Article
Astronomy & Astrophysics
Tom Shachar, Erez Zohar
Summary: We present a continuous tensor-network construction called continuous projected entangled pair state (cPEPS) for quantum fields, which has the same symmetries as the ground states of relativistic field theories. We demonstrate how this state can approximate and converge to the vacuum state of the free field theory, and propose a regularization-independent method for estimating the convergence. Additionally, we provide a detailed bottom-up construction of cPEPS as the continuum limit of the conventional lattice projected entangled pair state.
Article
Materials Science, Multidisciplinary
Daniel Azses, David F. Mross, Eran Sela
Summary: Symmetry-resolved entanglement is a useful tool for characterizing symmetry-protected topological states. We developed tensor-network methods to study the entanglement data of two-dimensional symmetry-protected topological states by constructing matrix product operators. We verified our approach using the Levin-Gu model and tracked the evolution of entanglement features and their symmetry resolution by using the cohomology formalism.
Article
Computer Science, Artificial Intelligence
Friederike Metz, Marin Bukov
Summary: Quantum many-body control is a key milestone in harnessing quantum technologies, and classically simulating these systems and designing optimal control protocols is challenging due to the exponential growth of the Hilbert space dimension. In this study, a framework combining reinforcement learning with matrix product states is proposed for efficient control of quantum many-body systems. The framework allows for control of larger systems than traditional neural network methods and retains the advantages of deep learning algorithms. The authors demonstrate that reinforcement learning agents can find universal controls, adapt control protocols, and learn to optimally steer previously unseen many-body states.
NATURE MACHINE INTELLIGENCE
(2023)
Article
Materials Science, Multidisciplinary
Philipp Schmoll, Augustine Kshetrimayum, Jan Naumann, Jens Eisert, Yasir Iqbal
Summary: We investigate the ground state of the spin S = 1/2 Heisenberg antiferromagnet on the shuriken lattice, and found that a valence bond crystal with resonances over length six loops emerges as the ground state, yielding the lowest reported estimate of the ground state energy for this model. We also study the model in the presence of an external magnetic field and find the emergence of 0, 1/3, and 2/3 magnetization plateaus, with the 1/3 and 2/3 plateau states respecting translation and point group symmetries and featuring loop-four plaquette resonances.
Article
Optics
Dheeraj Peddireddy, Utkarsh Priyam, Vaneet Aggarwal
Summary: This research proposes an improved VQE algorithm by utilizing a classical gradient computation method that uses tensor-ring approximation. By truncating singular values and preserving the structure of the tensor ring, this method allows for faster evaluation of gradients on classical simulators, addressing the scalability challenge of VQE.
Article
Materials Science, Multidisciplinary
Song Cheng, Lei Wang, Pan Zhang
Summary: Tensor networks, originating from quantum physics, have been generalized in machine learning, with PEPS showing superior performance in image classification compared to treelike tensor networks. Furthermore, our model performs as well as multilayer perceptron classifiers with fewer parameters and increased stability.
Article
Materials Science, Multidisciplinary
Feng-Feng Song, Guang-Ming Zhang
Summary: Using tensor network representation, the phase diagram of a generalized two-dimensional XY spin model with topological vortex excitations has been accurately determined, revealing continuous topological phase transitions and transitions between different vortex binding phases. Additionally, a hybrid BKT and Ising universality class of topological phase transition has been established, along with the discovery of a multicritical point where three phase transition lines intersect.
Article
Materials Science, Multidisciplinary
Miguel Frias-Perez, Mari Carmen Banuls
Summary: This paper presents a tensor network method, called the transverse folding algorithm, for computing time-dependent local observables in out-of-equilibrium quantum spin chains. The method overcomes the limitations of matrix product states when the entanglement grows slower in time than in space. A contraction strategy based on the exact light cone structure of the tensor network is proposed, which can be combined with the hybrid truncation approach to improve the efficiency of the method. The performance of this strategy is demonstrated for transport coefficients and potential extensions to other dynamical quantities are discussed.
