Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Approximation Techniques for Large Finite Quantum Many-body Systems

Shen Yong Ho

Published Mar 3, 2010• 0 citations
OpenAlex

Abstract

In this thesis, we will show how certain classes of quantum many-body Hamiltonians with $\\su{2}_1 \\oplus \\su{2}_2 \\oplus \\ldots \\oplus \\su{2}_k$ spectrum generating algebras can be approximated by multi-dimensional shifted harmonic oscillator Hamiltonians. The dimensions of the Hilbert spaces of such Hamiltonians usually depend exponentially on $k$. This can make obtaining eigenvalues by diagonalization computationally challenging. The Shifted Harmonic Approximation (SHA) developed here gives good predictions of properties such as ground state energies, excitation energies and the most probable states in the lowest eigenstates. This is achieved by solving only a system of $k$ equations and diagonalizing $k\\times k$ matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions.\n\nThe SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have $\\su{2}$ spectrum generating algebras. Next, we extend the technique to the non-compact $\\su{1,1}$ algebra, using the five-dimensional quartic oscillator (5DQO) as an example. Finally, the SHA is applied to a $k$-level Bardeen-Cooper-Shrieffer (BCS) pairing Hamiltonian with fixed particle number. The BCS model has a $\\su{2}_1 \\oplus \\su{2}_2 \\oplus \\ldots \\oplus \\su{2}_k$ spectrum generating algebra.\n\nAn attractive feature of the SHA is that it also provides information to construct basis states which yield very accurate eigenvalues for low-lying states by diagonalizing Hamiltonians in small subspaces of huge Hilbert spaces. For Hamiltonians that involve a smaller number of operators, accurate eigenvalues can be obtained using another technique developed in this thesis: the generalized Rowe-Rosensteel-Kerman-Klein equations-of-motion method (RRKK). The RRKK is illustrated using the LMG and the 5DQO. In RRKK, solving unknowns in a set of $10\\times 10$ matrices typically gives estimates of the lowest few eigenvalues to an accuracy of at least eight significant figures. The RRKK involves optimization routines which require initial guesses of the matrix representations of the operators. In many cases, very good initial guesses can be obtained using the SHA. \n\nThe thesis concludes by exploring possible future developments of the SHA.

Related papers

Eigenstate Estimation for the Bardeen-Cooper-Schrieffer (BCS) Hamiltonian

Sung Ho, DAVID J. ROWE, Stijn De Baerdemacker

We show how multi-level BCS Hamiltonians of finite systems in the strong pairing interaction regime can be accurately approximated using multi-dimensional shifted harmonic oscillator Hamiltonians. In the Shifted Harmonic Approximation (SHA), discrete quantum state variables are approximated as continuous ones and algebraic Hamiltonians are replaced by differential operators. Using the SHA, the results of the BCS theory, such as the gap equations, can be easily derived without the BCS approximation. In addition, the SHA preserves the symmetries associated with the BCS Hamiltonians. Lastly, for all interaction strengths, the SHA can be used to identify the most important basis states -- allowing accurate computation of low-lying eigenstates by diagonalizing BCS Hamiltonians in small subspaces of what may otherwise be vastly larger Hilbert spaces.

