We discuss a novel form of the variational approach in Quantum Field Theory in which the trial quantum configuration is represented directly in terms of relevant expectation values rather than, e.g., increasingly complicated structure from Fock space. The quantum algebra imposes constraints on such expectation values so that the variational problem is formulated here as an optimization under constraints. As an example of application of such approach we consider the study of ground state and critical properties in a variant of nonlinear sigma model.
We study a model of a two-level system (i.e. a qubit) in interaction with the electromagnetic field. By means of homodyne detection, one field-quadrature is observed continuously in time. Due to the interaction, information about the initial state of the qubit is transferred into the field, thus influencing the homodyne measurement results. We construct random variables (pointers) on the probability space of homodyne measurement outcomes having distributions close to the initial distributions of sigma-x and sigma-z. Using variational calculus, we find the pointers that are optimal. These optim…
The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potentials. As an example, the method is applied to discuss the ground state solution of Yukawa potential, using Hulthen solution as the trial function.
We evaluate perturbatively the density matrix in the low-temperature limit and thus the ground-state wave function of the anharmonic oscillator up to second order in the coupling constant. We then employ Kleinert's variational perturbation theory to determine the ground-state wave function for all coupling strengths.
Ground-state preparation for a given Hamiltonian is a common quantum-computing task of great importance and has relevant applications in quantum chemistry, computational material modeling, and combinatorial optimization. We consider an approach to simulate dissipative non-Hermitian Hamiltonian quantum dynamics using Hamiltonian simulation techniques to efficiently recover the ground state of a target Hamiltonian. The proposed method facilitates the energy transfer by repeatedly projecting ancilla qubits to the desired state, rendering the effective non-Hermitian Hamiltonian evolution on the sy…
Preparing the ground states of a many-body system is essential for evaluating physical quantities and determining the properties of materials. This work provides a quantum ground state preparation scheme with shallow variational warm-start to tackle the bottlenecks of current algorithms, i.e., demand for prior ground state energy information and lack of demonstration of efficient initial state preparation. Particularly, our methods would not experience the instability for small spectral gap $Δ$ during pre-encoding the phase factors since our methods involve only $\widetilde{O}(1)$ factors whil…
Finding and probing the ground states of spin lattices, such as the antiferromagnetic Heisenberg model on the kagome lattice (KAFH), is a very challenging problem on classical computers and only possible for relatively small systems. We propose using the Variational Quantum Eigensolver (VQE) to find the ground state of the KAFH on a quantum computer. We find efficient ansatz circuits and show how physically interesting observables can be measured efficiently. To investigate the expressiveness and scaling of our ansatz circuits we used classical, exact simulations of VQE for the KAFH for differ…
One of the key applications for the emerging quantum simulators is to emulate the ground state of many-body systems, as it is of great interest in various fields from condensed matter physics to material science. Traditionally, in an analog sense, adiabatic evolution has been proposed to slowly evolve a simple Hamiltonian, initialized in its ground state, to the Hamiltonian of interest such that the final state becomes the desired ground state. Recently, variational methods have also been proposed and realized in quantum simulators for emulating the ground state of many-body systems. Here, we …
A scalable quantum-neural hybrid variational algorithm for ground state estimation
Seoul National University（首尔大学）；NextQuantum, Seoul National University（NextQuantum，首尔大学）；Automation and Systems Research Institute, Seoul National University（自动化与系统研究所，首尔大学）；Dept. of Computer Science and Engineering, Seoul National University（计算机科学与工程系，首尔大学）；Dept. of Electrical and Computer Engineering, Seoul National University（电气与计算机工程系，首尔大学）；Institute of Computer Technology, Seoul National University（计算机技术研究所，首尔大学）
Quantum computers have been widely speculated to offer significant advantages in obtaining the ground state of difficult Hamiltonian in chemistry and physics. In this work, we first propose a Lyapunov control-inspired strategy to accelerate the well-established imaginary-time method for ground-state preparation. We also dig for the source of acceleration of the imaginary-time process under Lyapunov control with theoretical understanding and dynamic process visualization. To make the method accessible in the noisy intermediate-scale quantum era, we further propose a variational form of the algo…