Article
Materials Science, Multidisciplinary
Nils Niggemann, Bjorn Sbierski, Johannes Reuther
Summary: The study introduces a general functional renormalization group approach based on Majorana fermions to improve the accuracy of treating frustrated quantum spin systems at finite temperatures. By implementing spin operators via an SO(3) symmetric Majorana representation, the method shows significantly enhanced accuracy compared to previous methods at finite temperatures. The development of functional renormalization group approaches with Majorana fermions expands the applicability of such methods in a broader scope.
Article
Physics, Multidisciplinary
Peng-Fei Zhou, Ying Lu, Jia-Hao Wang, Shi-Ju Ran
Summary: Efficient methods for accessing quantum entanglement in many-body systems have been a long-standing concern due to exponential scaling complexity. In this study, a Schmidt tensor network state (Schmidt TNS) is proposed, which efficiently represents the Schmidt decomposition of quantum states of finite and infinite sizes with nontrivial bipartition boundary. The key idea is to represent the Schmidt coefficients and transformations as tensor networks with linearly scaled complexity. Simulation results demonstrate the validity of the Schmidt TNS, showing that the encoded Schmidt coefficients are weakly entangled, supporting the efficiency of using matrix product states (MPS) for encoding.
PHYSICAL REVIEW LETTERS
(2023)
Article
Quantum Science & Technology
Yuchen Guo, Shuo Yang
Summary: Quantum decoherence caused by imperfect manipulation of quantum devices is a crucial issue in the NISQ era. The traditional method of using error rates to parameterize quantum noise channels does not provide an explicit relationship between the decoherence effect and the error rate. This study proposes characterizing the decoherence effect of a noise channel based on the physical implementability of its inverse, which quantifies the difficulty of simulating the noise inverse using accessible quantum channels. Two concise inequalities are established to connect the decrease in state purity and logarithmic negativity after a noise channel to the physical implementability of the noise inverse, which should be decomposed as mutually orthogonal unitaries or product channels. Numerical demonstrations are conducted on commonly adopted two-qubit noise models, and the results contribute to the theoretical research on the entanglement properties of noise channels and provide guiding principles for quantum circuit design.
NPJ QUANTUM INFORMATION
(2023)
Article
Chemistry, Physical
Tobias Serwatka, Pierre-Nicholas Roy
Summary: In this study, a density matrix-based optimization procedure is used to generate customized basis functions for describing rotating water molecule chains at different intermolecular distances. The procedure provides a compact basis with a clear truncation criterion based on the population of single particle basis functions. For the water trimer, the convergence behavior of various properties is discussed and shown to be superior to an energy-based truncated basis. It is demonstrated that the optimized basis reduces the required number of basis functions by at least an order of magnitude. Finally, the optimization procedure is applied to larger chains of up to ten water molecules, investigating the formation of hydrogen bonds and their impact on the net polarization of the chain.
JOURNAL OF CHEMICAL PHYSICS
(2023)
Article
Materials Science, Multidisciplinary
Manuel Schneider, Johann Ostmeyer, Karl Jansen, Thomas Luu, Carsten Urbach
Summary: Tensor networks are a powerful tool for simulating various physical models, overcoming sign problems in Monte Carlo simulations. Using imaginary-time evolution, accurate estimators for ground states of models like the Hubbard model have been provided. A method to directly simulate the subspace with an odd number of fermions has also been presented.
Article
Optics
Jack Y. Araz, Michael Spannowsky
Summary: This paper compares classical Tensor Networks (TN) with TN-inspired quantum circuits in the context of machine learning. The study shows that classical TNs require larger dimensions and result in a flat loss landscape, making optimization challenging. By using quantitative metrics, the paper also demonstrates that classical TNs require more training samples compared to TN-inspired quantum circuits. Additionally, the study explores the possibility of hybrid classical-quantum TNs and presents different TN ansatzes.
Article
Quantum Science & Technology
Jose Garre-Rubio, Laurens Lootens, Andras Molnar
Summary: We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This characterization provides a set of quantities satisfying the coupled pentagon equations, which give complete invariants for an MPO symmetry protected phase. Our techniques facilitate the numerical study and match the known renormalization fixed point classifications. Additionally, we recover the symmetry protected topological order classification and explore the interplay between time reversal symmetry and MPO symmetries.
