Multiscale Entanglement Renormalization Ansatz Research Articles

A hybrid quantum-classical classification model based on branching multi-scale entanglement renormalization ansatz

Tensor networks are emerging architectures for implementing quantum classification models. The branching multi-scale entanglement renormalization ansatz (BMERA) is a tensor network known for its enhanced entanglement properties. This paper introduces a hybrid quantum-classical classification model based on BMERA and explores the correlation between circuit layout, expressiveness, and classification accuracy. Additionally, we present an autodifferentiation method for computing the cost function gradient, which serves as a viable option for other hybrid quantum-classical models. Through numerical experiments, we demonstrate the accuracy and robustness of our classification model in tasks such as image recognition and cluster excitation discrimination, offering a novel approach for designing quantum classification models.

Entanglement Renormalization for Quantum Field Theories with Discrete Wavelet Transforms

We propose an adaptation of Entanglement Renormalization for quantum field theories that, through the use of discrete wavelet transforms, strongly parallels the tensor network architecture of the Multiscale Entanglement Renormalization Ansatz (a.k.a. MERA). Our approach, called wMERA, has several advantages of over previous attempts to adapt MERA to continuum systems. In particular, (i) wMERA is formulated directly in position space, hence preserving the quasi-locality and sparsity of entanglers; and (ii) it enables a built-in RG flow in the implementation of real-time evolution and in computations of correlation functions, which is key for efficient numerical implementations. As examples, we describe in detail two concrete implementations of our wMERA algorithm for free scalar and fermionic theories in (1+1) spacetime dimensions. Possible avenues for constructing wMERAs for interacting field theories are also discussed.

Inhomogeneous quenches as state preparation in two-dimensional conformal field theories

The nonequilibrium process where the system does not evolve to the featureless state, that without quantum properties, is one of the new central objects in the nonequilibrium phenomena. In this paper, starting from the short-range entangled state in the two-dimensional conformal field theories (2D CFTs), the boundary state with a regularization, we evolve the system with the inhomogeneous Hamiltonians called Möbius/sine-square-deformed (SSD) ones. Regardless of the details of CFTs considered in this paper, during the Möbius evolution, the entanglement entropy exhibits the periodic motion called quantum revival. During SSD time evolution, except for some subsystems, in the large time regime, entanglement entropy and mutual information are approximated by those for the vacuum state. We argue the time regime for the subsystem to cool down to vacuum one is t1≫O(LlA), where t1, L, and lA are time, system, and subsystem sizes. This finding suggests the inhomogeneous quench induced by the SSD Hamiltonian may be used as the preparation for the approximately vacuum state. We propose the gravity dual of the systems considered in this paper, furthermore, and generalize it. In addition to them, we discuss the relation between the inhomogeneous quenches and continuous multiscale entanglement renormalization ansatz (cMERA). Published by the American Physical Society 2024

From digital to quantum epidemiology: The Quantum Data Lake concept for big data related to viral infectious diseases

The development of quantum epidemiology represents the next anticipated phase in epidemiology transformation, driven by the emergence of new quantum technologies. Epidemiology is currently transitioning into the digital era and undergoing a paradigm shift from a data-driven to a value-driven strategy. Epidemiology data are characterized by uncertainty, multidimensionality, and disconnection, thereby correlating with the preferential quantum approach for data exposition, value creation, and modeling. Examples of such complex epidemiology data include the data on DNA viruses with associated symptoms and diseases. The Quantum Data Lake concept is proposed and consists of several layers and quantum tools, including Robson semantic triples, Quantum Universal Exchange Language, Hyperbolic Dirac Net, “quantum ribosome” structure, quantum random access memory, teleportation, Quantum Query Language, non-Hermitian gates, and tensor networks (e.g., matrix product state, projected entangled pair state, and multiscale entanglement renormalization ansatz [MERA]), alongside PT-symmetry properties. PT-symmetry can serve as an intuitive modeling tool, and PT-symmetry breaking can detect the hidden shift in the information that is permanently updated in the Data Lake. The computational output is presented as PT-symmetry gain/loss equilibrium breaking in the form of a complex number, i.e., two possible variants of epidemic modeling. For MERA, non-Hermiticity with spontaneous PT-symmetry breaking can theoretically appear as a violation of the entanglement monotonicity and local entanglement gain, leading to a non-reversible character of the coarse-graining transformation. The duality of PT-symmetry equilibrium breaking can be compared to, for example, the estimation of the best and worst scenarios simultaneously, or the gain of entanglement can display a significant correlation between some studied parameters embedded into the data. The fundamental difference between digital and quantum epidemiology is the implementation of quantum logic and reliance on a quantum theory.

