original constrained model, provided that the penalty parameter is larger than a certain threshold. By avoiding the orthogonality constraint, we can use a gradient-type method to solve this model. Specifically, we use the gradient descent method with Barzilai–Borwein step size. Moreover, we develop a restarting strategy for the proposed algorithm whereby higher accuracy and faster convergence can be achieved. This is verified by preliminary experimental results.</p>

Finiteness of entanglement entropy in collective field theory

We explore the question of finiteness of the entanglement entropy in gravitational theories whose emergent space is the target space of a holographic dual. In the well studied duality of two-dimensional non-critical string theory and c = 1 matrix model, this question has been studied earlier using fermionic many-body theory in the space of eigenvalues. The entanglement entropy of a subregion of the eigenvalue space, which is the target space entanglement in the matrix model, is finite, with the scale being provided by the local Fermi momentum. The Fermi momentum is, however, a position dependent string coupling, as is clear in the collective field theory formulation. This suggests that the finiteness is a non-perturbative effect. We provide evidence for this expectation by an explicit calculation in the collective field theory of matrix quantum mechanics with vanishing potential. The leading term in the cumulant expansion of the entanglement entropy is calculated using exact eigenstates and eigenvalues of the collective Hamiltonian, yielding a finite result, in precise agreement with the fermion answer. Treating the theory perturbatively, we show that each term in the perturbation expansion is UV divergent. However the series can be resummed, yielding the exact finite result. Our results indicate that the finiteness of the entanglement entropy for higher dimensional string theories is non-perturbative as well, with the scale provided by Newton’s constant.