We show that the standard Lanczos algorithm can be efficiently implemented statistically and self-consistently improved, using the stochastic reconfiguration method, which has been recently introduced to stabilize the Monte Carlo sign problem instability. With this scheme a few Lanczos steps over a given variational wave function are possible even for large size as a particular case of a more general and more accurate technique that allows to obtain lower variational energies. This method has been tested extensively for a strongly correlated model like the $t\ensuremath{-}J$ model. With the standard Lanczos technique it is possible to compute any kind of correlation functions, with no particular computational effort. By using the fact that the variance $〈{H}^{2}〉\ensuremath{-}〈H{〉}^{2}$ is zero for an exact eigenstate, we show that the approach to the exact solution with few Lanczos iterations is indeed possible even for $\ensuremath{\sim}100$ electrons for reasonably good initial wave functions. The variational stochastic reconfiguration technique presented here allows in general a many-parameter energy optimization of any computable many-body wave function, including for instance generic long-range Jastrow factors and arbitrary site-dependent orbital determinants. This scheme improves further the accuracy of the calculation, especially for long-distance correlation functions.
The steepest descent perturbation theory is extended to the calculation of the energy and eigenfunction of the excited state of a quantum system. In case of the orthogonality of the trial function for the excited state to those for lower-energy state or ground state in the same symmetry class is preserved, the variational collapse to lower energy state can be avoided in this proposal. An iterative procedure is given for generating better eigenvalue and eigenfunction of the excited state without requiring an infinite summation over reference states as in conventional perturbation theory. This new perturbation method can be applied to calculate the excitation energy and wave function of excited states for any many-body quantum system to a high degree of accuracy without so much computational effort as in conventional method.
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.
Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME)
dtdρt=Lρt, where
L is a linear (super-)operator and
ρt is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state
ρ∞ (such that
Lρ∞=0), the first few excited states, and trajectories
t↦ρt. In finite dimension,
ρt is an
n×n matrix,
L thus typically costs
n4 to store explicitly as a dense matrix, and
O(n6) to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However,
L usually costs only
O(n3) to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution
S, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map
Φ whose fixed point is directly related to
ρ∞, all the other eigenvalues having smaller magnitude. The map
Φ is thus well suited to iterative methods, and
ρ∞ can be found in a few Arnoldi iterations. Using the same inverse map
S−1 as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like
O(n3) per iteration and offer state-of-the-art performance on CPU and GPU.
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 1