Quantum Monte Carlo Study of Frustrated Spin Chain under External Field

In this paper a stochastic series expansion quantum Monte Carlo algorithm is used to study a frustratedspin chain with diagonal next-nearest-neighbor interactions.The detailed balance conditions are carefully analyzed toimprove the efficiency of simulation process.As an application of this algothrim,the total magnetization,the staticstructure factor and spin-stiffness are calculated for a certain set of system parameters as a function of external fieldstrength.

Quantum Monte Carlo study of hard-core bosons in Creutz ladder with zero flux

The quantum phase of hard-core bosons in Creutz ladder with zero flux is studied. For a specific regime of the parameters （tx = tp,ty 〈 0）, the exact ground-state is found analytically, which is a dimerized insulator with one electron bound in each rung of the ladder. For the case tx, ty, tp 〉 0, the system is exactly studied using quantum Monte Carlo （QMC） method without a sign problem. It is found that the system is a Mott insulator for small tp and a quantum phase transition to a superfluid phase is driven by increasing tp. The critical t~ is determined precisely by a scaling analysis. Since it is possible that the Creutz ladder is realized experimentally, the theoretical results are interesting to the cold-atom experiments.

A Novel Exact Fixed-node Quantum Monte Carlo Algorithm

In this paper we proposed a novel exact fixed-node quantum Monte Carlo (EFNQMC) algorithm, which is a self-optimizing and self-improving procedure. In contrast to the previous EFNQMC method, the trial function is optimized synchronistically in the diffusion procedure, but not before the beginning of EFNQMC computation. In order to optimize the trial function, the improved steepest descent technique is used, in which the step size is automatically adjustable. The procedure is quasi-Newton and converges super linearly. We also use a novel trial function, which has correct electron-electron and electron-nucleus cusp conditions. The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 11 A1 state of CH2, 1Ag state of C8 and the ground-states of H2, LiH, Li2, H2O, respectively. The test results show that both the novel algorithm and the trial function proposed in the present paper are very excellent.

Fixed-node Quantum Monte Carlo: A Novel Approach

In this paper, a novel method for fixed-node quantum Monte Carlo is given. We have derived an expansion of the eigenvalue of the energy for a system and proved that the value of the energy calculated using the traditional fixed-node quantum Monte Carlo method is only the zero order approximation of the eigenvalue of the energy. But when using our novel method, in the case of only increasing less computing amounts (<1%), we can obtain conveniently the first order approximation, second order approximation, and so on. We have calculated the values of the zero, first and second approximation (0, 1 and 2) of the energies of 11A1 state of CH2, 1Ag (C4h, acet) state of C8 and the ground-state of H2O using this novel method. The results indicate that for 11A1 state of CH2, 1Ag (C4h, acet) state of C8 and the ground-state of H2O it needs only the second order approximation to obtain electronic correlation energy with over 97%. This demonstrates that this novel method is very excellent in both the computing accuracy and the amount of calculation required.

Validation of the Ability of Full Configuration Interaction Quantum Monte Carlo for Studying the 2D Hubbard Model

To validate the ability of full configuration interaction quantum Monte Carlo （FCIQMC） for studying the 2D Hubbard model near half-filling regime, the ground state energies of a 4×44×4 square lattice system with various interaction strengths are calculated. It is found that the calculated results are in good agreement with those obtained by exact diagonalization （i.e., the exact values for a given basis set） when the population of psi particles （psips） is higher than the critical population required to correctly sample the ground state wave function. In addition, the variations of the average computational time per 20 Monte Carlo cycles with the coupling strength and the number of processors are also analyzed. The calculated results show that the computational efficiency of an FCIQMC calculation is mainly affected by the total population of psips and the communication between processors. These results can provide useful references for understanding the FCIQMC algorithm, studying the ground state properties of the 2D Hubbard model for the larger system size by the FCIQMC method and using a computational budget as effectively as possible.

Study on Quantum Finance Algorithm:Quantum Monte Carlo Algorithm based on European Option Pricing

As one of the major methods for the simulation of option pricing,Monte Carlo method assumes random fluctuations in the distribution of asset prices.Under certain uncertainties process,different evolution paths could be simulated so as to finally yield the expectation value of the asset price,which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations.In order to solve the above computational problem,quantum Monte Carlo(QMC)has been established and applied in the relevant systems such as European call options.In this work,both MC and QM methods are adopted to simulate European call options.Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer,the amplitude estimation(AE)algorithm is found to play a secondary role in accelerating the pricing of European options.More importantly,the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.