Surface growth approach for bulk reconstruction in the AdS/BCFT correspondence

Abstract In this paper, we extend the surface growth approach for bulk reconstruction into the AdS spacetime with a boundary in the AdS/BCFT correspondence. We show that the geometry in the entanglement wedge with a boundary can be constructed from the direct growth of bulk extremal surfaces layer by layer. In addition, we find that the surface growth configuration in BCFT can be connected with the defect multi scale entanglement renormalization ansatz (MERA) tensor network. Furthermore, we also investigate the entanglement of purification within the surface growth process, which not only reveals more refined structure of entanglement entropy in the entanglement wedge but also suggests a selection rule for surface growth in the bulk reconstruction.

Multiscale Entanglement Renormalization Ansatz: Causality and Error Correction

Computational complexity reduction is at the basis of a new formulation of many-body quantum states according to tensor network ansatz, originally framed in one-dimensional lattices. In order to include long-range entanglement characterizing phase transitions, the multiscale entanglement renormalization ansatz (MERA) defines a sequence of coarse-grained lattices, obtained by targeting the map of a scale-invariant system into an identical coarse-grained one. The quantum circuit associated with this hierarchical structure includes the definition of causal relations and unitary extensions, leading to the definition of ground subspaces as stabilizer codes. The emerging error correcting codes are referred to logical indices located at the highest hierarchical level and to physical indices yielded by redundancy, framed in the AdS-CFT correspondence as holographic quantum codes with bulk and boundary indices, respectively. In a use-case scenario based on errors consisting of spin erasure, the correction is implemented as the reconstruction of a bulk local operator.

Tensor networks for quantum machine learning

Once developed for quantum theory, tensor networks (TNs) have been established as a successful machine learning (ML) paradigm. Now, they have been ported back to the quantum realm in the emerging field of quantum ML to assess problems that classical computers are unable to solve efficiently. Their nature at the interface between physics and ML makes TNs easily deployable on quantum computers. In this review article, we shed light on one of the major architectures considered to be predestined for variational quantum ML. In particular, we discuss how layouts like matrix product state, projected entangled pair states, tree tensor networks and multi-scale entanglement renormalization ansatz can be mapped to a quantum computer, how they can be used for ML and data encoding and which implementation techniques improve their performance.

Variational quantum simulation of the critical Ising model with symmetry averaging

Here we investigate the use of deep multiscale entanglement renormalization ansatz (DMERA) circuits as a variational ansatz. We use the exactly solvable one-dimensional critical transverse-field Ising model as a test bed. Numerically exact simulation of the quantum circuit ansatz can in this case be carried out to hundreds of qubits by exploiting efficient classical algorithms for simulating matchgate circuits. We find that, for this system, the DMERA strongly outperforms a standard quantum approximate optimization algorithm (QAOA)--style ansatz, and that a major source of systematic error in correlation functions approximated using the DMERA is the breaking of the translational and Kramers-Wannier symmetries of the transverse-field Ising model. We are able to reduce this error by up to four orders of magnitude by symmetry averaging, without incurring additional cost in qubits or circuit depth. We propose that this technique for mitigating systematic error could be applied to noisy intermediate-scale quantum (NISQ) simulations of physical systems with other symmetries.

