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International Conference for High Performance Computing, Networking, Storage and Analysis (SC)

Taking a quantum leap in time to solution for simulations of high-Tc superconductors

Peter Staar, Thomas Maier, Michael S. Summers, Gilles Fourestey, Raffaele Solcà, T. C. Schulthess

Published Oct 30, 2013• 8 citations
Publisher

Abstract

We present a new quantum cluster algorithm to simulate models of high-Tc superconductors. This algorithm extends current methods with continuous lattice self-energies, thereby removing artificial long-range correlations. This cures the fermionic sign problem in the underlying quantum Monte Carlo solver for large clusters and realistic values of the Coulomb interaction in the entire temperature range of interest. We find that the new algorithm improves time-to-solution by nine orders of magnitude compared to current, state of the art quantum cluster simulations. An efficient implementation is given, which ports to multi-core as well as hybrid CPU-GPU systems. Running on 18,600 nodes on ORNL's Titan supercomputer enables us to compute a converged value of Tc/t = 0.053 ± 0.0014 for a 28 site cluster in the 2D Hubbard model with U/t = 7 at 10% hole doping. Typical simulations on Titan sustain between 9.2 and 15.4 petaflops (double precision measured over full run), depending on configuration and parameters used.

DOI
10.1145/2503210.2503282

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We present a new quantum cluster algorithm to simulate models of high-Tc superconductors. This algorithm extends current methods with continuous lattice self-energies, thereby removing artificial long-range correlations. This cures the fermionic sign problem in the underlying quantum Monte Carlo solver for large clusters and realistic values of the Coulomb interaction in the entire temperature range of interest. We find that the new algorithm improves time-to-solution by nine orders of magnitude compared to current, state of the art quantum cluster simulations. An efficient implementation is given, which ports to multi-core as well as hybrid CPU-GPU systems. Running on 18,600 nodes on ORNL's Titan supercomputer enables us to compute a converged value of Tc/t = 0.053 ± 0.0014 for a 28 site cluster in the 2D Hubbard model with U/t = 7 at 10% hole doping. Typical simulations on Titan sustain between 9.2 and 15.4 petaflops (double precision measured over full run), depending on configuration and parameters used.
The Cray X1 in the Center for Computational Sciences at Oak Ridge National Laboratory as well as algorithmic improvements over the past decade enable significant new science in the simulation of high-temperature cuprate superconductors. We describe the method of dynamic cluster approximation with quantum Monte Carlo, along with its computational requirements. We then show the unique capabilities of the X1 for supporting this method and delivering near optimal performance. This allows us to study systematically the cluster size dependence of the superconductivity in the conventional two-dimensional Hubbard model, which is commonly believed to describe high-temperature superconductors. Due to the non-locality of the d-wave superconducting order parameter, the results on small clusters show large size and geometry effects. In large enough clusters, converged results are found that display a finite temperature instability to d-wave superconductivity. The results we report here demonstrate for the first time that superconductivity is possible in a system of strongly correlated electrons without the need of a phonon mediated attractive interaction.
Abstract We have developed a strong-coupling perturbation scheme for a general doped Hubbard model around a particle-hole-symmetric reference system, which is free from the fermionic sign problem. Our approach is based on the lattice determinantal Quantum Monte Carlo (QMC) method in both continuous and discrete time versions for large periodic clusters in a fermionic bath. By considering the first-order perturbation in the shift of the chemical potential and the second-neighbor hopping, we are able to obtain an accurate electronic spectral function for a range of parameters that correspond to optimally doped cuprate systems at temperatures of up to T = 0.1 t , which are challenging to access using straightforward lattice QMC calculations. We also discuss the formation of the pseudogap and the nodal-antinodal dichotomy for a doped Hubbard system with the interaction parameter U equal to the bandwidth and the optimal value of the next-nearest-neighbor hopping parameter t′{t}^{{\prime} }t′ t ′ for high-temperature superconducting cuprates.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of (Nε−1)O(log⁡Δ)(N\varepsilon^{-1})^{\mathcal{O}(\logΔ)}(Nε−1)O(logΔ), where NNN is the system size, ε\varepsilonε the desired precision, and ΔΔΔ the interaction-graph degree. This becomes particularly severe in long-range systems, where Δ≡NΔ\equiv NΔ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime O~(N4ε−3)\tilde{\mathcal{O}}(N^4\varepsilon^{-3})O~(N4ε−3). For quantum partition functions at temperatures β≤βsampling≡O(1)β\leqβ_{\mathrm{sampling}}\equiv\mathcal{O}(1)β≤βsampling​≡O(1), this improves to O~(N3+1/Dε−3)\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})O~(N3+1/Dε−3), where DDD is the lattice dimension. The framework directly applies to long-range quantum systems with interactions ∝1/rα\propto1/r^α∝1/rα, including 222-local models with α>2Dα>2Dα>2D.
The dynamical cluster approximation (DCA) is a systematic extension beyond the single-site approximation in dynamical mean field theory, to include spatially nonlocal correlations in quantum many-body simulations of strongly correlated systems. We extend the DCA with a continuous lattice self-energy in order to achieve better convergence with cluster size. This method, which we call DCA+{}^{+}+, cures the cluster-shape dependence problems of the DCA, without suffering from causality violations of previous attempts to interpolate the cluster self-energy. A practical approach based on standard inference techniques is given to deduce the continuous lattice self-energy from an interpolated cluster self-energy. We study the pseudogap region of a hole-doped two-dimensional Hubbard model and find that, in the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm, the self-energy and pseudogap temperature T∗{T}^{*}T∗ converge monotonously with cluster size. Introduction of a continuous lattice self-energy eliminates artificial long-range correlations and thus significantly reduces the sign problem of the quantum Monte Carlo cluster solver in the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm compared to the normal DCA. Simulations with much larger cluster sizes thus become feasible, which, along with the improved convergence in cluster size, raises hope that precise extrapolations to the exact infinite cluster size limit can be reached for other physical quantities as well.
Many phenomena of strongly correlated materials are encapsulated in the Fermi-Hubbard model whose thermodynamic properties can be computed from its grand-canonical potential. In general, there is no closed-form expression of the grand-canonical potential for lattices of more than one spatial dimension, but solutions can be numerically approximated using cluster methods. To model long-range effects such as order parameters, a powerful method to compute the cluster's Green's function consists of finding its self-energy through a variational principle. This allows the possibility of studying various phase transitions at finite temperature in the Fermi-Hubbard model. However, a classical cluster solver quickly hits an exponential wall in the memory (or computation time) required to store the computation variables. Here it is shown theoretically that the cluster solver can be mapped to a subroutine on a quantum computer whose quantum memory usage scales linearly with the number of orbitals in the simulated cluster and the number of measurements scales quadratically. A quantum computer with a few tens of qubits could therefore simulate the thermodynamic properties of complex fermionic lattices inaccessible to classical supercomputers.
Efficient continuous-time quantum Monte Carlo (CT-QMC) algorithms that do not suffer from time discretization errors have become the state of the art for most discrete quantum models. They have not been widely used yet for fermionic quantum lattice models, such as the Hubbard model, nor other fermionic lattice systems due to a suboptimal scaling of $O({\ensuremath{\beta}}^{3})$ with inverse temperature $\ensuremath{\beta}$, compared to the linear scaling of discrete-time algorithms. Here we present a CT-QMC algorithm for fermionic lattice systems that matches the scaling of discrete-time methods but is more efficient and free of time discretization errors. This provides an efficient simulation scheme that is free from the systematic errors opening an avenue to more precise studies of large systems at low and zero temperature.