arXiv subjects

Bryan K. Clark

Publications and source records attributed to Bryan K. Clark.

87 records · Page 5Linked to original sources

arXiv2015

Solving strongly correlated electron models on a quantum computer

One of the main applications of future quantum computers will be the simulation of quantum models. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground state of the Hubbard model starting from various mean-field states with broken symmetry. We present an efficient procedure to prepare arbitrary Slater determinants as initial states and present the complete set of quantum circuits needed to evolve from these to the ground state of the Hubbard model. We show that, using efficient nesting of the various terms each time step in the evolution can be performed with just $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ circuit depth. We give explicit circuits to measure arbitrary local observables and static and dynamic correlation functions, both in the time and frequency domain. We further present efficient non-destructive approaches to measurement that avoid the need to re-prepare the ground state after each measurement and that quadratically reduce the measurement error.

quant-ph↗

arXiv2014

Encoding the structure of many-body localization with matrix product operators

Anderson insulators are non-interacting disordered systems which have localized single particle eigenstates. The interacting analogue of Anderson insulators are the Many-Body Localized (MBL) phases. The natural language for representing the spectrum of the Anderson insulator is that of product states over the single-particle modes. We show that product states over Matrix Product Operators of small bond dimension is the corresponding natural language for describing the MBL phases. In this language all of the many-body eigenstates are encode by Matrix Product States (i.e. DMRG wave function) consisting of only two sets of low bond-dimension matrices per site: the $G_i$ matrix corresponding to the local ground state on site $i$ and the $E_i$ matrix corresponding to the local excited state. All $2^n$ eigenstates can be generated from all possible combinations of these matrices.

cond-mat.str-el↗

arXiv2014

Gate count estimates for performing quantum chemistry on small quantum computers

As quantum computing technology improves and quantum computers with a small but non-trivial number of N > 100 qubits appear feasible in the near future the question of possible applications of small quantum computers gains importance. One frequently mentioned application is Feynman's original proposal of simulating quantum systems, and in particular the electronic structure of molecules and materials. In this paper, we analyze the computational requirements for one of the standard algorithms to perform quantum chemistry on a quantum computer. We focus on the quantum resources required to find the ground state of a molecule twice as large as what current classical computers can solve exactly. We find that while such a problem requires about a ten-fold increase in the number of qubits over current technology, the required increase in the number of gates that can be coherently executed is many orders of magnitude larger. This suggests that for quantum computation to become useful for quantum chemistry problems, drastic algorithmic improvements will be needed.

quant-ph↗

arXiv2014

Stochastically Projecting Tensor Networks

We apply a series of projection techniques on top of tensor networks to compute energies of ground state wave functions with higher accuracy than tensor networks alone with minimal additional cost. We consider both matrix product states as well as tree tensor networks in this work. Building on top of these approaches, we apply fixed-node quantum Monte Carlo, Lanczos steps, and exact projection. We demonstrate these improvements for the triangular lattice Heisenberg model, where we capture up to 57 percent of the remaining energy not captured by the tensor network alone. We conclude by discussing further ways to improve our approach.

cond-mat.str-el↗

arXiv2013

Searching for Topological Degeneracy in the Hubbard Model with Quantum Monte Carlo

Z2 spin liquids have topological order. One manifestation of this is that a Z2 spin liquid on a torus exhibits a four-fold degeneracy. Recent numerical evidence has argued for the existence of a spin liquid ground state in the Hubbard model on a honeycomb lattice near U approximately 4. The evidence for this claim involves the presence of a gapped state that lacks any identifiable order. This argument relies on being able to distinguish small order from no order which is notoriously difficult. In this paper we demonstrate an approach which uses quantum Monte Carlo to search for one of the key features that positively identify the topological spin liquid: the topological degeneracy. For any finite system, this topological degeneracy is split where the splitting decays exponentially with system size. We search for low lying states in the energy spectrum that could be identified as these topologically degenerate states. We show that, for system sizes up to N=162 sites, there is no evidence for these states being significantly below the first excited state giving evidence against the existence of a topological phase. We discuss the possible options for the Hubbard model on the honeycomb lattice in the absence of such degeneracy.

