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Physical Review C

Numerical realization of the fermion dynamical symmetry model

Hua Wu, Mathilde Vallières

Published Mar 1, 1989• 11 citations
Publisher

Abstract

A new and simple algorithm is presented to solve the fermion dynamical symmetry model numerically in the heritage-zero subspace for neutron-proton systems. The algorithm is based on a shell-model approach using the fermion pair operators as building blocks for the many-body wave functions. The same algorithm applies to all possible neutron and proton Hamiltonians. The results of all known dynamical symmetries are reproduced. Typical approximate vibrational and rotational spectra are presented for cases where dynamical symmetries cannot be found from group chain decompositions.

DOI
10.1103/physrevc.39.1066

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A new and simple algorithm is presented to solve the fermion dynamical symmetry model numerically in the heritage-zero subspace for neutron-proton systems. The algorithm is based on a shell-model approach using the fermion pair operators as building blocks for the many-body wave functions. The same algorithm applies to all possible neutron and proton Hamiltonians. The results of all known dynamical symmetries are reproduced. Typical approximate vibrational and rotational spectra are presented for cases where dynamical symmetries cannot be found from group chain decompositions.
A method for the calculation of translationally invariant wave functions for systems of identical fermions with arbitrary potential of pair interaction is developed.It is based on the well-known result that the essential dynamic part of Hamiltonian for the system of identical particles is the reduced Hamiltonian operator describing a relative movement of two particles inside the system.The eigenfunctions of this operator take into account all correlations caused by interaction.These eigenfunctions are basic elements for building the components (i.e., the functions with a lower degree of antisymmetry) used to construct the total antisymmetric wave function of the system.The main problem of this approach appears to be the antisymmetrization of the components.The developed universal algorithm for antisymmetrization gives a possibility to carry out this operation in a simple way and to keep numerical approximations under control.
Simulating noninteracting fermion systems is a common task in computational many-body physics. In the absence of translational symmetries, modeling free fermions on N modes usually requires poly ( N ) computational resources. While often moderate, these costs can be prohibitive in practice when large systems are considered. We present several free fermion problems that can be solved by a quantum algorithm with substantially reduced computational costs. The memory costs are exponentially improved, poly log ( N ) . The run-time improvement, compared to the best known classical algorithms, is either exponential or significantly polynomial, depending on the geometry of the problem. The simulation of free fermion dynamics belongs to the BQP-hard complexity class (i.e., as hard as any (decision) problem that can be solved on a quantum computer). This implies (under standard assumptions) that our algorithm yields an exponential speedup for classical algorithm at least for some geometries. The key technique in our algorithm is the block encoding of objects such as correlation matrices and Green's functions into a unitary. We demonstrate how such unitaries can be efficiently realized as quantum circuits, in the context of dynamics and thermal states of tight-binding Hamiltonians. The special cases of disordered and inhomogeneous lattices, as well as large nonlattice graphs, are presented in detail. Finally, we show that our simulation algorithm generalizes to other promising targets, including free-boson systems.
The validity of the essential ingredients of the microscopic formulation of the interacting boson model: (i) truncation of the shell-model or fermion space and (ii) the procedures for mapping the truncated fermion space on to the boson space, is tested over various (spherical, deformed and transitional) regions both in the simplistic single j-shell, identical boson systems and the more realistic multi j-shell, proton-neutron systems depending on the feasibility of such calculations. In the situations where the truncation and mapping procedures fail to reproduce the exact shell-model results, it is demonstrated that renormalization and higher order mapping procedures cure the associated problems, respectively and provide excellent agreements. In addition, a Hamiltonian of the interacting boson-fermion model which is relevant for a study of odd mass nuclei is derived for a single j-shell case in the shell-model framework, which yields results identical to those in the shell-model for the monopole pairing interaction.
The nuclear shell model is a general way of expressing the many-body nuclear Hamiltonian and deciphering the underlying nuclear structure. In today's era of modern and high-power computation, the primary limitation of the nuclear shell model is the enormous dimensionality of its Hilbert space, which far exceeds available storage capacity and prevents the diagonalization of the full Hamiltonian matrix in that space. Quantum computing offers a scalable solution to bypass this curse of dimensionality. In this work, we introduce a single-run quantum simulation capable of obtaining multiple shell-model eigenstates simultaneously. The nuclear Hamiltonian is transformed from a bit to a qubit basis using the Jordan-Wigner transformation, explicitly preserving fermionic anti-commutation. We employ a Subspace Search Variational Quantum Eigensolver (SSVQE) along with an Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT) ansatz to construct the quantum circuit required to solve the shell-model problem. The ADAPT-SSVQE algorithm uses a symmetry-preserving single and double-excitation operator pool and optimizes a weighted energy sum to obtain the simultaneous convergence of all eigenstates within a targeted MJ subspace, eliminating the need for post-processing efforts to extract excited spectra. We benchmark this approach by solving the problem for two and three identical nucleons in a j = 9/2 orbital, successfully extracting five and ten mutually orthogonal states, respectively, within a 10-qubit active space. The algorithm achieves spectroscopic accuracy, in simulation, relative to exact diagonalization and intrinsically restores total angular momentum (\hat{J}^2) symmetry.