Most quantum algorithms designed to generate or probe properties of the ground state of a quantum many-body system require as input an initial state with a large overlap with the desired ground state. One approach for preparing such a ground state is Imaginary Time Evolution (ITE). Recent work by [Motta, M., Sun, C., Tan, A.T.K. et al. (2020)] introduced an algorithm -- which we will refer to as Quantum Imaginary Time Evolution (QITE) -- that shows how ITE can be approximated by a sequence of unitary operators, making QITE potentially implementable on early fault-tolerant quantum computers. In this work, we provide a heuristic study of the capabilities of the QITE algorithm in approximating the ITE of lattice and molecular electronic structure Hamiltonians. We numerically study the performance of the QITE algorithm when provided with a good classical initial state for a large class of systems, some of which are of interest to industrial applications, and check if QITE is able to qualitatively replicate the ITE behavior and improve over a classical mean-field solution. The systems we consider in this work range from one- and two-dimensional lattice systems of various lattice geometries displaying short- and long-range interactions, to active spaces of molecular electronic structure Hamiltonians. In addition to the comparison of QITE and ITE, we explicitly show how imaginary time evolved fermionic Gaussian states can serve as initial states which can be efficiently computed on classical computers and efficiently implemented on quantum computers for generic spin Hamiltonians in arbitrary lattice geometries and dimensions, which can be of independent interest.
Directly in the thermodynamic limit, we show how to combine imaginary and real time evolution of tensor networks to efficiently and accurately find the nonequilibrium steady states (NESS) of one-dimensional dissipative quantum lattices governed by the Lindblad master equation. The imaginary time evolution first bypasses any highly correlated portions of the real-time evolution trajectory by directly converging to the weakly correlated subspace of the NESS, after which real time evolution completes the convergence to the NESS with high accuracy. We demonstrate the power of the method with the d…
For a system strongly coupled to a heat bath, the quantum coherence of the system and the heat bath plays an important role in the system dynamics. This is particularly true in the case of non-Markovian noise. We rigorously investigate the influence of system-bath coherence by deriving the reduced hierarchal equations of motion (HEOM), not only in real time, but also in imaginary time, which represents an inverse temperature. It is shown that the HEOM in real time obtained when we include the system-bath coherence of the initial thermal equilibrium state possess the same form as those obtained…
For quantum (quasi)particles living on complex toboggan-shaped curves which spread over N Riemann sheets the approximate evaluation of topology-controlled bound-state energies is shown feasible. In a cubic-oscillator model the low-lying spectrum is shown decreasing with winding number N.
Double-bracket quantum algorithms for quantum imaginary-time evolution
Nanyang Technological University（南洋理工大学）；Fraunhofer Institute FOKUS（弗劳恩霍夫FOKUS研究所）；Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Keio University（庆应义塾大学）；National University of Singapore（新加坡国立大学）
This thesis describes several topics related to finite temperature studies of strongly correlated systems: finite temperature density matrix embedding theory (FT-DMET), finite temperature metal-insulator transition, and quantum algorithms including quantum imaginary time evolution (QITE), quantum Lanczos (QLanczos), and quantum minimally entangled typical thermal states (QMETTS) algorithms. While the absolute zero temperature is not reachable, studies of physical and chemical problems at finite temperatures, especially at low temperature, is essential for understanding the quantum behaviors of…
The accurate theoretical description of the dynamic properties of correlated quantum many-body systems such as the dynamic structure factor $S(\mathbf{q},ω)$ constitutes an important task in many fields. Unfortunately, highly accurate quantum Monte Carlo methods are usually restricted to the imaginary time domain, and the analytic continuation of the imaginary time density--density correlation function $F(\mathbf{q},τ)$ to real frequencies is a notoriously hard problem. In this work, we argue that no such analytic continuation is required as $F(\mathbf{q},τ)$ contains, by definition, the same …
A projected entangled pair state (PEPS) with ancillas can be evolved in imaginary time to obtain thermal states of a strongly correlated quantum system on a 2D lattice. Every application of a Suzuki-Trotter gate multiplies the PEPS bond dimension $D$ by a factor $k$. It has to be renormalized back to the original $D$. In order to preserve the accuracy of the Suzuki-Trotter (S-T) decomposition, the renormalization has in principle to take into account full environment made of the new tensors with the bond dimension $k\times D$. Here we propose a self-consistent renormalization procedure operati…
Most of the research done on quantum error correction studies an error model in which each qubit is affected by noise, independently of the other qubits. In this paper we study a different noise model -- one in which the noise may be correlated with the qubits it acts upon. We show both positive and negative results. On the one hand, we show controlled-X errors cannot be perfectly corrected, yet can be approximately corrected with sub-constant approximation error. On the other hand, we show that no non-trivial quantum error correcting code can approximately correct controlled phase error with …
A probabilistic imaginary-time evolution quantum algorithm for advection-diffusion equation: Explicit gate-level implementation and comparisons to quantum linear system algorithms
Imaginary Hamiltonian variational ansatz for combinatorial optimization problems
Peking University（北京大学）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学项目（iTHEMS））；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；Computation-Based Science and Technology Research Center, The Cyprus Institute（塞浦路斯研究所计算科学与技术研究中心）；Institut für Physik, Humboldt-Universität zu Berlin（柏林洪堡大学物理研究所）
