Symmetry Subspaces Research Articles

Ansatz optimization of the variational quantum eigensolver tested on the atomic Anderson model

We present a detailed analysis and optimization of the variational quantum algorithms required to find the ground state of a correlated electron model, using several types of variational ansatz. Specifically, we apply our approach to the atomic limit of the Anderson model, which is widely studied in condensed matter physics since it can simulate fundamental physical phenomena, ranging from magnetism to superconductivity. The method is developed by presenting efficient state preparation circuits that exhibit total spin, spin projection, particle number and time-reversal symmetries. These states contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace allowing to avoid irrelevant sectors of Hilbert space. Then, we show how to construct quantum circuits, providing explicit decomposition and gate count in terms of standard gate sets. We test these quantum algorithms looking at ideal quantum computer simulations as well as implementing quantum noisy simulations. We finally perform an accurate comparative analysis among the approaches implemented, highlighting their merits and shortcomings.

Quantum phase transition in skewed ladders: an entanglement entropy and fidelity study

Entanglement entropy (EE) of a state is a measure of correlation or entanglement between two parts of a composite system and it may show appreciable change when the ground state (GS) undergoes a qualitative change in a quantum phase transition (QPT). Therefore, the EE has been extensively used to characterise the QPT in various correlated Hamiltonians. Similarly fidelity also shows sharp changes at a QPT. We characterized the QPT of frustrated antiferromagnetic Heisenberg spin-1/2 systems on 3/4, 3/5 and 5/7 skewed ladders using the EE and fidelity analysis. It is noted that all the non-magnetic to magnetic QPT boundary in these systems can be accurately determined using the EE and fidelity, and the EE exhibits a discontinuous change, whereas fidelity shows a sharp dip at the transition points. It is also noted that in case of the degenerate GS, the unsymmetrized calculations show wild fluctuations in the EE and fidelity even without actual phase transition, however, this problem is resolved by calculating the EE and the fidelity in the lowest energy state of the symmetry subspaces, to which the degenerate states belong.

Symmetry-induced many-body quantum interference in chaotic bosonic systems: an augmented truncated Wigner method

Although highly successful, the truncated Wigner approximation (TWA) does not account for genuine many-body (MB) quantum interference between different solutions of the mean-field equations of a bosonic MB system. This renders the TWA essentially classical, where a large number of particles formally takes the role of the inverse of Planck’s constant ℏ. The failure to describe genuine interference phenomena, such as localization and scarring in Fock space, can be seen as a virtue of this quasiclassical method, which thereby allows one to identify genuine quantum effects when being compared with ‘exact’ quantum calculations that do not involve any a priori approximation. A rather prominent cause for such quantum effects that are not accounted for by the TWA is the constructive interference between the contributions of symmetry-related trajectories, which would occur in the presence of discrete symmetries provided the phase-space distribution of the initial state and the observable to be evaluated feature a strong localization about the corresponding symmetry subspaces. Here we show how one can conceive an augmented version of the TWA which can account for this particular effect. This augmented TWA effectively amounts to complementing conventional TWA calculations by separate truncated Wigner simulations that are restricted to symmetric subspaces and involve weight factors that account for the dynamical stability of sampling trajectories with respect to perpendicular deviations from those subspaces. We illustrate the validity of this method at pre- as well as post-Ehrenfest time scales in prototypical Bose–Hubbard systems displaying chaotic classical dynamics, where it also reveals the existence of additional MB interference effects.

Variational Quantum Circuits to Prepare Low Energy Symmetry States

We explore how to build quantum circuits that compute the lowest energy state corresponding to a given Hamiltonian within a symmetry subspace by explicitly encoding it into the circuit. We create an explicit unitary and a variationally trained unitary that maps any vector output by ansatz A(α→) from a defined subspace to a vector in the symmetry space. The parameters are trained varitionally to minimize the energy, thus keeping the output within the labelled symmetry value. The method was tested for a spin XXZ Hamiltonian using rotation and reflection symmetry and H2 Hamiltonian within Sz=0 subspace using S2 symmetry. We have found the variationally trained unitary gives good results with very low depth circuits and can thus be used to prepare symmetry states within near term quantum computers.