Barren plateaus in quantum tensor network optimization

We analyze the barren plateau phenomenon in the variational optimization of quantum circuits inspired by matrix product states (qMPS), tree tensor networks (qTTN), and the multiscale entanglement renormalization ansatz (qMERA). We consider as the cost function the expectation value of a Hamiltonian that is a sum of local terms. For randomly chosen variational parameters we show that the variance of the cost function gradient decreases exponentially with the distance of a Hamiltonian term from the canonical centre in the quantum tensor network. Therefore, as a function of qubit count, for qMPS most gradient variances decrease exponentially and for qTTN as well as qMERA they decrease polynomially. We also show that the calculation of these gradients is exponentially more efficient on a classical computer than on a quantum computer.

Hierarchy of entanglement renormalization and long-range entangled states

As a quantum-informative window into quantum many-body physics, the concept and application of the entanglement renormalization group (ERG) have been playing a vital role in the study of novel quantum phases of matter, especially long-range entangled (LRE) states in topologically ordered systems. For instance, by recursively applying local unitaries as well as adding and removing qubits that form product states, the 2D toric code ground states, i.e., the fixed point of ${\mathbb{Z}}_{2}$ topological order, are efficiently coarse-grained with respect to the system size. As a further improvement, the addition and removal of 2D toric codes in the ground states of the 3D X-cube model is shown to be indispensable and, remarkably, leads to well-defined fixed points of a large class of fracton orders that are non-liquid-like. Here, we present a substantially unified ERG framework in which general degrees of freedom are allowed to be recursively added or removed. Specifically, we establish an exotic hierarchy of ERG and LRE states in Pauli stabilizer codes, where the 2D toric code and 3D X-cube models are naturally included. In the hierarchy, LRE states like 3D X-cube and 3D toric code ground states can be added or removed in ERG processes of more complex LRE states. In this way, a large group of Pauli stabilizer codes are categorized into a series of state towers; within each tower, in addition to local unitaries including cnot (controlled-NOT) gates, lower LRE states of level-$n$ are added or removed in the level-$n$ ERG process of an upper LRE state of level $(n+1)$, connecting LRE states of different levels and unveiling complex relations among LRE states. As future directions, we expect this hierarchy can be applied to more general LRE states, leading to a unified ERG scenario of LRE states and exact tensor-network representations in the form of more generalized branching multiscale entanglement renormalization ansatz.

Robustness and Explainability of Image Classification Based on QCNN

In this paper, we propose a multiscale entanglement renormalization ansatz (MERA) feature extraction method based on a novel quantum convolutional neural network (QCNN) for binary scanning tunneling microscopy (STM) image classification. We design QCNN quantum circuits for state preparation, quantum convolution, and quantum pooling in the TensorFlow quantum framework and compare the performance of QCNN classifier and two hybrid quantum-classical QCNN models. Adversarial attacks are considered as a type of interpretable method to evaluate the robustness of QCNN models. The similarity between the pixels of image bitplane slicing and Ising phase transition opens up new ways for exploring classification performance enhancement by QCNN classifiers. Classification performance of different bitplanes of QCNN also shows that they can robustly resist adversarial attacks such as FGSM, CW, JSMA, and DEEPFOOL.

Measurement as a Shortcut to Long-Range Entangled Quantum Matter

The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback (``adaptive circuits'') can evade such restrictions. We introduce three classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs). The three classes are inspired by distinct physical insights, including tensor-network constructions, multiscale entanglement renormalization ansatz (MERA), and parton constructions. A large class of topological orders, including chiral topological order, can be prepared in constant depth or time, and one-dimensional CFT states and non-abelian topological orders with both solvable and non-solvable groups can be prepared in depth scaling logarithmically with system size. We also build on a recently discovered correspondence between symmetry-protected topological phases and long-range entanglement to derive efficient protocols for preparing symmetry-enriched topological order and arbitrary CSS (Calderbank-Shor-Steane) codes. Our work illustrates the practical and conceptual versatility of measurement for state preparation.