A variational upper bound on the ground state energy $E_{\rm gs}$ of a quantum system, $E_{\rm gs} \leqslant \langle Ψ|H| Ψ\rangle$, is well-known (here $H$ is the Hamiltonian of the system and $Ψ$ is an arbitrary wave function). Much less known are variational {\it lower} bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as $H=\sum_{i=1}^M H_i$, where a term $H_i$ is supported on the $i$'th…
A key goal of digital quantum computing is the simulation of fermionic systems such as molecules or the Hubbard model. Unfortunately, for present and near-future quantum computers the use of quantum error correction schemes is still out of reach. Hence, the finite error rate limits the use of quantum computers to algorithms with a low number of gates. The variational Hamiltonian ansatz (VHA) has been shown to produce the ground state in good approximation in a manageable number of steps. Here we study explicitly the effect of gate errors on its performance. The VHA is inspired by the adiabatic…
We introduce a simple, physically-motivated variational ground state for the quantum Rabi model, and demonstrate that it provides a high-fidelity approximation of the true ground state in all parameter regimes (including intermediate and strong coupling regimes). Our variational state is constructed using Gaussian cavity states and nonorthogonal qubit pointer states, and contains only three variational parameters. We use our state to develop a heuristic understanding of how the ground state evolves with increasing coupling, and find a previously unexplored regime where the ground state corresp…
We use the Variational Quantum Eigensolver (VQE) as implemented in the Qiskit software package to compute the ground state energy of small molecules derived from water, H$_2$O, and hydrogen cyanide, HCN. The work aims to benchmark algorithms for calculating the electronic structure and energy surfaces of molecules of relevance to prebiotic chemistry, beginning with water and hydrogen cyanide, and to run them on the available simulated and physical quantum hardware. The numerical calculations of the algorithms for small quantum processors allow us to design more efficient protocols to be run in…
Accelerated spin-adapted ground state preparation with non-variational quantum algorithms
The University of Tokyo（东京大学）；Quemix Inc.（Quemix公司）；Institute for Physics of Intelligence, The University of Tokyo（东京大学智能物理研究所）；Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）；National Institutes for Quantum Science and Technology (QST)（国立量子科学技术研究院）
We propose the simulation of quantum-optical systems in the ultrastrong-coupling regime using a variational quantum algorithm. More precisely, we introduce a short-depth variational form to prepare the groundstate of the multimode Dicke model on a quantum processor and present proof-of-principle results obtained via cloud access to an IBM device. We moreover provide an algorithm for characterizing the groundstate by Wigner state tomography. Our work is a first step towards digital quantum simulation of quantum-optical systems with potential applications to the spin-boson, Kondo and Jahn-Teller…
We present a physically motivated variational wave function for the ground state of the asymmetric quantum Rabi model (AQRM). The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states, and relies only on three variational parameters in the regimes of interest where the squeezing effect becomes negligible. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. We use the variational result to calculate various relevant physical observables of the ground state…
We propose a variational quantum attack algorithm (VQAA) for classical AES-like symmetric cryptography, as exemplified the simplified-data encryption standard (S-DES). In the VQAA, the known ciphertext is encoded as the ground state of a Hamiltonian that is constructed through a regular graph, and the ground state can be found using a variational approach. We designed the ansatz and cost function for the S-DES's variational quantum attack. It is surprising that sometimes the VQAA is even faster than Grove's algorithm as demonstrated by our simulation results. The relationships of the entanglem…
Hamiltonian-reconstruction distance as a success metric for the Variational Quantum Eigensolver
School of Applied and Engineering Physics, Cornell University（康奈尔大学应用工程物理学院）；Laboratory of Atomic and Solid State Physics, Cornell University（康奈尔大学原子与固态物理实验室）；Kavli Institute at Cornell for Nanoscale Science, Cornell University（康奈尔大学卡弗里纳米科学研究所）
We present and analyze large-scale simulation results of a hybrid quantum-classical variational method to calculate the ground state energy of the anti-ferromagnetic Heisenberg model. Using a massively parallel universal quantum computer simulator, we observe that a low-depth-circuit ansatz advantageously exploits the efficiently preparable Néel initial state, avoids potential barren plateaus, and works for both one- and two-dimensional lattices. The analysis reflects the decisive ingredients required for a simulation by comparing different ansätze, initial parameters, and gradient-based versu…