^(4)He monolayer on graphene:a quantum Monte Carlo study

We revisit the problem of adsorption of a single^(4)He layer on graphene,focusing on the commensurate(C_(1/3))crystalline phase,specifically on whether it may possess a nonzero superfluid response,and on the existence of superfluid phases,either(metastable)liquid or vacancy-doped crystalline.We make use of canonical quantum Monte Carlo simulations at zero and finite temperature,based on a realistic microscopic model of the system.Our results confirm the absence of any superfluid response in the commensurate crystal,and that no thermodynamically stable uniform phase exists at lower coverage.No evidence of a possibly long-lived,metastable superfluid phase at C_(1/3)coverage is found.Altogether,the results of ground-state projection methods and finite-temperature simulations are entirely consistent.

Exact Fixed-node Quantum Monte Carlo： Differential Approach

A differential approach for exact fixed-node quantum Monte Carlo calculation was proposed in this paper. This new algorithm can be used to directly compute the energy differential between two systems in exact fixed-node quantum Monte Carlo process, making the statistical error of calculation reduce to order of 10^-2 kJ/mol and recover about more than 90% of the correlation energy. The approach was employed to set up a potential energy surface of a molecule, through a model of rigid move, and Jacobi transformation utilized to make energy calculation for two configurations of a molecule having good positive correlation. So, an accurate energy differential could be obtained, and the potential energy surface with good quality depicted. This novel algorithm was used to study the potential energy curve of the ground state of BH and the potential energy surface of H3, and could be also applied to study other related fields such as molecular spectroscopy and the energy variation of chemical reactions.

Exact Fixed-node Quantum Monte Carlo: Self-optimizing Procedure

In this paper, a novel exact fixed-node quantum Monte Carlo (EFNQMC) algorithm was proposed, which is a self-optimizing and self-improving procedure. In contrast to the previous EFNQMC method, the importance function of this method is optimized synchronistically in the diffusion procedure, but not before beginning the EFNQMC computation. In order to optimize the importance function, the improved steepest descent technique is used, in which the step size is automatically adjustable. The procedure is quasi-Newton type and converges super linearly. The present method also uses a novel trial function, which has correct electron-electron and electron-nucleus cusp conditions. The novel EFNQMC algorithm and the novel trial function are employed to calculate the energies of 1 1A 1 state of CH 2, 1A g state of C 8 and the ground-states of H 2, LiH, Li 2 and H 2O.

The Lognormal Distribution and Quantum Monte Carlo Data

Quantum Monte Carlo data are often afflicted with distributions that resemble lognormal probability distributions and consequently their statistical analysis cannot be based on simple Gaussian assumptions.To this extent a method is introduced to estimate these distributions and thus give better estimates to errors associated with them.This method entails reconstructing the probability distribution of a set of data,with given mean and variance,that has been assumed to be lognormal prior to undergoing a blocking or renormalization transformation.In doing so,we perform a numerical evaluation of the renormalized sum of lognormal random variables.This technique is applied to a simple quantum model utilizing the single-thread Monte Carlo algorithm to estimate the ground state energy or dominant eigenvalue of a Hamiltonian matrix.

Differential diffusion quantum Monte Carlo method: determination of potential energy surfaces of molecules

A differential approach for self-optimizing diffusion Monte Carlo calculation was proposed in this paper, which is a new algorithm combining three techniques such as optimizing, diffusion and correlation sampling. This method can be used to directly compute the energy differential between two system in the diffusion process, making the statistical error of calculation be reduced to Order of 10?-5 hartree, and recover about more than 80% of the correlation. We employed this approach to set up a potential energy surface of a molecule, used a “rigid move” model, and utilized Jacobi transformation to make energy calculation for two configurations of a molecule having good positive correlation. So, an accurate energy differential could be obtained, and the potential energy surface with good quality can be depicted. In calculation, a technique called “post-equilibrium remaining sample” was set up firstly, which can save about 50% of computation expense. This novel algorithm was used to study the potential as molecular spectroscopy and the energy variation in chemical reactions.