cond-mat.str-el↗

arXiv2013

The effect of quantization on the FCIQMC sign problem

The sign problem in Full Configuration Interaction Quantum Monte Carlo (FCIQMC) without annihilation can be understood as an instability of the psi-particle population to the ground state of the matrix obtained by making all off-diagonal elements of the Hamiltonian negative. Such a matrix, and hence the sign problem, is basis dependent. In this paper we discuss the properties of a physically important basis choice: first versus second quantization. For a given choice of single-particle orbitals, we identify the conditions under which the fermion sign problem in the second quantized basis of antisymmetric Slater determinants is identical to the sign problem in the first quantized basis of unsymmetrized Hartree products. We also show that, when the two differ, the fermion sign problem is always less severe in the second quantized basis. This supports the idea that FCIQMC, even in the absence of annihilation, improves the sign problem relative to first quantized methods. Finally, we point out some theoretically interesting classes of Hamiltonians where first and second quantized sign problems differ, and others where they do not.

physics.comp-ph↗

arXiv2012

Path Integral Monte Carlo Simulation of the Warm-Dense Homogeneous Electron Gas

We perform calculations of the {3D} finite-temperature homogeneous electron gas (HEG) in the warm-dense regime ({r_{s} \equiv (3/4πn)^{1/3}a_{B}^{- 1} = 1.0- 40.0} and {Θ\equiv T/T_{F} = 0.0625- 8.0}) using restricted path integral Monte Carlo (RPIMC). Precise energies, pair correlation functions, and structure factors are obtained. For all densities, we find a significant discrepancy between the ground state parameterized local density approximation (LDA) and our results around {T_{F}} . These results can be used as a benchmark for improved functionals, as well as input for orbital-free DFT formulations.

cond-mat.str-el↗

arXiv2012

A second-quantized red herring in full configuration-interaction Monte Carlo

This paper deals with the sign problem in full configuration-interaction quantum Monte Carlo. After putting this article on the arxiv, it was pointed out to us that our argument applies to a number of model Hamiltonians we considered numerically but not to the most generic case. Please see arxiv: 1209.3044, where we have a corrected comprehensive discussion of the necessary and sufficient conditions for when the sign problem for a given Hamiltonian (and basis) differ between first and second quantization.

physics.comp-ph↗

arXiv2012

Non-equilibrium dynamic critical scaling of the quantum Ising chain

We solve for the time-dependent finite-size scaling functions of the 1D transverse-field Ising chain during a linear-in-time ramp of the field through the quantum critical point. We then simulate Mott-insulating bosons in a tilted potential, an experimentally-studied system in the same equilibrium universality class, and demonstrate that universality holds for the dynamics as well. We find qualitatively athermal features of the scaling functions, such as negative spin correlations, and show that they should be robustly observable within present cold atom experiments.

cond-mat.stat-mech↗

arXiv2012

FCI-QMC approach to the Fermi polaron

Finding the ground state of a fermionic Hamiltonian using quantum Monte Carlo is a very difficult problem, due to the Fermi sign problem. While still scaling exponentially, full configuration-interaction Monte Carlo (FCI-QMC) mitigates some of the exponential variance by allowing annihilation of noise -- whenever two walkers arrive at the same configuration with opposite signs, they are removed from the simulation. While FCI-QMC has been quite successful for quantum chemistry problems, its application to problems in condensed systems has been limited. In this paper, we apply FCI-QMC to the Fermi polaron problem, which provides an ideal test-bed for improving the algorithm. In its simplest form, FCI-QMC is unstable for even a fairly small system sizes. However, with a series of algorithmic improvements, we are able to significantly increase its effectiveness. We modify fixed node QMC to work in these systems, introduce a well chosen importance sampled trial wave function, a partial node approximation, and a variant of released node. Finally, we develop a way to perform FCI-QMC directly in the thermodynamic limit.

