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arXiv:1410.1877 · Physical Review A

Clock quantum Monte Carlo technique: An imaginary-time method for real-time quantum dynamics

Jarrod R. McClean, Alán Aspuru‐Guzik

Published Jan 9, 2015• 20 citations
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Abstract

In quantum information theory, there is an explicit mapping between general unitary dynamics and Hermitian ground-state eigenvalue problems known as the Feynman-Kitaev clock Hamiltonian. A prominent family of methods for the study of quantum ground states is quantum Monte Carlo methods, and recently the full configuration interaction quantum Monte Carlo (FCIQMC) method has demonstrated great promise for practical systems. We combine the Feynman-Kitaev clock Hamiltonian with FCIQMC to formulate a technique for the study of quantum dynamics problems. Numerical examples using quantum circuits are provided as well as a technique to further mitigate the sign problem through time-dependent basis rotations. Moreover, this method allows one to combine the parallelism of Monte Carlo techniques with the locality of time to yield an effective parallel-in-time simulation technique.

DOI
10.1103/physreva.91.012311

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In quantum information theory, there is an explicit mapping between general unitary dynamics and Hermitian ground-state eigenvalue problems known as the Feynman-Kitaev clock Hamiltonian. A prominent family of methods for the study of quantum ground states is quantum Monte Carlo methods, and recently the full configuration interaction quantum Monte Carlo (FCIQMC) method has demonstrated great promise for practical systems. We combine the Feynman-Kitaev clock Hamiltonian with FCIQMC to formulate a technique for the study of quantum dynamics problems. Numerical examples using quantum circuits are provided as well as a technique to further mitigate the sign problem through time-dependent basis rotations. Moreover, this method allows one to combine the parallelism of Monte Carlo techniques with the locality of time to yield an effective parallel-in-time simulation technique.
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Full configuration interaction quantum Monte Carlo for accurate ab initio\textit{ab initio}ab initio nuclear structure calculations: algorithms and calculation details
Full configuration interaction quantum Monte Carlo (FCIQMC) is a stochastic many-body solver that has been widely applied to electronic, molecular, and condensed-matter systems. In this work we apply FCIQMC to ab initio\textit{ab initio}ab initio nuclear structure calculations using interactions derived from chiral effective field theory. We describe the algorithm in detail, including imaginary-time propagation, excitation generation, estimator choices, the initiator approximation with adaptive shift correction, and reduced-density-matrix (RDM) sampling. Benchmark calculations in small model spaces, where deterministic full configuration interaction (FCI) results are available, validate the stochastic calculation of energies, radii, and RDM-based pure estimators. For large model spaces, we analyze the residual finite-walker bias through systematic walker-number convergence and infinite-walker extrapolations. We also demonstrate that FCIQMC can be extended beyond ground-state calculations by computing the low-lying spectrum of 6^66Li.
Two distinct types of Quantum Monte Carlo (QMC) calculations are applied to electronic structure problems such as calculating potential energy curves and producing benchmark values for reaction barriers. First, Variational and Diffusion Monte Carlo (VMC and DMC) methods using a trial wavefunction subject to the fixed node approximation were tested using the CASINO code.[1] Next, Full Configuration Interaction Quantum Monte Carlo (FCIQMC), along with its initiator extension (i-FCIQMC) were tested using the NECI code.[2] FCIQMC seeks the FCI energy for a specific basis set. At a reduced cost, the efficient i-FCIQMC method can be applied to systems in which the standard FCIQMC approach proves to be too costly. Since all of these methods are statistical approaches, uncertainties (error-bars) are introduced for each calculated energy. This study tests the performance of the methods relative to traditional quantum chemistry for some benchmark systems. References: [1] R. J. Needs et al., J. Phys.: Condensed Matter 22, 023201 (2010). [2] G. H. Booth et al., J. Chem. Phys. 131, 054106 (2009).
On the basis of a Feynman–Kac-type formula involving Poisson stochastic processes, a Monte Carlo algorithm has recently been introduced, which describes exactly the real- or imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and for the Heisenberg model.
A novel scheme to solve the quantum eigenvalue problem through the imaginary-time Green function Monte Carlo method is presented. This method is applicable not only to the ground states, but also to the excited states of generic system. We demonstrate the actual utility of the method with the numerical examples on three simple systems. Typeset using REVTEX 1 The stochastic quantization has been successfully applied in recent years to the various problems of physics ranging from quantum chromo dynamics to the plasmon oscillation [1-3]. While it can handle very complex systems which is usually beyond the approach either by the direct diagonalization or by the perturbation, it has been only applicable to the calculation of the ground state properties of the system. In this paper, we propose a new method based on the imaginary-time Green function Monte Carlo approach, which is extendable to the excited states of Hamiltonian system. We have primarily in mind the non-relativistic quantum few body problem such as encountered in nuclear physics [4-6].
The convergence of full configuration interaction quantum Monte Carlo (FCIQMC) is accelerated using a quasi-Newton propagation (QN) which can also be applied to coupled cluster Monte Carlo (CCMC). The computational scaling of this optimized propagation is, keeping the additional computational cost to a bare minimum. Its effects are investigated deterministically and stochastically on a model system, the uniform electron gas, with Hilbert space size up to 10 40, and shown to accelerate convergence of the instantaneous projected energy by over an order of magnitude in the FCIQMC test case. Its capabilities are then demonstrated with FCIQMC on an archetypical quantum chemistry problem, the chromium dimer, in an all-electron basis set with Hilbert space size of about 10 22 yielding highly accurate FCI energies.