Tensor Network Representation Research Articles

State diagrams to determine tree tensor network operators

This work is concerned with tree tensor network operators (TTNOs) for representing quantum Hamiltonians. We first establish a mathematical framework connecting tree topologies with state diagrams. Based on these, we devise an algorithm for constructing a TTNO given a Hamiltonian. The algorithm exploits the tensor product structure of the Hamiltonian to add paths to a state diagram, while combining local operators if possible. We test the capabilities of our algorithm on random Hamiltonians for a given tree structure. Additionally, we construct explicit TTNOs for nearest neighbour interactions on a tree topology. Furthermore, we derive a bound on the bond dimension of tensor operators representing arbitrary interactions on trees. Finally, we consider an open quantum system in the form of a Heisenberg spin chain coupled to bosonic bath sites as a concrete example. We find that tree structures allow for lower bond dimensions of the Hamiltonian tensor network representation compared to a matrix product operator structure. This reduction is large enough to reduce the number of total tensor elements required as soon as the number of baths per spin reaches 3.

Tensor network representation of non-abelian gauge theory coupled to reduced staggered fermions

We show how to construct a tensor network representation of the path integral for reduced staggered fermions coupled to a non-abelian gauge field in two dimensions. The resulting formulation is both memory and computation efficient because reduced staggered fermions can be represented in terms of a minimal number of tensor indices while the gauge sector can be approximated using Gaussian quadrature with a truncation. Numerical results obtained using the Grassmann TRG algorithm are shown for the case of SU(2) lattice gauge theory and compared to Monte Carlo results.

Tensor network representation and entanglement spreading in many-body localized systems: a novel approach

Tensor network representation and entanglement spreading in many-body localized systems: a novel approach

Area law for steady states of detailed-balance local Lindbladians

We study steady-states of quantum Markovian processes whose evolution is described by local Lindbladians. We assume that the Lindbladian is gapped and satisfies quantum detailed balance with respect to a unique full-rank steady state σ. We show that under mild assumptions on the Lindbladian terms, which can be checked efficiently, the Lindbladian can be mapped to a local Hamiltonian on a doubled Hilbert space that has the same spectrum and a ground state that is the vectorization of σ1/2. Consequently, we can use Hamiltonian complexity tools to study the steady states of such open systems. In particular, we show an area-law in the mutual information for the steady state of such 1D systems, together with a tensor-network representation that can be found efficiently.

Linear to multi-linear algebra and systems using tensors

In the past few decades, multi-linear algebra also known as tensor algebra has been adapted and employed as a tool for various engineering applications. Recent developments in tensor algebra have indicated that several well-known concepts from linear algebra can be extended to a multi-linear setting with the help of a special form of tensor contracted product, known as the Einstein product. Thus, the tensor contracted product and its properties can be harnessed to define the notions of multi-linear system theory where the input, output signals, and the system are inherently multi-domain or multi-modal. This study provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the differences and advantages that the multi-linear setting brings forth. In particular, the notions of tensor inversion, tensor singular value, and tensor eigenvalue decomposition using the Einstein product are explained. In addition, this study also introduces the notion of contracted convolution for both discrete and continuous multi-linear system tensors. Tensor network representation of various tensor operations is also presented. In addition, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA system, is presented. This study provides a foundation for professionals whose research involves multi-domain or multi-modal signals and systems.

Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance.

A recent quantum simulation of observables of the kicked Ising model on 127 qubits implemented circuits that exceed the capabilities of exact classical simulation. We show that several approximate classical methods, based on sparse Pauli dynamics and tensor network algorithms, can simulate these observables orders of magnitude faster than the quantum experiment and can also be systematically converged beyond the experimental accuracy. Our most accurate technique combines a mixed Schrödinger and Heisenberg tensor network representation with the Bethe free entropy relation of belief propagation to compute expectation values with an effective wave function-operator sandwich bond dimension >16,000,000, achieving an absolute accuracy, without extrapolation, in the observables of <0.01, which is converged for many practical purposes. We thereby identify inaccuracies in the experimental extrapolations and suggest how future experiments can be implemented to increase the classical hardness.

General tensor network theory for frustrated classical spin models in two dimensions

General tensor network theory for frustrated classical spin models in two dimensions

Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories.

Treating the infinite-dimensional Hilbert space of non-Abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we employ q-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, this approach simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. This enables the development of both quantum as well as quantum-inspired classical spin-network algorithms for q-deformed gauge theories. To be explicit, we focus on SU(2)_{k} gauge theories with k∈N that are controlled by the deformation parameter q=e^{2πi/(k+2)}, a root of unity, and converge to the standard SU(2) Kogut-Susskind model as k→∞. In particular, we demonstrate that this formulation is well suited for efficient tensor network representations by variational ground-state simulations in 2D, providing first evidence that the continuum limit can be reached with k=O(10). Finally, we develop a scalable quantum algorithm for Trotterized real-time evolution by analytically diagonalizing the SU(2)_{k} plaquette interactions. Our work gives a new perspective for the application of tensor network methods to high-energy physics and paves the way for quantum simulations of non-Abelian gauge theories far from equilibrium where no other methods are currently available.