Probing Symmetries of Quantum Many-Body Systems through Gap Ratio Statistics

The statistics of gap ratios between consecutive energy levels is a widely used tool, in particular in the context of many-body physics, to distinguish between chaotic and integrable systems, described respectively by Gaussian ensembles of random matrices and Poisson statistics. In this work we extend the study of the gap ratio distribution P(r) to the case where discrete symmetries are present. This is important, since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place. Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks. We check our formulae against simulations from large random matrices, showing excellent agreement. We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically-driven spin systems. In all these models the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces. We finally discuss the relevance of our analysis for existing results in the literature, as well as its practical usefulness, and point out possible future applications and extensions.

Periodic orbits exhibit oblique stripe patterns in plane Couette flow

Spatiotemporally chaotic dynamics of transitional plane Couette flow may give rise to regular turbulent-laminar stripe patterns with a large-scale pattern wavelength and an oblique orientation to the laminar flow direction. How the stripe pattern forms remains an open problem. To understand stripe pattern spatiotemporal dynamics we identify unstable periodic orbits of the 3D Naiver-Stokes equations that show oblique large-scale spatial amplitude modulation and a temporal evolution of standing waves. The unstable periodic orbits are embedded in the edge of chaos in a symmetry subspace of plane Couette flow and thereby may mediate transition to and from turbulent flows with oblique patterns.

Quantum phases of spin-1 system on 3/4 and 3/5 skewed ladders

We study the quantum phase transitions of frustrated antiferromagnetic Heisenberg spin-1 systems on the 3/4 and 3/5 skewed two leg ladder geometries. These systems can be viewed as arising by periodically removing rung bonds from a zigzag ladder. We find that in large systems, the ground state (gs) of the 3/4 ladder switches from a singlet to a magnetic state for J1≥1.82; the gs spin corresponds to the ferromagnetic alignment of effective S=2 objects on each unit cell. The gs of antiferromagnetic exchange Heisenberg spin-1 system on a 3/5 skewed ladder is highly frustrated and has spiral spin arrangements. The amplitude of the spin density wave in the 3/5 ladder is significantly larger compared to that in the magnetic state of the 3/4 ladder. The gs of the system switches between singlet state and low spin magnetic states multiple times on tuning J1 in a finite size system. The switching pattern is nonmonotonic as a function of J1 and depends on the system size. It appears to be the consequence of a higher J1 favoring a higher spin magnetic state and the finite system favoring a standing spin wave. For some specific parameter values, the magnetic gs in the 3/5 system is doubly degenerate in two different mirror symmetry subspaces. This degeneracy leads to spontaneous spin-parity and mirror symmetry breaking, giving rise to spin current in the gs of the system.

Quantization and interference of a quantum billiard with fourfold rotational symmetry.

Systems with discrete symmetries are highly important in quantum mechanics. We consider a two-dimensional quantum billiard with fourfold rotational symmetry, where the eigenenergies and eigenstates can be grouped into four symmetry subspaces. Unlike the threefold rotational symmetry case, here the interference of the scarring states on the fundamental domain orbits (FDO) is clean, that they either interfere constructively or annihilate completely. We shall show the complex behavior of the interference revealed in the length spectra for eigenenergies that belong to a particular symmetry subspace and combinations of different symmetry subspaces. We then provide detailed analysis of phase accumulation along the FDOs, which are the keys to determine the interference and could explain the enhancement or annihilation of the peaks well. The quantization condition for the scarring states belonging to different symmetry subspaces is discussed and used to reveal the time-reversal symmetry broken for a particular subspace. An experimental scheme to observe such complex behaviors is also proposed.