An analytical variational method for the ground state of the biased quantum Rabi model in the ultra-strong coupling regime is presented. This analytical variational method can be obtained by a unitary transformation or alternatively by assuming the form of ground state wave function. The key point of the method is to introduce a variational parameter $λ$, which can be determined by minimizing the energy functional. Using this method, we calculate physical observables with high accuracy in comparison with the numerical exact one. Our method evidently improves the results from the widely used ge…
We propose a ground-state ansatz for the Ohmic spin-boson model that improves upon the variational treatment of Silbey and Harris for biased systems in the scaling limit. In particular, it correctly captures the smooth crossover behaviour expected for the ground-state magnetisation when moving between the delocalised and localised regimes of the model, a feature that the variational treatment is unable to properly reproduce, while it also provides a lower ground-state energy estimate in the crossover region. We further demonstrate the validity of our intuitive ground-state by showing that it l…
We propose a variational quantum algorithm to study the real time dynamics of quantum systems as a ground-state problem. The method is based on the original proposal of Feynman and Kitaev to encode time into a register of auxiliary qubits. We prepare the Feynman-Kitaev Hamiltonian acting on the composed system as a qubit operator and find an approximate ground state using the Variational Quantum Eigensolver. We apply the algorithm to the study of the dynamics of a transverse field Ising chain with an increasing number of spins and time steps, proving a favorable scaling in terms of the number …
We present an ansatz for the ground states of the Quantum Sherrington-Kirkpatrick model, a paradigmatic model for quantum spin glasses. Our ansatz, based on the concept of generalized coherent states, very well captures the fundamental aspects of the model, including the ground state energy and the position of the spin glass phase transition. It further enables us to study some previously unexplored features, such as the non-vanishing longitudinal field regime and the entanglement structure of the ground states. We find that the ground state entanglement can be captured by a simple ensemble of…
An adaptive variational quantum imaginary time evolution (AVQITE) approach is introduced that yields efficient representations of ground states for interacting Hamiltonians on near-term quantum computers. It is based on McLachlan's variational principle applied to imaginary time evolution of variational wave functions. The variational parameters evolve deterministically according to equations of motions that minimize the difference to the exact imaginary time evolution, which is quantified by the McLachlan distance. Rather than working with a fixed variational ansatz, where the McLachlan dista…
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）
Counterdiabatic (CD) protocols enable fast driving of quantum states by invoking an auxiliary adiabatic gauge potential (AGP) that suppresses transitions to excited states throughout the driving process. Usually, the full spectrum of the original unassisted Hamiltonian is a prerequisite for constructing the exact AGP, which implies that CD protocols are extremely difficult for many-body systems. Here, we apply a variational CD protocol recently proposed by P. W. Claeys et al. [Phys. Rev. Lett. 123, 090602 (2019)] to a two-component fermionic Hubbard model in one spatial dimension. This protoco…
We introduce a new discrete-time variational principle inspired by the quantum clock originally proposed by Feynman, and use it to write down quantum evolution as a ground state eigenvalue problem. The construction allows one to apply ground state quantum many-body theory to quantum dynamics, extending the reach of many highly developed tools from this fertile research area. Moreover this formalism naturally leads to an algorithm to parallelize quantum simulation over time. We draw an explicit connection between previously known time-dependent variational principles and the new time embedded v…
As quantum computing approaches its first commercial implementations, quantum simulation emerges as a potentially ground-breaking technology for several domains, including Biology and Chemistry. However, taking advantage of quantum algorithms in Quantum Chemistry raises a number of theoretical and practical challenges at different levels, from the conception to its actual execution. We go through such challenges in a case study of a quantum simulation for the hydrogen (H2) and lithium hydride (LiH) molecules, at an actual commercially available quantum computer, the IBM Q. The former molecule …
We provide an algorithm for preparing the thermofield double (TFD) state of the Sachdev-Ye-Kitaev model without the need for an auxiliary bath. Following previous work, the TFD can be cast as the approximate ground state of a Hamiltonian, $H_{\text{TFD}}$. Using variational quantum circuits, we propose and implement a gradient-based algorithm for learning parameters that find this ground state, an application of the variational quantum eigensolver. Concretely, we find quantum circuits that prepare the ground state of $H_{\text{TFD}}$ for the $q=4$ SYK model up to $N=12$.