Quantum phase recognition using quantum tensor networks

Machine learning (ML) has recently facilitated many advances in solving problems related to many-body physical systems. Given the intrinsic quantum nature of these problems, it is natural to speculate that quantum-enhanced machine learning will enable us to unveil even greater details than we currently have. With this motivation, this paper examines a quantum machine learning approach based on shallow variational ansatz inspired by tensor networks for supervised learning tasks. In particular, we first look at the standard image classification tasks using the Fashion-MNIST dataset and study the effect of repeating tensor network layers on ansatz’s expressibility and performance. Finally, we use this strategy to tackle the problem of quantum phase recognition for the transverse-field Ising and Heisenberg spin models in one and two dimensions, where we were able to reach \(\ge 98\%\) test-set accuracies with both multi-scale entanglement renormalization ansatz (MERA) and tree tensor network (TTN) inspired parametrized quantum circuits.

Noise-robust optimization of quantum machine learning models for polymer properties using a simulator and validated on the IonQ quantum computer

Quantum machine learning for predicting the physical properties of polymer materials based on the molecular descriptors of monomers was investigated. Under the stochastic variation of the expected predicted values obtained from quantum circuits due to finite sampling, the methods proposed in previous works did not make sufficient progress in optimizing the parameters. To enable parameter optimization despite the presence of stochastic variations in the expected values, quantum circuits that improve prediction accuracy without increasing the number of parameters and parameter optimization methods that are robust to stochastic variations in the expected predicted values, were investigated. The multi-scale entanglement renormalization ansatz circuit improved the prediction accuracy without increasing the number of parameters. The stochastic gradient descent method using the parameter-shift rule for gradient calculation was shown to be robust to sampling variability in the expected value. Finally, the quantum machine learning model was trained on an actual ion-trap quantum computer. At each optimization step, the coefficient of determination R^{2} improved equally on the actual machine and simulator, indicating that our findings enable the training of quantum circuits on the actual quantum computer to the same extent as on the simulator.

Dynamic hierarchical quantum secret sharing based on the multiscale entanglement renormalization ansatz

Tensor networks offer a novel and powerful tool for solving a variety of problems in mathematics, data science, and engineering. One such network is the multiscale entanglement renormalization ansatz (MERA). The MERA exhibits a hierarchical structure of layers, where each layer corresponds to a particular length (or energy) scale. The structure can be easily constructed using isometric and disentangler transformations. The following question arises: Is it possible to use the MERA to build hierarchical quantum secret sharing (HQSS)? The paper answers the question in the affirmative. In particular, it shows how a hierarchy of participant trust and authority relates to a MERA structure. The structure consists of binary and ternary MERA modules, which generate secret shares for participants. Because a binary MERA can be replaced by its ternary sibling and vice versa, our HQSS scheme is dynamic, allowing promotion and demotion of participants from the different layers but also enrollment of new ones and disenrollment of old ones from the same layer. The correctness and security of our dynamic hierarchical quantum secret sharing scheme are discussed.

Entanglement distillation toward minimal bond cut surface in tensor networks

In tensor networks, a geometric operation of pushing a bond cut surface toward a minimal surface corresponds to entanglement distillation. Cutting bonds defines a reduced transition matrix on the bond cut surface and the associated quantum state naturally emerges from it. We justify this picture quantitatively by evaluating the trace distance between the maximally entangled states and the states on bond cut surfaces in the multi-scale entanglement renormalization ansatz (MERA) and matrix product states in a canonical form. Our numerical result for the random MERA is in a reasonable agreement with our proposal. The result sheds new light on a deeper understanding of the Ryu-Takayanagi formula for entanglement entropy in holography and the emergence of geometry from the entanglement structure.