Security Simulation of Continuous-Variable Quantum Key Distribution over Air-to-Water Channel Using Monte Carlo Method

Considering the ocean water＇s optical attenuation and the roughness of the sea surface, we analyze the security of continuous-variable （CV） quantum key distribution （QKD） based Mr-to-water channel. The effects of the absorp- tion and scattering on the transmittance of underwater quantum channel and the maximum secure transmission distance are studied. Considering the roughness of the sea surface, we simulate the performance bounds of CV QKD with different wind speeds using the Monte Carlo method. The results show that even if the secret key rate gradually reduces as the wind speed increases, the maximum transmission distance will not be affected obviously. Compared to the works regarding short-distance underwater optical communication, our research represents a significant step towards establishing secure communication between air platform and submarine vehicle.

Neural network analytic continuation for Monte Carlo:Improvement by statistical errors

This study explores the use of neural network-based analytic continuation to extract spectra from Monte Carlo data.We apply this technique to both synthetic and Monte Carlo-generated data.The training sets for neural networks are carefully synthesized without“data leakage”.We find that the training set should match the input correlation functions in terms of statistical error properties,such as noise level,noise dependence on imaginary time,and imaginary time-displaced correlations.We have developed a systematic method to synthesize such training datasets.Our improved algorithm outperforms the widely used maximum entropy method in highly noisy situations.As an example,our method successfully extracted the dynamic structure factor of the spin-1/2 Heisenberg chain from quantum Monte Carlo simulations.

A sport and a pastime:Model design and computation in quantum many-body systems

We summarize the recent developments in the model design and computation for a few representative quantum manybody systems,encompassing quantum critical metals beyond the Hertz-Millis-Moriya framework with pseudogap and superconductivity,SYK non-Fermi-liquid with self-tuned quantum criticality and fluctuation induced superconductivity,and the flat-band quantum Moirélattice models in continuum where the interplay of quantum geometry of flat-band wave function and the long-range Coulomb interactions gives rise to novel insulating phases at integer fillings and superconductivity away from them.Although the narrative choreography seems simple,we show how important the appropriate model design and their tailor-made algorithmic developments-in other words,the scientific imagination inspired by the corresponding fast experimental developments in the aforementioned systems-compel us to invent and discover new knowledge and insights in the sport and pastime of quantum many-body research.

Emergent O(4) symmetry at the phase transition from plaquette-singlet to antiferromagnetic order in quasi-two-dimensional quantum magnets

Recent experiments [Guo et al., Phys. Rev. Lett. 124 206602(2020)] on thermodynamic properties of the frustrated layered quantum magnet SrCu_(2)(BO_(3))_(2)-the Shastry–Sutherland material-have provided strong evidence for a lowtemperature phase transition between plaquette-singlet and antiferromagnetic order as a function of pressure. Further motivated by the recently discovered unusual first-order quantum phase transition with an apparent emergent O(4) symmetry of the antiferromagnetic and plaquette-singlet order parameters in a two-dimensional "checkerboard J-Q" quantum spin model[Zhao et al., Nat. Phys. 15 678(2019)], we here study the same model in the presence of weak inter-layer couplings. Our focus is on the evolution of the emergent symmetry as the system crosses over from two to three dimensions and the phase transition extends from strictly zero temperature in two dimensions up to finite temperature as expected in SrCu_(2)(BO_(3))_(2).Using quantum Monte Carlo simulations, we map out the phase boundaries of the plaquette-singlet and antiferromagnetic phases, with particular focus on the triple point where these two ordered phases meet the paramagnetic phase for given strength of the inter-layer coupling. All transitions are first-order in the neighborhood of the triple point. We show that the emergent O(4) symmetry of the coexistence state breaks down clearly when the interlayer coupling becomes sufficiently large, but for a weak coupling, of the magnitude expected experimentally, the enlarged symmetry can still be observed at the triple point up to significant length scales. Thus, it is likely that the plaquette-singlet to antiferromagnetic transition in SrCu_(2)(BO_(3))_(2) exhibits remnants of emergent O(4) symmetry, which should be observable due to additional weakly gapped Goldstone modes.

