An Attempt to Calculate Energy Eigenvalues in Quantum Systems of Large Sizes

We report an attempt to calculate energy eigenvalues of large quantum systems by the diagonalization of an effectively truncated Hamiltonian matrix. For this purpose we employ a specific way to systematically make a set of orthogonal states from a trial wavefunction and the Hamiltonian. In comparison with the Lanczos method, which is quite powerful if the size of the system is within the memory capacity of computers, our method requires much less memory resources at the cost of the extreme accuracy. In this paper we demonstrate that our method works well in the systems of one-dimensional frustrated spins up to 48 sites, of bosons on a chain up to 32 sites and of fermions on a ladder up to 28 sites. We will see this method enables us to study eigenvalues of these quantum systems within reasonable accuracy.We report an attempt to calculate energy eigenvalues of large quantum systems by the diagonalization of an effectively truncated Hamiltonian matrix. For this purpose we employ a specific way to systematically make a set of orthogonal states from a trial wavefunction and the Hamiltonian. In comparison with the Lanczos method, which is quite powerful if the size of the system is within the memory capacity of computers, our method requires much less memory resources at the cost of the extreme accuracy. In this paper we demonstrate that our method works well in the systems of one-dimensional frustrated spins up to 48 sites, of bosons on a chain up to 32 sites and of fermions on a ladder up to 28 sites. We will see this method enables us to study eigenvalues of these quantum systems within reasonable accuracy.

arXiv:cond-mat/9912295Journal of the Physical Society of JapanJul 15, 20004 citations

The Stochastic State Selection Method Combined with the Lanczos Approach to Eigenvalues in Quantum Spin Systems

Tomo Munehisa, Yasuko Munehisa

We describe a further development of the stochastic state selection method, a new Monte Carlo method we have proposed recently to make numerical calculations in large quantum spin systems. Making recursive use of the stochastic state selection technique in the Lanczos approach, we estimate the ground state energy of the spin-1/2 quantum Heisenberg antiferromagnet on a 48-site triangular lattice. Our result for the upper bound of the ground state energy is −0.1833 ± 0.0003 per bond. This value, being compatible with values from other work, indicates that our method is efficient in calculating energy eigenvalues of frustrated quantum spin systems on large lattices. PACS number: 05.10.Ln, 02.70.Ss, 75.10.Jm 1We describe a further development of the stochastic state selection method, a new Monte Carlo method we have proposed recently to make numerical calculations in large quantum spin systems. Making recursive use of the stochastic state selection technique in the Lanczos approach, we estimate the ground state energy of the spin-1/2 quantum Heisenberg antiferromagnet on a 48-site triangular lattice. Our result for the upper bound of the ground state energy is −0.1833 ± 0.0003 per bond. This value, being compatible with values from other work, indicates that our method is efficient in calculating energy eigenvalues of frustrated quantum spin systems on large lattices. PACS number: 05.10.Ln, 02.70.Ss, 75.10.Jm 1

Jan 1, 20053 citations

The stochastic state selection method combined with the Lanczos approach to eigenvalues in quantum spin systems

Tomo Munehisa, Yasuko Munehisa

We describe a further development of the stochastic state selection method, a new Monte Carlo method we have proposed recently to make numerical calculations in large quantum spin systems. Making recursive use of the stochastic state selection technique in the Lanczos approach, we estimate the ground state energy of the spin-1/2 quantum Heisenberg antiferromagnet on a 48-site triangular lattice. Our result for the upper bound of the ground state energy is -0.1833 +/- 0.0003 per bond. This value, being compatible with values from other work, indicates that our method is efficient in calculating energy eigenvalues of frustrated quantum spin systems on large lattices.We describe a further development of the stochastic state selection method, a new Monte Carlo method we have proposed recently to make numerical calculations in large quantum spin systems. Making recursive use of the stochastic state selection technique in the Lanczos approach, we estimate the ground state energy of the spin-1/2 quantum Heisenberg antiferromagnet on a 48-site triangular lattice. Our result for the upper bound of the ground state energy is -0.1833 +/- 0.0003 per bond. This value, being compatible with values from other work, indicates that our method is efficient in calculating energy eigenvalues of frustrated quantum spin systems on large lattices.

arXiv:cond-mat/0505661Journal of Physics Condensed MatterFeb 3, 20065 citations

Renormalization group method for weakly coupled quantum chains: comparison with exact diagonalization