Recently, Singh and Chandrasekharan showed that fixed points of the non-linear O(3) sigma model can be reproduced near a quantum phase transition of a spin model with just two qubits per lattice site. In a paper by the NuQS collaboration, the proposal is made to simulate such field theories on a quantum computer using the universal properties of a similar model. In this paper, following that direction, we demonstrate how to prepare the ground state of the model from and measure a dynamical quantity of interest, the O(3) Noether charge, on a quantum computer. In particular, we apply Trotter met…
Warm Starts, Cold States: Exploiting Adiabaticity for Variational Ground-States
Institute of Physics, Ecole Polytechnique F\' e d\' e rale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland；Centre for Quantum Science；Engineering, Ecole Polytechnique F\' e d\' e rale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland；Barcelona Supercomputing Center, Plaça Eusebi G\"uell, 1-3, 08034 Barcelona, Spain；Institute for Theoretical Physics, ETH Zürich, 8093, Zürich, Switzerland
We investigate an approach for studying the ground state of a quantum many-body Hamiltonian that is based on treating the correlation functions as variational parameters. In this approach, the challenge set by the exponentially-large Hilbert space is circumvented by approximating the positivity of the density matrix, order-by-order, in a way that keeps track of a limited set of correlation functions. In particular, the density-matrix description is replaced by a correlation matrix whose dimension is kept linear in system size, to all orders of the approximation. Unlike the conventional variati…
Recent research has demonstrated the usefulness of neural networks as variational ansatz functions for quantum many-body states. However, high-dimensional sampling spaces and transient autocorrelations confront these approaches with a challenging computational bottleneck. Compared to conventional neural networks, physical-model devices offer a fast, efficient and inherently parallel substrate capable of related forms of Markov chain Monte Carlo sampling. Here, we demonstrate the ability of a neuromorphic chip to represent the ground states of quantum spin models by variational energy minimizat…
By Using the variational Monte Carlo (VMC) method, we calculate the 1sσ_g state energies, the dissociation energies and the binding energies of the hydrogen molecule and its molecular ion in the presence of an aligned magnetic field regime between 0 a.u. and 10 a.u. The present calculations are based on using two types of compact and accurate trial wave functions, which are put forward for consideration in calculating energies in the absence of magnetic field. The obtained results are compared with the most recent accurate values. We conclude that the applications of VMC method can be extended…
We present a variational method for approximating the ground state of spin models close to (Richardson-Gaudin) integrability. This is done by variationally optimizing eigenstates of integrable Richardson-Gaudin models, where the toolbox of integrability allows for an efficient evaluation and minimization of the energy functional. The method is shown to return exact results for integrable models and improve substantially on perturbation theory for models close to integrability. For large integrability-breaking interactions, it is shown how (avoided) level crossings necessitate the use of excite…
We propose a qubit efficient scheme to study ground state properties of quantum many-body systems on near-term noisy intermediate scale quantum computers. One can obtain a tensor network representation of the ground state using a number of qubits smaller than the physical degrees of freedom. By increasing the qubits number, one can exponentially increase the bond dimension of the tensor network variational ansatz on a quantum computer. Moreover, we construct circuits blocks which respect U(1) and SU(2) symmetries of the physical system and show that they can significantly speed up the training…
We benchmark the accuracy of a variational quantum eigensolver based on a finite-depth quantum circuit encoding ground state of local Hamiltonians. We show that in gapped phases, the accuracy improves exponentially with the depth of the circuit. When trying to encode the ground state of conformally invariant Hamiltonians, we observe two regimes. A finite-depth regime, where the accuracy improves slowly with the number of layers, and a finite-size regime where it improves again exponentially. The cross-over between the two regimes happens at a critical number of layers whose value increases lin…