Generating tensor-network states via combination of phonons and qubits in a trapped-ion platform

Tensor-network (TN) states supply one of the most powerful variational tools to simulate quantum many-body systems. Though classical simulation of TN states (with approximation) is efficient, the required computational (classical) resources are still beyond our current capability when the size of the system and bond dimension becomes large (which is necessary for studying complicated quantum many-body systems). The TN contraction, which is the dominant cost in TN algorithms, can be replaced by measuring the corresponding physical observables directly in the experimentally prepared TN states. The computational cost is thus dramatically reduced. Here, we propose a scheme to generate TN states by combining multiple phonon modes and qubits in the ion trap platform. With the full connectivity and parallelism of the phonon modes operation, we can generate TN states with rough but complex entanglement structures [such as the multiscale entanglement renormalization Ansatz on the one-dimensional (1D) lattice and the projected entangled pair states on the kagome lattice] by shallow generation circuits. With the abundant local free parameters of the qubits operation, we can refine the local details of the generated TN states. We further benchmark the expressive power of the generated TN states by optimizing the ground energy, which is very close to the exact results of the 1D transverse-field Ising model near the critical point and the 1D Heisenberg model. Our method supplies a quantum-classical hybrid way to simulate the ground states of many-body systems and paves a promising way to demonstrate the advantage of quantum simulation.

Variational convolutional neural networks classifiers

Convolutional neural networks have been shown to extract features better than traditional algorithms in the fields such as image classification, object detection, and speech recognition. In parallel, a variational quantum algorithm incorporating parameterized quantum circuits has higher performance on near-term quantum processors. In this paper, we propose a classification algorithm called variational convolutional neural networks (VCNN), allowing for efficient training and implementation on near-term quantum devices. The VCNN algorithm combines the multi-scale entanglement renormalization ansatz. We deploy the VCNN algorithm on the TensorFlow Quantum platform with the numerical simulator backends using the MNIST and Fashion MNIST datasets. Experimental results show that the average accuracy of VCNN on classification tasks can reach up to 96.41%. Our algorithm has higher learning accuracy and fewer training epochs than quantum neural network algorithms. Moreover, we conclude that circuit-based models have excellent resilience to noise by numerical simulations.

On the closedness and geometry of tensor network state sets

Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse of dimensionality is addressed by parametrizing the many-body state in terms of a network of partially contracted tensors. These tensors form a substantially reduced set of effective degrees of freedom. In practical algorithms, functionals like energy expectation values or overlaps are optimized over certain sets of TNS. Concerning algorithmic stability, it is important whether the considered sets are closed because, otherwise, the algorithms may approach a boundary point that is outside the TNS set and tensor elements diverge. We discuss the closedness and geometries of TNS sets, and we propose regularizations for optimization problems on non-closed TNS sets. We show that sets of matrix product states (MPS) with open boundary conditions, tree tensor network states, and the multiscale entanglement renormalization ansatz are always closed, whereas sets of translation-invariant MPS with periodic boundary conditions (PBC), heterogeneous MPS with PBC, and projected entangled pair states are generally not closed. The latter is done using explicit examples like the W state, states that we call two-domain states, and fine-grained versions thereof.

Developments in the Tensor Network — from Statistical Mechanics to Quantum Entanglement

Tensor networks (TNs) have become one of the most essential building blocks for various fields of theoretical physics such as condensed matter theory, statistical mechanics, quantum information, and quantum gravity. This review provides a unified description of a series of developments in the TN from the statistical mechanics side. In particular, we begin with the variational principle for the transfer matrix of the 2D Ising model, which naturally leads us to the matrix product state (MPS) and the corner transfer matrix (CTM). We then explain how the CTM can be evolved to such MPS-based approaches as density matrix renormalization group (DMRG) and infinite time-evolved block decimation. We also elucidate that the finite-size DMRG played an intrinsic role for incorporating various quantum information concepts in subsequent developments in the TN. After surveying higher-dimensional generalizations like tensor product states or projected entangled pair states, we describe tensor renormalization groups (TRGs), which are a fusion of TNs and Kadanoff-Wilson type real-space renormalization groups, focusing on their fixed point structures. We then discuss how the difficulty in TRGs for critical systems can be overcome in the tensor network renormalization and the multi-scale entanglement renormalization ansatz.