Jose Vicente Alvarez, S. Moukouri

We show that numerical quasi-one-dimensional renormalization group allows accurate study of weakly coupled chains with modest computational effort. We perform a systematic comparison with exact diagonalization results in two and three-leg spin ladders with a transverse Hamiltonian that can involve frustration. Due to the variational nature of the algorithm, the accuracy can be arbitrarily improved enlarging the basis of eigenstates of the density matrix defined in the transverse direction. We observe that the precision of the algorithm is directly correlated to the binding of the chains. We also show that the method performs especially well in frustrated systems.We show that numerical quasi-one-dimensional renormalization group allows accurate study of weakly coupled chains with modest computational effort. We perform a systematic comparison with exact diagonalization results in two and three-leg spin ladders with a transverse Hamiltonian that can involve frustration. Due to the variational nature of the algorithm, the accuracy can be arbitrarily improved enlarging the basis of eigenstates of the density matrix defined in the transverse direction. We observe that the precision of the algorithm is directly correlated to the binding of the chains. We also show that the method performs especially well in frustrated systems.

arXiv:cond-mat/0402530Feb 20, 20040 citations

Lanczos-type algorithm for excited states of very-large-scale quantum systems

Giuseppe Grosso, Liana Martinelli, Giuseppe Pastori Parravicini

We provide a very efficient procedure for obtaining the excited states of a quantum operator H, in any arbitrary chosen energy range, independently from the knowledge of the states at lower energies. Our procedure consists in determining, within the Lanczos algorithm, the ground state of the auxiliary operator A=(H- ${\mathit{E}}_{\mathit{t}}$ ${)}^{2}$ , and hence the eigenvalue of H nearest in energy to ${\mathit{E}}_{\mathit{t}}$ , where ${\mathit{E}}_{\mathit{t}}$ is any chosen trial energy in the energy range of interest. We show that a variational method exploiting diagonalization of 2\ifmmode\times\else\texttimes\fi{}2 Lanczos matrices, combined with a two-pass Lanczos procedure of relatively small number of iterations, produces eigenvalues of H within any desired accuracy, as well as the corresponding eigenfunctions. We discuss in particular the T\ensuremath{\bigotimes}\ensuremath{\tau} Jahn-Teller model, as a realistic prototype of a quantum system with a very large number of degrees of freedom.We provide a very efficient procedure for obtaining the excited states of a quantum operator H, in any arbitrary chosen energy range, independently from the knowledge of the states at lower energies. Our procedure consists in determining, within the Lanczos algorithm, the ground state of the auxiliary operator A=(H-${\mathit{E}}_{\mathit{t}}$${)}^{2}$, and hence the eigenvalue of H nearest in energy to ${\mathit{E}}_{\mathit{t}}$, where ${\mathit{E}}_{\mathit{t}}$ is any chosen trial energy in the energy range of interest. We show that a variational method exploiting diagonalization of 2\ifmmode\times\else\texttimes\fi{}2 Lanczos matrices, combined with a two-pass Lanczos procedure of relatively small number of iterations, produces eigenvalues of H within any desired accuracy, as well as the corresponding eigenfunctions. We discuss in particular the T\ensuremath{\bigotimes}\ensuremath{\tau} Jahn-Teller model, as a realistic prototype of a quantum system with a very large number of degrees of freedom.

Physical review. B, Condensed matterMay 15, 199541 citations

Exact and efficient Lanczos method on a quantum computer

William Kirby, Mário Motta, Antonio Mezzacapo

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.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.

arXiv:2208.00567QuantumMay 23, 202366 citations

{\mathit{E}}_{\mathit{t}}

{)}^{2}

, and hence the eigenvalue of H nearest in energy to

{\mathit{E}}_{\mathit{t}}

, where

{\mathit{E}}_{\mathit{t}}

is any chosen trial energy in the energy range of interest. We show that a variational method exploiting diagonalization of 2\ifmmode\times\else\texttimes\fi{}2 Lanczos matrices, combined with a two-pass Lanczos procedure of relatively small number of iterations, produces eigenvalues of H within any desired accuracy, as well as the corresponding eigenfunctions. We discuss in particular the T\ensuremath{\bigotimes}\ensuremath{\tau} Jahn-Teller model, as a realistic prototype of a quantum system with a very large number of degrees of freedom.