Hardware-Efficient Ground-State Preparation using Variational Imaginary-Time Majorana Evolution
IQM Quantum Computers GmbH（IQM量子计算机有限公司）；RPTU Kaiserslautern-Landau（莱布尼茨技术大学凯泽斯劳滕-兰道分校）；Universidad Autónoma de Madrid（马德里自治大学）；Technical University of Munich（慕尼黑工业大学）
We introduce an efficient and numerically stable technique to make use of a BCS trial wave function in the computation of correlation functions of strongly correlated quantum fermion systems. The technique is applicable to any projection approach involving paths of independent-fermion propagators, for example in mean-field or auxiliary-field quantum Monte Carlo (AFQMC) calculations. Within AFQMC, in the absence of the sign problem, the methodology allows the use of a BCS reference state which can greatly reduce the required imaginary time of projection, and improves Monte Carlo sampling effici…
Iterative warm-start optimization with quantum imaginary time evolution
Oak Ridge National Laboratory（橡树岭国家实验室）；Quantum Artificial Intelligence Laboratory, USRA RIACS, NASA Ames Research Center（量子人工智能实验室，USRA RIACS，NASA艾姆斯研究中心）
The connection between quantum state recovery and quantum conditional mutual information (QCMI) is studied for the class of purely generated finitely correlated states (pgFCS) of one-dimensional quantum spin chains. For a tripartition of the chain into two subsystems separated by a buffer region, it is shown that a pgFCS is an approximate quantum Markov chain, and stronger, may be approximated by a quantum Markov chain in trace distance, with an error exponentially small in the buffer size. This implies that, (1) a locally corrupted state can be approximately recovered by action of a quantum c…
We give a broad generalisation of the mapping, originally due to Dennis, Kitaev, Landahl and Preskill, from quantum error correcting codes to statistical mechanical models. We show how the mapping can be extended to arbitrary stabiliser or subsystem codes subject to correlated Pauli noise models, including models of fault tolerance. This mapping connects the error correction threshold of the quantum code to a phase transition in the statistical mechanical model. Thus, any existing method for finding phase transitions, such as Monte Carlo simulations, can be applied to approximate the threshold…
Imaginary time evolution and ground state preparation using unitary multi-copy protocols
ITAMP, Center for Astrophysics | Harvard & Smithsonian（哈佛大学与史密森尼天体物理中心）；Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Institute for Theoretical Physics, University of Innsbruck（因斯布鲁克大学理论物理研究所）
We analyze vacuum tunneling in quantum field theory in a general formalism by using the Wigner representation. In the standard instanton formalism, one usually approximates the initial false vacuum state by an eigenstate of the field operator, imposes Dirichlet boundary conditions on the initial field value, and evolves in imaginary time. This approach does not have an obvious physical interpretation. However, an alternative approach does have a physical interpretation: in quantum field theory, tunneling can happen via classical dynamics, seeded by initial quantum fluctuations in both the fiel…
We analyze the many-particle Schrodinger equation for fermions in a thermal ensemble by introducing an exponential operator expansion, defined in the context of thermofield dynamics. The expansion is optimized variationally at each time step through changes in the basis of excitations, which leads to a method of generating approximate differential equations to solve the time dependent problem, and can also be used to cool the system in imaginary time. The method is applied for a specific set of basis transformations and truncation scheme, leading to an explicit set of differential equations th…
Double-bracket quantum algorithms for thermal state preparation
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Centre for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院量子科学与工程中心）；Chula Intelligent and Complex Systems Lab, Department of Physics, Faculty of Science, Chulalongkorn University（朱拉隆功大学理学院物理系智能与复杂系统实验室）；Quantum Computing Center, Keio University（庆应义塾大学量子计算中心）
A macroscopic realization of the strange virtual particles is presented. The classical Helmholtz and the quantum mechanical Schrödinger equations are analogous differential equations. Their imaginary solutions are called evanescent modes in the case of elastic and electromagnetic fields. In the case of non-relativistic quantum mechanical fields they are called tunneling solutions. The imaginary solutions of this differential equation point to strange consequences: They are non local, they are not observable, and they described as virtual particles. During the last two decades QED calculations …
An approximate method is suggested to obtain analytical expressions for the eigenvalues and eigenfunctions of the some quantum optical models. The method is based on the Lie-type transformation of the Hamiltonians. In a particular case it is demonstrated that $E\times ε$ Jahn-Teller Hamiltonian can easily be solved within the framework of the suggested approximation. The method presented here is conceptually simple and can easily be extended to the other quantum optical models. We also show that for a purely imaginary coupling the $E\times ε$ Hamiltonian becomes non-Hermitian but $Pσ_{0}$-symm…
Suzuki-Trotter decomposition is a well-known technique used to calculate the partition function of quantum spin systems, in which the imaginary-time dependence of the partition function occurs inevitably. Since it is very difficult to explicitly treat the imaginary-time dependence of the partition function, we usually neglect the imaginary-time dynamical effect, which is called the static approximation. Although the static approximation is the first approach, it is not even clear when the static approximation is justified for mean-field quantum spin systems, that is, mean-field quantum spin sy…
Sparse-QITE: Discovering Compact Generators for Quantum Imaginary-Time Evolution via Regularization Methods
Universitat de València-CSIC（瓦伦西亚大学-西班牙国家科学研究委员会）；Universidad CEU Cardenal Herrera（塞乌红衣主教埃雷拉大学）；Instituto Universitario de Investigación de Matemática Pura y Aplicada, Universitat Politècnica de València（瓦伦西亚理工大学纯数学与应用数学研究所）
