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arXiv:1212.4815 · Physical Review B

Quasi-adiabatic quantum Monte Carlo algorithm for quantum evolution in imaginary time

Chengwei Liu, Anatoli Polkovnikov, Anders W. Sandvik

Published May 23, 2013• 32 citations
arXivHTMLPDF

Abstract

We propose a quantum Monte Carlo (QMC) algorithm for nonequilibrium dynamics in a system with a parameter varying as a function of imaginary time. The method is based on successive applications of an evolving Hamiltonian to an initial state and delivers results for a whole range of the tuning parameter in a single run, allowing for access to both static and dynamic properties of the system. This approach reduces to the standard Schr\"odinger dynamics in imaginary time for quasi-adiabatic evolutions, i.e., including the leading nonadiabatic correction to the adiabatic limit. We here demonstrate this quasi-adiabatic QMC (QAQMC) method for linear ramps of the transverse-field Ising model across its quantum-critical point in one and two dimensions. The critical behavior can be described by generalized dynamic scaling. For the two-dimensional square-lattice system, we use the method to obtain a high-precision estimate of the quantum-critical point ${(h/J)}_{c}=3.04458(7)$, where $h$ is the transverse magnetic field and $J$ is the nearest-neighbor Ising coupling. The QAQMC method can also be used to extract the Berry curvature and the metric tensor.

DOI
10.1103/physrevb.87.174302

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We propose a quantum Monte Carlo (QMC) algorithm for nonequilibrium dynamics in a system with a parameter varying as a function of imaginary time. The method is based on successive applications of an evolving Hamiltonian to an initial state and delivers results for a whole range of the tuning parameter in a single run, allowing for access to both static and dynamic properties of the system. This approach reduces to the standard Schr"odinger dynamics in imaginary time for quasi-adiabatic evolutions, i.e., including the leading nonadiabatic correction to the adiabatic limit. We here demonstrate this quasi-adiabatic QMC (QAQMC) method for linear ramps of the transverse-field Ising model across its quantum-critical point in one and two dimensions. The critical behavior can be described by generalized dynamic scaling. For the two-dimensional square-lattice system, we use the method to obtain a high-precision estimate of the quantum-critical point (h/J)c=3.04458(7){(h/J)}_{c}=3.04458(7)(h/J)c​=3.04458(7), where hhh is the transverse magnetic field and JJJ is the nearest-neighbor Ising coupling. The QAQMC method can also be used to extract the Berry curvature and the metric tensor.
We propose a method to study dynamical response of a quantum system by evolving it with an imaginary-time-dependent Hamiltonian. The leading nonadiabatic response of the system driven to a quantum-critical point is universal and characterized by the same exponents in real and imaginary time. For a linear quench protocol, the fidelity susceptibility and the geometric tensor naturally emerge in the response functions. Beyond linear response, we extend the finite-size scaling theory of quantum phase transitions to nonequilibrium setups. This allows, e.g., for studies of quantum phase transitions in systems of fixed finite size by monitoring expectation values as a function of the quench velocity. Nonequilibrium imaginary-time dynamics is also amenable to quantum Monte Carlo (QMC) simulations, with a scheme that we introduce here and apply to quenches of the transverse-field Ising model to quantum-critical points in one and two dimensions. The QMC method is generic and can be applied to a wide range of models and nonequilibrium setups.
We present a general approach to describe slowly driven quantum systems both in real and imaginary time. We highlight many similarities, qualitative and quantitative, between real and imaginary time evolution. We discuss how the metric tensor and the Berry curvature can be extracted from both real and imaginary time simulations as a response of physical observables. For quenches ending at or near the quantum critical point, we show the utility of the scaling theory for detecting the location of the quantum critical point by comparing sweeps at different velocities. We briefly discuss the universal relaxation to equilibrium of systems after a quench. We finally review recent developments of quantum Monte Carlo methods for studying imaginary time evolution. We illustrate our findings with explicit calculations using the transverse-field Ising model in one dimension.
Phase transitions and their associated critical phenomena are of fundamental importance and play a crucial role in the development of statistical physics for both classical and quantum systems. Phase transitions embody diverse aspects of physics and also have numerous applications outside physics, e.g., in chemistry, biology, and combinatorial optimization problems in computer science. Many problems can be reduced to a system consisting of a large number of interacting agents, which under some circumstances (e.g., changes of external parameters) exhibit collective behavior; this type of scenario also underlies phase transitions. The theoretical understanding of equilibrium phase transitions was put on a solid footing with the establishment of the renormalization group. In contrast, non-equilibrium phase transition are relatively less understood and currently a very active research topic. One important milestone here is the Kibble-Zurek (KZ) mechanism, which provides a useful framework for describing a system with a transition point approached through a non-equilibrium quench process. I developed two efficient Monte Carlo techniques for studying phase transitions, one is for classical phase transition and the other is for quantum phase transitions, both are under the framework of KZ scaling. For classical phase transition, I develop a non-equilibrium quench (NEQ) simulation that can completely avoid the critical slowing down problem. For quantum phase transitions, I develop a new algorithm, named quasi-adiabatic quantum Monte Carlo (QAQMC) algorithm for studying quantum quenches. I demonstrate the utility of QAQMC quantum Ising model and obtain high-precision results at the transition point, in particular showing generalized dynamic scaling in the quantum system. To further extend the methods, I study more complex systems such as spin-glasses and random graphs. The techniques allow us to investigate the problems efficiently. From the classical perspective, using the NEQ approach I verify the universality class of the 3D Ising spin-glasses. I also investigate the random 3-regular graphs in terms of both classical and quantum phase transitions. I demonstrate that under this simulation scheme, one can extract information associated with the classical and quantum spin-glass transitions without any knowledge prior to the simulation.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at (h/J)c=1.00003(8){(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)(h/J)c​=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length L=32−−256L=32--256L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by 3.04451(7)3.044\phantom{\rule{0.16em}{0ex}}51(7)3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size L=24L=24L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
The quantum dynamics of many-qubit systems is an outstanding problem that has recently driven significant advances in both numerical methods and programmable quantum processing units. In this work, we employ a comprehensive toolbox of state-of-the-art numerical approaches to classically simulate the dynamics of the two-dimensional transverse field Ising model. Our methods include three different tensor network techniques -- matrix product states, tree-tensor networks, and two-dimensional tensor-networks under the belief propagation approximation -- as well as time-dependent variational Monte Carlo with Neural Quantum States. We focus on two paradigmatic dynamical protocols: (i) quantum annealing through a critical point and (ii) post-quench dynamics. Our extensive results show the quantitative predictions of various state-of-the-art numerical methods providing a benchmark for future numerical investigations and experimental studies with the aim to push the limitations on classical and QPUs. In particular, our work connects classical simulability to different regimes associated with quantum dynamics in Rydberg arrays - namely, quasi-adiabatic dynamics, the Kibble-Zurek mechanism, and quantum quenches.
We develop an efficient Monte Carlo algorithm, which accelerates slow Monte Carlo dynamics in quasi-one-dimensional Ising spin systems. The loop algorithm of the quantum Monte Carlo method is applied to the classical spin models with highly anisotropic exchange interactions. Both correlation time and real CPU time are reduced drastically. The algorithm is demonstrated in the layered triangular-lattice antiferromagnetic Ising model. We have obtained the relation between the transition temperature and the exchange interaction parameters, which modifies the result of the chain-mean-field theory.