Article
Physics, Multidisciplinary
Benoit Tuybens, Jacopo De Nardis, Jutho Haegeman, Frank Verstraete
Summary: Matrix product states and continuous matrix product states are faithful representations of ground states of one-dimensional quantum spin systems and interacting field theories in one spatial dimension, respectively. By constructing a piecewise linear parameterization and using high-order Taylor expansions, we are able to optimize the continuous matrix product states efficiently for systems with inhomogeneous external potentials. This method allows for the calculation of exact reduced density matrices and exact computation of energy and its backwards derivative.
PHYSICAL REVIEW LETTERS
(2022)
Article
Physics, Multidisciplinary
Robijn Vanhove, Laurens Lootens, Hong-Hao Tu, Frank Verstraete
Summary: We explore the topological defects of the critical three-state Potts spin system on different surfaces and analyze their characteristics using the lattice setting and tensor network descriptions. By utilizing matrix product operators and intertwiners, we gain a comprehensive understanding of the system's symmetries and boundary conditions.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Physics, Multidisciplinary
Zongping Gong, Tommaso Guaita, J. Ignacio Cirac
Summary: In this paper, we study free fermions on lattices in arbitrary dimensions with hopping amplitudes that decay with a power-law. We provide a comprehensive set of constraints on the equilibrium and nonequilibrium properties of these fermions in the regime where the power-law decay is larger than the spatial dimension. Our results include the derivation of an optimal Lieb-Robinson bound and a clustering property for the Green's function. We also discuss the implications of these results on topological phases in long-range free-fermion systems.
PHYSICAL REVIEW LETTERS
(2023)
Article
Physics, Multidisciplinary
Bram Vanhecke, David Devoogdt, Frank Verstraete, Laurens Vanderstraeten
Summary: We propose an efficient approximation method using tensor-network representations to calculate the exponential of a local operator in quantum spin systems. Through benchmarking, we demonstrate its effectiveness for large time steps in one-dimensional systems. We apply this method to represent the thermal density operator of a two-dimensional spin system and successfully obtain a continuous phase transition in the correct universality class.
Editorial Material
Physics, Applied
Frank Verstraete, Tomotoshi Nishino, Ulrich Schollwoeck, Mari Carmen Banuls, Garnet K. Chan, Miles E. Stoudenmire
Summary: The density matrix renormalization group (DMRG) algorithm, developed in 1992, is a variational optimization algorithm used by physicists to find the ground states of quantum many-body systems in low dimensions. It not only serves as a powerful numerical method, but also brings together ideas from theoretical condensed matter physics and quantum information, leading to advancements in quantum chemistry and the study of dissipative systems. DMRG also popularized the use of tensor networks as mathematical representations of quantum many-body states, extending its applications beyond quantum systems. Six researchers discuss the early history of DMRG and its impact over the past three decades.
NATURE REVIEWS PHYSICS
(2023)
Article
Physics, Mathematical
Jacob C. Bridgeman, Laurens Lootens, Frank Verstraete
Summary: The study utilizes a generalization to weak Hopf algebras to provide a new and verifiable condition on skeletal data for determining the invertibility of a given bimodule category, which defines a Morita equivalence. The condition is derived from Schur orthogonality relations on the characters of the annular algebra associated with a module category. As applications, an algorithm is provided for constructing the complete skeletal data of the invertible bimodule category related to a given module category, and the condition for invertibility is shown to be equivalent to MPO-injectivity in tensor network representations of string-net models with topological order. The study also discusses applications to generalized symmetries, including a generalized Wigner-Eckart theorem.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Quantum Science & Technology
Laurens Lootens, Clement Delcamp, Gerardo Ortiz, Frank Verstraete
Summary: We propose a systematic approach for generating and classifying duality transformations in one-dimensional quantum lattice systems. Our construction focuses on the role of global symmetries, including Abelian and non-Abelian groups, as well as categorical symmetries. These symmetries can be realized as matrix product operators, allowing us to extract a fusion category that characterizes the algebra of symmetric operators. We define a duality using pairs of module categories that give rise to dual realizations of the bond algebra and Hamiltonians.