arXiv:1011.4304•Nov 18, 2010•2 citations

EXTENSIONS FOR THE RIGID-BODY QUANTUM MONTE CARLO METHOD

Chris Smith

Solving the Schrodinger equation and finding excited states for quantum mechanical many-body systems is a fundamental problem. If this problem is formulated in terms of integrals, the dimensionality of the configuration space is often too high to perform the integrations required directly. This is because the volume of configuration space increases exponentially with dimension quickly, making its complete exploration impossible. Monte Carlo methods provide a way to estimate these integrals by statistically sampling a subset of configuration space, and these methods provide 1/√ N convergence regardless of dimension. For many problems involving the quantum mechanics of molecules there exists a large time scale separation between the high-frequency internal vibrations, and low-frequency intermolecular motions. This separation motivates the rigid-body approximation which freezes internal degrees of freedom in order to study intermolecular effects. The reduction in the dimensionality of configuration space further increases the range of accessible problems. This thesis is devoted to the construction and implementation of algorithms which incorporate the rigid-body approximation into existing Monte Carlo methods for solving the quantum mechanical many-body problem. Monte Carlo estimators are constructed as averages over samples drawn from some probability distribution. If this distribution is known in closed form the samples can be generated by application of the Metropolis algorithm. In most cases the distribution is not known in closed form or even representable with a finite number of variables. If this distribution is the dominant eigenstate of some known operator then a stochastic implementation of the power method can be used to generate the required samples. For the quantum mechanical problem this operator can be taken to be the imaginary-time evolution operator and its application can be represented in terms of random walks. For rigid bodies, this method involves the implementation of rotational Brownian motion. The use of quaternion algebra to implement this rotational motion enhances simplicity, performance, and numerical stability. The method described above can be generalized to investigate excited state properties using correlation function Monte Carlo which is a Monte Carlo implementation of the Rayleigh-Ritz variational method. This generalization requires the construction of a trial subspace which is then subjected to the variational method resulting in a generalized eigenvalue problem. It is important to represent optimized trial states accurately and efficiently. For this purpose, we write trial states as functions of invariant polynomials of the interparticle distances. By reducing the number of free interparticle distances, the rigid-body approximation greatly simplifies

•Jan 1, 2010•0 citations

Exact Diagonalization Techniques for Quantum Spin Systems

Jürgen Schnack

The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert spaceHilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert space of the spin system with the number of spins: $$\dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right)$$ . Nowadays, complex Hermitian matricesHermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size $$s_{i}$$ to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos methodFinite-Temperature Lanczos Method (FTLM), allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.

Challenges and advances in computational chemistry and physics•Jan 1, 2023•2 citations

Symmetry breaking and restoration for many-body problems treated on quantum computers

Ruiz, Andres

This thesis explores the application of the Symmetry-Breaking/Symmetry-Restoration methodology on quantum computers to better approximate a Hamiltonian's ground state energy within a variational framework in many-body physics. This involves intentionally breaking and restoring the symmetries of the wave function ansatz at different stages of the variational search for the ground state. The Variational Quantum Eigensolver (VQE) is utilized for the variational component together with an ansatz inspired by the Bardeen-Cooper-Schrieffer (BCS) theory. The applications were demonstrated using the pairing and Hubbard Hamiltonians. Two approaches were identified with the VQE method: varying the symmetry-breaking ansatz parameters before or after symmetry restoration, termed Quantum Projection After Variation and Quantum Variation After Projection, respectively. The main contribution of this thesis was the development of a variety of symmetry restoration techniques based on the principles of the Quantum Phase Estimation algorithm, the notion of a Quantum "Oracle," and the Classical Shadow formalism. In the final part, hybrid quantum-classical techniques were introduced to extract an approximation of the low-lying spectrum of a Hamiltonian. Assuming accurate Hamiltonian moment extraction from their generating function with a quantum computer, two methods were presented for spectral analysis: the t-expansion method and the Krylov method, which provides, in particular, information about the evolution of the survival probability. Furthermore, the Quantum Krylov method was introduced, offering similar insights without the need to estimate Hamiltonian moments, a task that can be difficult on near-term quantum computers.

arXiv:2310.17996•Oct 27, 2023•0 citations

Reduced Density Matrices and Moduli of Many-Body Eigenstates

Chaoming Song

Many-body wavefunctions usually lie in high-dimensional Hilbert spaces. However, physically relevant states, i.e, the eigenstates of the Schrödinger equation are rare. For many-body systems involving only pairwise interactions, these eigenstates form a low-dimensional subspace of the entire Hilbert space. The geometry of this subspace, which we call the eigenstate moduli problem is parameterized by a set of 2-particle Hamiltonian. This problem is closely related to the $N$-representability conditions for 2-reduced density matrices, a long-standing challenge for quantum many-body systems. Despite its importance, the eigenstate moduli problem remains largely unexplored in the literature. In this Letter, we propose a comprehensive approach to this problem. We discover an explicit set of algebraic equations that fully determine the eigenstate spaces of $m$-interaction systems as projective varieties, which in turn determine the geometry of the spaces for representable reduced density matrices. We investigate the geometrical structure of these spaces, and validate our results numerically using the exact diagonalization method. Finally, we generalize our approach to the moduli problem of the arbitrary family of Hamiltonians parameterized by a set of real variables.

arXiv:2301.01419•Jan 4, 2023•0 citations

Many-body dynamics on a time-dependent basis

E. Susana Hernández, Dora M. Jezek

We propose a method of solution of the many-body Schr\"odinger equation that involves an expansion of the wave function in terms of a finite, time-dependent basis of a relevant subspace. The equations of motion for the expansion coefficients generalize previous proposals of approximate dynamics. The method is illustrated in the case of an N-particle system with an SU(2) Hamiltonian, and it is shown that it improves the approximation that disregards off-diagonal elements of the dynamical matrices.