Nonlinear form-finding of symmetric cable–strut structures using stiffness submatrices associated with full symmetry subspace

An efficient and nonlinear form-finding method is proposed for symmetric cable–strut structures with complex geometry or many nodes. Expressed in the symmetry-adapted coordinate system, the first block matrix of the symmetry-adapted tangent stiffness matrix is extracted using the full symmetry subspace, which is much smaller-sized and associated with the first irreducible representation of a symmetry group. Then, this stiffness submatrix and the principle of minimum potential energy are adopted for the fast but stable convergence of the initial configuration to the stable configuration. During the form-finding process, the generalized inverse of a matrix and modification for the minimum eigenvalues are employed, to guarantee the positive definiteness of the stiffness submatrix. The form-finding process can start from an arbitrary initial configuration, whereas only certain symmetry group, and the connectivity pattern and the initial lengths of the members should be given in advance. A few numerical examples are illustrated to show the efficiency and accuracy of the form-finding method for cable–strut structures with complex geometry and different symmetry.

Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm

The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediate-scale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and time-reversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the {H}_{2} and LiH molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth.

Stiffness contributions of tension structures evaluated from the levels of components and symmetry subspaces

Tension structures are kinematically indeterminate and contain flexible cables. Geometric stiffness of each member induced by its internal force and involved symmetry subspace play an important role in maintaining structural stability. In this paper, stiffness contributions of tension structures are systematically evaluated from the levels of both structural components and symmetry subspaces. Starting from the level of components, we transform the modified elastic stiffness matrix, the geometric stiffness matrix and the tangent stiffness matrix of a structure in combinational contributions from all the members. Then, on the basis of symmetry-adapted stiffness blocks obtained by group theory, all the stiffness characters and directions are neatly extracted from different symmetry subspaces. Three types of stiffness contribution indexes are denoted, and a Geiger cable dome is presented as the illustrative example. It shows that the contribution index evaluated from the level of components clarifies the importance of different members. On the other hand, contribution indexes evaluated from different symmetry subspaces can predict symmetry of lower-order stiffness, and reveal the importance of prestresses in different symmetry spaces.

Symmetry representations and elastic redundancy for members of tensegrity structures

Unexpectable damage of a member may cause a significant change in the load-bearing capacity or collapse of a tensegrity structure, thus redundancy analysis is important. Nevertheless, conventional approaches generally neglect the inherent symmetry and material properties. Here, we propose a method for symmetry representations and elastic redundancy for members of tensegrity structures. The symmetry and member stiffness are taken into account. To evaluate the contribution of each member to the total static indeterminacy, irreducible representation and full symmetry subspace of every type of members are computed. Then, distributed static indeterminacy of the members can be independently obtained by a simplified process. The accuracy and efficiency of the presented method are demonstrated through examples of some well-known tensegrity structures with different symmetries. This method can offer reasonable design on tensegrity structures, which are assembled by many cables and struts with identical sensitivity to initial elongations and equal importance to load-bearing capacity.

Group-theoretical form-finding of cable-strut structures based on irreducible representations for rigid-body translations

A group-theoretical method is proposed for form-finding of symmetric cable-strut structures. Only the given topology and structural symmetry are necessary for implementing this form-finding process. Irreducible representations for rigid-body translations along d directions are used to get the corresponding symmetry subspaces, which transform the original force density matrix into d small-sized block matrices. Therefore, significant simplification on the form-finding process is achieved. Numerical decomposition of these blocks is performed to find analytical sets of force densities and nodal coordinates, which satisfy the minimum required rank deficiencies of the blocks and indicate meaningful null spaces. On condition that the tangent stiffness matrix of the obtained configuration is positive definite, a non-degenerate d-dimensional stable cable-strut structure is built. Some examples specified by Cnv or Dn symmetry are presented. It shows that the proposed method is effective for searching feasible configurations of cable-strut structures with specific symmetries.

