Combined use of translational and spin-rotational invariance for spin systems

Heitmann, Tjark; Schnack, Jürgen

doi:10.1103/physrevb.99.134405

Cited by 18 publications

(9 citation statements)

References 73 publications

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“…On classical computers, with a sophisticated and elaborated algorithm that incorporates the spatial symmetry, such as the lattice translational symmetry, and S z conservation [73,92], one can obtain the numerically exact ground state of the spin-1/2 Heisenberg model up to 50 spins [93], which is far larger than the case of 16 qubits studied here. However, S 2 conservation is usually not implemented because the programming of a total-spin-preserved code is, although possible [94,95], not easy and often computationally demanding on classical computers. We expect that the circuit that operates eSWAP gates on a singlet-pair product state or on a pair-product state with higher spin-quantum numbers might be useful for studying spin-liquid states including the RVB state as well as excited states on quantum computers in the near future.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Symmetry-adapted variational quantum eigensolver

Seki,

Shirakawa,

Yunoki

2019

Preprint

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We propose a scheme to restore spatial symmetry of Hamiltonian in the variational-quantum-eigensolver (VQE) algorithm for which quantum circuit structures used usually break the Hamiltonian symmetry. The symmetry-adapted VQE scheme introduced here simply applies the projection operator, which is hermitian but not unitary, to restore the spatial symmetry in a desired irreducible representation of the spatial group. The entanglement of a quantum state is still represented in a quantum circuit but the nonunitarity of the projection operator is treated classically as postprocessing in the VQE framework. By numerical simulations for a spin-1/2 Heisenberg model on a one-dimensional ring, we demonstrate that the symmetry-adapted VQE scheme with a shallower quantum circuit can achieve significant improvement in terms of the fidelity of the ground state and has a great advantage in terms of the ground state energy with decent accuracy, as compared to the non-symmetry-adapted VQE scheme. We also demonstrate that the present scheme can approximate low-lying excited states that can be specified by symmetry sectors.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Symmetry-adapted variational quantum eigensolver

Seki,

Shirakawa,

Yunoki

2019

Preprint

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“…Due to impractical implementations, the total SU(2) spin symmetry of the Heisenberg model is usually not used in numerical approaches [39,97], except the work from Flocke and Karwowski [11,[98][99][100][101][102] using the symmetric group approach (SGA) [103][104][105][106][107], spin-symmetry adapted MPS/DMRG studies [108][109][110][111][112][113][114][115][116] and occasional ED [97,117,118] and real-space renormalization group studies [119]. Nevertheless the theoretical advantages of using a description conserving both total spin projection, m s , the total spin, S, are striking: (a) further reduction of the Hilbert space size (by additional block diagonalization of Ĥ), (b) optimization of electronic states of desired spin, and (c) separation of nearly degenerate states of different total spin.…”

Section: The Heisenberg Modelmentioning

confidence: 99%

Combined unitary and symmetric group approach applied to low-dimensional spin systems

Dobrautz,

Katukuri,

Bogdanov

et al. 2021

Preprint

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A novel combined unitary and symmetric group approach is used to study the spin-1 2 Heisenberg model and related Fermionic systems in a spin-adapted representation, using a linearlyparameterised Ansatz for the many-body wave function. We show that a more compact ground state wave function representation is obtained when combining the symmetric group, Sn, in the form of permutations of the underlying lattice site ordering, with the cumulative spin-coupling based on the unitary group, U(n). In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, in this work exemplified by a one-dimensional hydrogen chain. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.

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“…While the action of H on |ψ can also be calculated on the fly, we save the sparse Hamiltonian matrix for the sake of run time. Moreover, symmetries of the system can be exploited in order to split the problem into smaller subproblems and to further reduce the computational effort [80]. In this paper, we exploit the particle number (magnetization) conservation for both particle species (legs) separately.…”

Section: Time Evolution Via Pure-state Propagationmentioning

confidence: 99%

Density dynamics in the mass-imbalanced Hubbard chain

et al. 2020

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We consider two mutually interacting fermionic particle species on a one-dimensional lattice and study how the mass ratio η between the two species affects the (equilibration) dynamics of the particles. Focusing on the regime of strong interactions and high temperatures, two well-studied points of reference are given by (i) the case of equal masses η = 1, i.e., the standard Fermi-Hubbard chain, where initial nonequilibrium density distributions are known to decay, and (ii) the case of one particle species being infinitely heavy, η = 0, leading to a localization of the lighter particles in an effective disorder potential. Given these two opposing cases, the dynamics in the case of intermediate mass ratios 0 < η < 1 is of particular interest. To this end, we study the real-time dynamics of pure states featuring a sharp initial nonequilibrium density profile. Relying on the concept of dynamical quantum typicality, the resulting nonequilibrium dynamics can be related to equilibrium correlation functions. Summarizing our main results, we observe that diffusive transport occurs for moderate values of the mass imbalance and manifests itself in a Gaussian spreading of real-space density profiles and an exponential decay of density modes in momentum space. For stronger imbalances, we provide evidence that transport becomes anomalous on intermediate timescales, and in particular, our results are consistent with the absence of strict localization in the long-time limit for any η > 0. Based on our numerical analysis, we provide an estimate for the "lifetime" of the effective localization as a function of η.

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Combined use of translational and spin-rotational invariance for spin systems

Cited by 18 publications

References 73 publications

Symmetry-adapted variational quantum eigensolver

Symmetry-adapted variational quantum eigensolver

Combined unitary and symmetric group approach applied to low-dimensional spin systems

Density dynamics in the mass-imbalanced Hubbard chain

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