Physical Review A•Nov 1, 1988•3 citations
In this thesis, we will show how certain classes of quantum many-body Hamiltonians with su21oplussu22oplusldotsoplussu2k\\su{2}_1 \\oplus \\su{2}_2 \\oplus \\ldots \\oplus \\su{2}_ksu21​oplussu22​oplusldotsoplussu2k​ spectrum generating algebras can be approximated by multi-dimensional shifted harmonic oscillator Hamiltonians. The dimensions of the Hilbert spaces of such Hamiltonians usually depend exponentially on kkk. This can make obtaining eigenvalues by diagonalization computationally challenging. The Shifted Harmonic Approximation (SHA) developed here gives good predictions of properties such as ground state energies, excitation energies and the most probable states in the lowest eigenstates. This is achieved by solving only a system of kkk equations and diagonalizing ktimeskk\\times kktimesk matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions.\n\nThe SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have su2\\su{2}su2 spectrum generating algebras. Next, we extend the technique to the non-compact su1,1\\su{1,1}su1,1 algebra, using the five-dimensional quartic oscillator (5DQO) as an example. Finally, the SHA is applied to a kkk-level Bardeen-Cooper-Shrieffer (BCS) pairing Hamiltonian with fixed particle number. The BCS model has a su21oplussu22oplusldotsoplussu2k\\su{2}_1 \\oplus \\su{2}_2 \\oplus \\ldots \\oplus \\su{2}_ksu21​oplussu22​oplusldotsoplussu2k​ spectrum generating algebra.\n\nAn attractive feature of the SHA is that it also provides information to construct basis states which yield very accurate eigenvalues for low-lying states by diagonalizing Hamiltonians in small subspaces of huge Hilbert spaces. For Hamiltonians that involve a smaller number of operators, accurate eigenvalues can be obtained using another technique developed in this thesis: the generalized Rowe-Rosensteel-Kerman-Klein equations-of-motion method (RRKK). The RRKK is illustrated using the LMG and the 5DQO. In RRKK, solving unknowns in a set of 10times1010\\times 1010times10 matrices typically gives estimates of the lowest few eigenvalues to an accuracy of at least eight significant figures. The RRKK involves optimization routines which require initial guesses of the matrix representations of the operators. In many cases, very good initial guesses can be obtained using the SHA. \n\nThe thesis concludes by exploring possible future developments of the SHA.
We show how multi-level BCS Hamiltonians of finite systems in the strong pairing interaction regime can be accurately approximated using multi-dimensional shifted harmonic oscillator Hamiltonians. In the Shifted Harmonic Approximation (SHA), discrete quantum state variables are approximated as continuous ones and algebraic Hamiltonians are replaced by differential operators. Using the SHA, the results of the BCS theory, such as the gap equations, can be easily derived without the BCS approximation. In addition, the SHA preserves the symmetries associated with the BCS Hamiltonians. Lastly, for all interaction strengths, the SHA can be used to identify the most important basis states -- allowing accurate computation of low-lying eigenstates by diagonalizing BCS Hamiltonians in small subspaces of what may otherwise be vastly larger Hilbert spaces.
Solving the Schrodinger equation and finding excited states for quantum mechanical many-body systems is a fundamental problem. If this problem is formulated in terms of integrals, the dimensionality of the configuration space is often too high to perform the integrations required directly. This is because the volume of configuration space increases exponentially with dimension quickly, making its complete exploration impossible. Monte Carlo methods provide a way to estimate these integrals by statistically sampling a subset of configuration space, and these methods provide 1/√ N convergence regardless of dimension. For many problems involving the quantum mechanics of molecules there exists a large time scale separation between the high-frequency internal vibrations, and low-frequency intermolecular motions. This separation motivates the rigid-body approximation which freezes internal degrees of freedom in order to study intermolecular effects. The reduction in the dimensionality of configuration space further increases the range of accessible problems. This thesis is devoted to the construction and implementation of algorithms which incorporate the rigid-body approximation into existing Monte Carlo methods for solving the quantum mechanical many-body problem. Monte Carlo estimators are constructed as averages over samples drawn from some probability distribution. If this distribution is known in closed form the samples can be generated by application of the Metropolis algorithm. In most cases the distribution is not known in closed form or even representable with a finite number of variables. If this distribution is the dominant eigenstate of some known operator then a stochastic implementation of the power method can be used to generate the required