Typicality at quantum-critical points

We discuss the concept of typicality of quantum states at quantum-critical points, using projector Monte Carlo simu- lations of an S ---- 1/2 bilayer Heisenberg antiferromagnet as an illustration. With the projection （imaginary） time t scaled as t= aLz, L being the system length and z the dynamic critical exponent （which takes the value z = 1 in the bilayer model studied here）, a critical point can be identified which asymptotically flows to the correct location and universality class with increasing L, independently of the prefactor a and the initial state. Varying the proportionality factor a and the initial state only changes the cross-over behavior into the asymptotic large-L behavior. In some cases, choosing an optimal factor a may also lead to the vanishing of the leading finite-size corrections. The observation of typicality can be used to speed up simulations of quantum criticality, not only within the Monte Carlo approach but also with other numerical methods where imaginary-time evolution is employed, e.g., tensor network states, as it is not necessary to evolve fully to the ground state but only for sufficiently long times to reach the typicality regime.

Tunable deconfined quantum criticality and interplay of different valence-bond solid phases

We use quantum Monte Carlo simulations to study an S = 1/2 spin model with competing multi-spin interactions. We find a quantum phase transition between a columnar valence-bond solid(cVBS) and a Néel antiferromagnet(AFM), as in the scenario of deconfined quantum-critical points, as well as a transition between the AFM and a staggered valence-bond solid(sVBS). By continuously varying a parameter, the sVBS–AFM and AFM–cVBS boundaries merge into a direct sVBS–cVBS transition. Unlike previous models with putative deconfined AFM–cVBS transitions, e.g., the standard J–Q model,in our extended J–Q model with competing cVBS and sVBS inducing terms the transition can be tuned from continuous to first-order. We find the expected emergent U(1) symmetry of the microscopically Z4 symmetric cVBS order parameter when the transition is continuous. In contrast, when the transition changes to first-order, the clock-like Z4 fluctuations are absent and there is no emergent higher symmetry. We argue that the confined spinons in the sVBS phase are fracton-like.We also present results for an SU(3) symmetric model with a similar phase diagram. The new family of models can serve as a useful tool for further investigating open questions related to deconfined quantum criticality and its associated emergent symmetries.

Analysis of pseudo-random number generators in QMC-SSE method

In the quantum Monte Carlo(QMC)method,the pseudo-random number generator(PRNG)plays a crucial role in determining the computation time.However,the hidden structure of the PRNG may lead to serious issues such as the breakdown of the Markov process.Here,we systematically analyze the performance of different PRNGs on the widely used QMC method known as the stochastic series expansion(SSE)algorithm.To quantitatively compare them,we introduce a quantity called QMC efficiency that can effectively reflect the efficiency of the algorithms.After testing several representative observables of the Heisenberg model in one and two dimensions,we recommend the linear congruential generator as the best choice of PRNG.Our work not only helps improve the performance of the SSE method but also sheds light on the other Markov-chain-based numerical algorithms.

Magnetic excitations of diagonally coupled checkerboards

By using quantum Monte Carlo based stochastic analytic continuation(QMC-SAC)and spin wave theory,we study magnetic excitations of Heisenberg models with diagonally coupled checkerboard structures.We consider three kinds of checkerboard models(DC 2×2,DC 3×3,and CDC 3×3)consisting nearest-neighbor strong J1 and weak J2 antiferromagnetic interactions.When the coupling ratio g=J2/J1 approaches 1,all three diagonal checkerboards have the same long-range antiferromagnetic Neel order at´T=0.When g decreases,the quantum fluctuation can drive DC 2×2 model to quantum paramagnetic state,while DC 3×3 and CDC 3×3 models still have the long-range Neel order.By calculating´the magnetic excitations at different coupling ratios,we find that the low-energy part of magnetic excitations calculated by QMC-SAC can be well explained by the spin wave theory.However,the high-energy parts even deep in the long-range antiferromagnetic phase are beyond the spin wave description.Compared to the g=1 uniform square lattice,the high-energy excitations are more rich in our models.Our study may also draw the attention to the high-energy exctitaions beyond the spin wave theory.

Backflow Transformation for A=3 Nuclei with Artificial Neural Networks

A novel variational wave function defined as a Jastrow factor multiplying a backflow transformed Slater determinant was developed for A=3 nuclei.The Jastrow factor and backflow transformation were represented by artificial neural networks.With this newly developed wave function,variational Monte Carlo calculations were carried out for3H and3He nuclei starting from a nuclear Hamiltonian based on the leadingorder pionless effective field theory.The obtained ground-state energy and charge radii were successfully benchmarked against the results of the highly-accurate hypersphericalharmonics method.The backflow transformation plays a crucial role in improving the nodal surface of the Slater determinant and,thus,providing accurate ground-state energy.