Article
Optics
Kerstin Beer, Megha Khosla, Julius Koehler, Tobias J. Osborne, Tianqi Zhao
Summary: In this paper, an approach is developed to improve learning efficiency by leveraging the graph structure of the quantum source for an arbitrary quantum neural network (QNN) ansatz. A self-supervised objective is devised and optimized to capture the information-theoretic closeness of quantum states during QNN training. Numerical simulations demonstrate that this approach enhances learning efficiency and generalization behavior of the base QNN. Moreover, scalable quantum implementations of the learning procedure described in this paper are likely feasible on the next generation of quantum computing devices.
Article
Optics
Ivana Kurecic, Tobias J. Osborne
Summary: The dynamics of interacting quantum systems in the presence of disorder is studied and an exact representation for disorder-averaged quantities via Ito stochastic calculus is obtained. The stochastic integral representation allows for analytic approximation, potential applicability to interacting systems, and compatibility with tensor network methods. The integral can be expanded to produce a series of approximations, the first of which includes all diffusive corrections and is manifestly completely positive. As examples, expressions for the density of states and spectral form factor for the Anderson model are obtained.
Article
Materials Science, Multidisciplinary
Bram Vancraeynest-De Cuiper, Jacob C. Bridgeman, Nicolas Dewolf, Jutho Haegeman, Frank Verstraete
Summary: In this study, we systematically investigate the symmetry-protected topological gapped phases of quantum spin chains using matrix product states. We consider the spatial symmetries of the one-dimensional lattice together with an additional vertical reflection. We identify seventeen distinct non-trivial phases and compare the topological indices obtained from the MPS analysis with the group cohomological predictions.
Article
Physics, Fluids & Plasmas
Lander Burgelman, Lukas Devos, Bram Vanhecke, Frank Verstraete, Laurens Vanderstraeten
Summary: Tensor-network methods are used to compare the scaling behavior of the two-dimensional classical Heisenberg and RP2 models. The study shows that uniform matrix product states with explicit SO(3) symmetry can accurately probe long correlation lengths. The results reveal fundamental differences in scaling behavior between the two models.
Article
Astronomy & Astrophysics
Bram Vanhecke, Frank Verstraete, Karel Van Acoleyen
Summary: In this study, we investigate the 2.04 model at criticality in 0 thorn 2 dimensions and focus on the scaling properties originating from UV and IR physics. We demonstrate that the entanglement entropy, correlation length, and order parameters exhibit distinctive double scaling properties that prove to be powerful tools in data analysis.
Article
Materials Science, Multidisciplinary
Lukas Devos, Laurens Vanderstraeten, Frank Verstraete
Summary: In this study, the Haldane gap of the SU(3) spin [3 0 0] Heisenberg model was calculated using variational uniform fully symmetric SU(3) matrix product states. The minimal gap was found in the [2 1 0] sector at momentum 2z/3. The symmetry protected topological order of the ground state was discussed, and the full dispersion relation of the elementary excitations and the correlation lengths of the system were determined.
Article
Materials Science, Multidisciplinary
Laurens Vanderstraeten, Lander Burgelman, Boris Ponsioen, Maarten Van Damme, Bram Vanhecke, Philippe Corboz, Jutho Haegeman, Frank Verstraete
Summary: In this paper, the authors reformulate the contraction of a subclass of PEPS as a variational problem that is independent of the algorithm. They use this variational feature to assess and compare the accuracy of CTMRG and VUMPS contractions, and also propose a new variational contraction scheme for computing general N-point correlation functions.
Article
Materials Science, Multidisciplinary
Laurens Lootens, Bram Vancraeynest-De Cuiper, Norbert Schuch, Frank Verstraete
Summary: The study constructs a constant depth quantum circuit for mapping between Morita-equivalent string-net models. The circuit, with its constant depth and unitarity, is unable to alter the topological order, thereby demonstrating that Morita-equivalent string nets are in the same phase.