A group-theoretic approach to the mobility and kinematic of symmetric over-constrained structures

Some symmetric structures can be finite mechanisms with at least one degree-of-freedom even though they are over-constrained. Exploring group representation theory, this study presents a group-theoretic approach to the mobility of symmetric over-constrained structures, which have both internal mechanism modes and self-stress states. To establish the compatibility equations for the structures, the constraint equations and elementary compatibility matrix for a general link member are given. Then, symmetry subspaces associated with different irreducible representations are established, which are important to derive the symmetry-adapted coordinate system. Using the Great orthogonality theorem, the original compatibility matrix is decomposed into a few independent block matrices along the diagonal. Investigation on the null-space and the left null-space of these blocks enables us to present the symmetry representations for the internal mechanisms and self-stress states. Mobility of some example structures is studied, including 3D C3v symmetric Bricard mechanisms, a 2D C4v symmetric retractable structure, and a 3D Ih symmetric polyhedral structure. The generality and effectiveness of this group-theoretic approach are validated. Kinematic behavior of the numerical models agrees with that of the physical models assembled by LEGO components. It turns out that all the illustrative structures can be transformable while retaining full symmetry.

Mobility of symmetric deployable structures subjected to external loads

Deployable structures can exhibit remarkable and continuous geometric transformations, however, they are likely to be rigid under certain external loads. This study adopts group theory to evaluate the mobility of symmetric deployable structures under external loads. Mobility analysis is expressed as determining the orthogonality of internal mechanism modes and external loads. Based on the symmetry groups, both the mechanism modes and external loads are associated with specific symmetry subspaces. Thus, it can be evaluated whether the external loads stiffen all the internal mechanism modes. Illustrative examples on pin-jointed structures and over-constrained mechanisms are given to verify the proposed method. It turns out that the product of the internal mechanisms and external loads is equivalent to that of the mechanisms and loads in the symmetry subspaces associated with different irreducible representations. A deployable structure will be immobile under external loads if symmetry order of the loads is higher than that of the mechanisms. In addition, the structure will be immobile, if the internal mechanisms and the external loads are equisymmetric and orthogonal to each other. The conclusions agree with published results, and need much fewer computations. The proposed method is efficient and applicable to most symmetric deployable structures.

The electronic states of a double carbon vacancy defect in pyrene: a model study for graphene.

The electronic states occurring in a double vacancy defect for graphene nanoribbons have been calculated in detail based on a pyrene model. Extended ab initio calculations using the MR configuration interaction (MRCI) method have been performed to describe in a balanced way the manifold of electronic states derived from the dangling bonds created by initial removal of two neighboring carbon atoms from the graphene network. In total, this study took into account the characterization of 16 electronic states (eight singlets and eight triplets) considering unrelaxed and relaxed defect structures. The ground state was found to be of (1)Ag character with around 50% closed shell character. The geometry optimization process leads to the formation of two five-membered rings in a pentagon-octagon-pentagon (5-8-5) structure. The closed shell character increases thereby to ∼70%; the analysis of unpaired density shows only small contributions confirming the chemical stability of that entity. For the unrelaxed structure the first five excited states ((3)B3g, (3)B2u, (3)B1u, (3)Au and (1)Au) are separated from the ground state by less than 2.5 eV. For comparison, unrestricted density functional theory (DFT) calculations using several types of functionals have been performed within different symmetry subspaces defined by the open shell orbitals. Comparison with the MRCI results gave good agreement in terms of finding the (1)Ag state as a ground state and in assigning the lowest excited states. Linear interpolation curves between the unrelaxed and relaxed defect structures also showed good agreement between the two classes of methods opening up the possibilities of using extended nanoflakes for multistate investigations at the DFT level.

Multi-electron systems in strong magnetic fields II: A fixed-phase diffusion quantum Monte Carlo application based on trial functions from a Hartree–Fock–Roothaan method