physics.comp-ph↗

arXiv2011

Non-equilibrium dynamics of Bosonic Mott insulators in an electric field

We study the non-equilibrium dynamics of one-dimensional Mott insulating bosons in the presence of a tunable effective electric field E which takes the system across a quantum critical point (QCP) separating a disordered and a translation symmetry broken ordered phase. We provide an exact numerical computation of the residual energy Q, the log-fidelity F, the excess defect density D, and the order parameter correlation function for a linear-in-time variation of E with a rate v. We discuss the temporal and spatial variation of these quantities for a range of v and for finite system sizes as relevant to realistic experimental setups [J. Simon et al., Nature 472, 307 (2011)]. We show that in finite-sized systems Q, F, and D obey Kibble-Zurek scaling, and suggest further experiments within this setup to test our theory.

cond-mat.quant-gas↗

arXiv2011

Computing the energy of a water molecule using MultiDeterminants: A simple, efficient algorithm

Quantum Monte Carlo (QMC) methods such as variational Monte Carlo and fixed node diffusion Monte Carlo depend heavily on the quality of the trial wave function. Although Slater-Jastrow wave functions are the most commonly used variational ansatz in electronic structure, more sophisticated wave-functions are critical to ascertaining new physics. One such wave function is the multiSlater-Jastrow wave function which consists of a Jastrow function multiplied by the sum of Slater determinants. In this paper we describe a method for working with these wavefunctions in QMC codes that is easy to implement, efficient both in computational speed as well as memory, and easily parallelized. The computational cost scales quadratically with particle number making this scaling no worse than the single determinant case and linear with the total number of excitations. Additionally we implement this method and use it to compute the ground state energy of a water molecule.

cond-mat.mtrl-sci↗

arXiv2011

Improved Scaling for Quantum Monte Carlo on Insulators

Quantum Monte Carlo (QMC) methods are often used to calculate properties of many body quantum systems. The main cost of many QMC methods, for example the variational Monte Carlo (VMC) method, is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators so that the cost of constructing Slater matrices in these systems is now linear in the number of particles, whereas computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing the determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution of this paper is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of ILUTP preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of QMC algorithms from O(n^3) per sweep to roughly O(n^2), where n is the number of particles, and a sweep is a sequence of n steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors. Our results show that preconditioned iterative solvers can dramatically reduce the cost of VMC for large(r) systems.

cond-mat.str-el↗

arXiv2009

Hexatic and mesoscopic phases in the 2D quantum Coulomb system

We study the Wigner crystal melting in a two dimensional quantum system of particles interacting via the 1/r Coulomb potential. We use quantum Monte Carlo methods to calculate its phase diagram, locate the Wigner crystal region, and analyze its instabilities towards the liquid phase. We discuss the role of quantum effects in the critical behavior of the system, and compare our numerical results with the classical theory of melting, and the microemulsion theory of frustrated Coulomb systems. We find a Pomeranchuk effect much larger then in solid helium. In addition, we find that the exponent for the algebraic decay of the hexatic phase differs significantly from the Kosterilitz-Thouless theory of melting. We search for the existence of mesoscopic phases and find evidence of metastable bubbles but no mesoscopic phase that is stable in equilibrium.

cond-mat.str-el↗

arXiv2005

Off-Diagonal Long-Range Order in Solid 4He

Measurements of the moment of inertia by Kim and Chan have found that solid 4He acts like a supersolid at low temperatures. To understand the order in solid 4He, we have used Path Integral Monte Carlo to calculate the off-diagonal long range order (ODLRO) (equivalent to Bose-Einstein condensation, BEC). We do not find ODLRO in a perfect hcp crystal at the melting density. We discuss these results in relation to proposed quantum solid trial functions. We conclude that the solid 4He wave function has correlations which suppress both vacancies and BEC.

cond-mat.other↗