samples. For the quantum mechanical problem this operator can be taken to be the imaginary-time evolution operator and its application can be represented in terms of random walks. For rigid bodies, this method involves the implementation of rotational Brownian motion. The use of quaternion algebra to implement this rotational motion enhances simplicity, performance, and numerical stability. The method described above can be generalized to investigate excited state properties using correlation function Monte Carlo which is a Monte Carlo implementation of the Rayleigh-Ritz variational method. This generalization requires the construction of a trial subspace which is then subjected to the variational method resulting in a generalized eigenvalue problem. It is important to represent optimized trial states accurately and efficiently. For this purpose, we write trial states as functions of invariant polynomials of the interparticle distances. By reducing the number of free interparticle distances, the rigid-body approximation greatly simplifies
The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert spaceHilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert space of the spin system with the number of spins: $$\dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right)$$ . Nowadays, complex Hermitian matricesHermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size sis_{i}si​ to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos methodFinite-Temperature Lanczos Method (FTLM), allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.
This thesis explores the application of the Symmetry-Breaking/Symmetry-Restoration methodology on quantum computers to better approximate a Hamiltonian's ground state energy within a variational framework in many-body physics. This involves intentionally breaking and restoring the symmetries of the wave function ansatz at different stages of the variational search for the ground state. The Variational Quantum Eigensolver (VQE) is utilized for the variational component together with an ansatz inspired by the Bardeen-Cooper-Schrieffer (BCS) theory. The applications were demonstrated using the pairing and Hubbard Hamiltonians. Two approaches were identified with the VQE method: varying the symmetry-breaking ansatz parameters before or after symmetry restoration, termed Quantum Projection After Variation and Quantum Variation After Projection, respectively. The main contribution of this thesis was the development of a variety of symmetry restoration techniques based on the principles of the Quantum Phase Estimation algorithm, the notion of a Quantum "Oracle," and the Classical Shadow formalism. In the final part, hybrid quantum-classical techniques were introduced to extract an approximation of the low-lying spectrum of a Hamiltonian. Assuming accurate Hamiltonian moment extraction from their generating function with a quantum computer, two methods were presented for spectral analysis: the t-expansion method and the Krylov method, which provides, in particular, information about the evolution of the survival probability. Furthermore, the Quantum Krylov method was introduced, offering similar insights without the need to estimate Hamiltonian moments, a task that can be difficult on near-term quantum computers.
Many-body wavefunctions usually lie in high-dimensional Hilbert spaces. However, physically relevant states, i.e, the eigenstates of the Schrödinger equation are rare. For many-body systems involving only pairwise interactions, these eigenstates form a low-dimensional subspace of the entire Hilbert space. The geometry of this subspace, which we call the eigenstate moduli problem is parameterized by a set of 2-particle Hamiltonian. This problem is closely related to the NNN-representability conditions for 2-reduced density matrices, a long-standing challenge for quantum many-body systems. Despite its importance, the eigenstate moduli problem remains largely unexplored in the literature. In this Letter, we propose a comprehensive approach to this problem. We discover an explicit set of algebraic equations that fully determine the eigenstate spaces of mmm-interaction systems as projective varieties, which in turn determine the geometry of the spaces for representable reduced density matrices. We investigate the geometrical structure of these spaces, and validate our results numerically using the exact diagonalization method. Finally, we generalize our approach to the moduli problem of the arbitrary family of Hamiltonians parameterized by a set of real variables.
We propose a method of solution of the many-body Schr\"odinger equation that involves an expansion of the wave function in terms of a finite, time-dependent basis of a relevant subspace. The equations of motion for the expansion coefficients generalize previous proposals of approximate dynamics. The method is illustrated in the case of an N-particle system with an SU(2) Hamiltonian, and it is shown that it improves the approximation that disregards off-diagonal elements of the dynamical matrices.