We present a quantum Monte Carlo application for the computation of energy eigenvalues for atoms and ions in strong magnetic fields. The required guiding wave functions are obtained with the Hartree–Fock–Roothaan code described in the accompanying publication (Schimeczek and Wunner, 2014). Our method yields highly accurate results for the binding energies of symmetry subspace ground states and at the same time provides a means for quantifying the quality of the results obtained with the above-mentioned Hartree–Fock–Roothaan method. Program summaryProgram title: MantecaCatalogue identifier: AETV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 72284No. of bytes in distributed program, including test data, etc.: 604948Distribution format: tar.gzProgramming language: C++.Computer: Cluster of 1–∼500 HP Compaq dc5750.Operating system: Linux.Has the code been vectorized or parallelized?: Yes. Code includes MPI directives.RAM: 500 MB per nodeClassification: 2.1.External routines: Boost::Serialization, Boost::MPI, LAPACK BLASNature of problem:Quantitative modelings of features observed in the X-ray spectra of isolated neutron stars are hampered by the lack of sufficiently large and accurate databases for atoms and ions up to the last fusion product iron, at high magnetic field strengths. The predominant amount of line data in the literature has been calculated with Hartree–Fock methods, which are intrinsically restricted in precision. Our code is intended to provide a powerful tool for calculating very accurate energy values from, and thereby improving the quality of, existing Hartree–Fock results.Solution method:The Fixed-phase quantum Monte Carlo method is used in combination with guiding functions obtained in Hartree–Fock calculations. The guiding functions are created from single-electron orbitals ψi which are either products of a wave function in the z-direction (the direction of the magnetic field) and an expansion of the wave function perpendicular to the direction of the magnetic field in terms of Landau states, ψi(ρ,φ,z)=Pi(z)∑n=0NLtinϕni(ρ,φ), or a full two-dimensional expansion using separate z-wave functions for each Landau level, i.e. ψi(ρ,φ,z)=∑n=0NLPni(z)ϕni(ρ,φ). In the first form, the tin are expansion coefficients, and the expansion is cut off at some maximum Landau level quantum number NL. In the second form, the expansion coefficients are contained in the respective Pni.Restrictions:The method itself is very flexible and not restricted to a certain interval of magnetic field strengths. However, it is only variational for the lowest-lying state in each symmetry subspace and the accompanying Hartree–Fock method can only obtain guiding functions in the regime of strong magnetic fields.Unusual features:The program needs approximate wave functions computed with another method as input.Running time:1 min–several days. The example provided takes approximately 50 min to run on 1 processor.

Efficient algorithm for asymptotics-based configuration-interaction methods and electronic structure of transition metal atoms

Asymptotics-based configuration-interaction (CI) methods [G. Friesecke and B. D. Goddard, Multiscale Model. Simul. 7, 1876 (2009)] are a class of CI methods for atoms which reproduce, at fixed finite subspace dimension, the exact Schrödinger eigenstates in the limit of fixed electron number and large nuclear charge. Here we develop, implement, and apply to 3d transition metal atoms an efficient and accurate algorithm for asymptotics-based CI. Efficiency gains come from exact (symbolic) decomposition of the CI space into irreducible symmetry subspaces at essentially linear computational cost in the number of radial subshells with fixed angular momentum, use of reduced density matrices in order to avoid having to store wave functions, and use of Slater-type orbitals (STOs). The required Coulomb integrals for STOs are evaluated in closed form, with the help of Hankel matrices, Fourier analysis, and residue calculus. Applications to 3d transition metal atoms are in good agreement with experimental data. In particular, we reproduce the anomalous magnetic moment and orbital filling of chromium in the otherwise regular series Ca, Sc, Ti, V, Cr.

Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems

The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations.

Reduced Gutzwiller formula with symmetry: Case of a finite group

We consider a classical Hamiltonian H on R2d, invariant by a finite group of symmetry G, whose Weyl quantization Ĥ is a self-adjoint operator on L2(Rd). If χ is an irreducible character of G, we investigate the spectrum of its restriction Ĥχ to the symmetry subspace Lχ2(Rd) of L2(Rd) coming from the decomposition of Peter-Weyl. We give reduced semiclassical asymptotics of a regularized spectral density describing the spectrum of Ĥχ near a noncritical energy E∊R. If ΣE≔{H=E} is compact, assuming that periodic orbits are nondegenerate in ΣE∕G, we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space ΣE∕G appear. The method is based upon the use of coherent states, whose propagation was given in the work of Combescure and Robert.