Validating Phase-Space Methods with Tensor Networks in Two-Dimensional Spin Models with Power-Law Interactions.

Using a recently developed extension of the time-dependent variational principle for matrix product states, we evaluate the dynamics of 2D power-law interacting XXZ models, implementable in a variety of state-of-the-art experimental platforms. We compute the spin squeezing as a measure of correlations in the system, and compare to semiclassical phase-space calculations utilizing the discrete truncated Wigner approximation (DTWA). We find the latter efficiently and accurately captures the scaling of entanglement with system size in these systems, despite the comparatively resource-intensive tensor network representation of the dynamics. We also compare the steady-state behavior of DTWA to thermal ensemble calculations with tensor networks. Our results open a way to benchmark dynamical calculations for two-dimensional quantum systems, and allow us to rigorously validate recent predictions for the generation of scalable entangled resources for metrology in these systems.

Faster variational quantum algorithms with quantum kernel-based surrogate models

Abstract We present a new optimization strategy for small-to-intermediate scale variational quantum algorithms (VQAs) on noisy near-term quantum processors which uses a Gaussian process surrogate model equipped with a classically-evaluated quantum kernel. VQAs are typically optimized using gradient-based approaches however these are difficult to implement on current noisy devices, requiring large numbers of objective function evaluations. Our approach shifts this computational burden onto the classical optimizer component of these hybrid algorithms, greatly reducing the number of quantum circuit evaluations required from the quantum processor. We focus on the variational quantum eigensolver (VQE) algorithm and demonstrate numerically that these surrogate models are particularly well suited to the algorithm’s objective function. Next, we apply these models to both noiseless and noisy VQE simulations and show that they exhibit better performance than widely-used classical kernels in terms of final accuracy and convergence speed. Compared to the typically-used stochastic gradient-descent approach to VQAs, our quantum kernel-based approach is found to consistently achieve significantly higher accuracy while requiring less than an order of magnitude fewer quantum circuit executions. We analyze the performance of the quantum kernel-based models in terms of the kernels’ induced feature spaces and explicitly construct their feature maps. Finally, we describe a scheme for approximating the best-performing quantum kernel using a classically-efficient tensor network representation of its input state and so provide a pathway for scaling this strategy to larger systems.

Boundary and domain wall theories of 2d generalized quantum double model

The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.

Quantum process tomography with unsupervised learning and tensor networks

The impressive pace of advance of quantum technology calls for robust and scalable techniques for the characterization and validation of quantum hardware. Quantum process tomography, the reconstruction of an unknown quantum channel from measurement data, remains the quintessential primitive to completely characterize quantum devices. However, due to the exponential scaling of the required data and classical post-processing, its range of applicability is typically restricted to one- and two-qubit gates. Here, we present a technique for performing quantum process tomography that addresses these issues by combining a tensor network representation of the channel with a data-driven optimization inspired by unsupervised machine learning. We demonstrate our technique through synthetically generated data for ideal one- and two-dimensional random quantum circuits of up to 10 qubits, and a noisy 5-qubit circuit, reaching process fidelities above 0.99 using several orders of magnitude fewer (single-qubit) measurement shots than traditional tomographic techniques. Our results go far beyond state-of-the-art, providing a practical and timely tool for benchmarking quantum circuits in current and near-term quantum computers.

Tensor network based finite-size scaling for two-dimensional Ising model

We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension $D=70$, the absolute errors of the critical temperature $T_c$ and the critical exponent $\nu$, $\beta$ are at the order of $10^{-7}, 10^{-5}$, $10^{-4}$ respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cut-off in bond dimension and elucidate its physical meaning in this context.

Isometric tensor network representations of two-dimensional thermal states

Tensor networks provide a useful tool to describe low-dimensional complex many-body systems. Finding efficient algorithms to use these methods for finite-temperature simulations in two dimensions is a continuing challenge. Here, we use the class of recently introduced isometric tensor network states, which can also be directly realized with unitary gates on a quantum computer. We utilize a purification ansatz to efficiently represent thermal states of the transverse field Ising model. By performing an imaginary-time evolution starting from infinite temperature, we find that this approach offers a different way with low computational complexity to represent states at finite temperatures.

Simulating thermal density operators with cluster expansions and tensor networks

We provide an efficient approximation for the exponential of a local operator in quantum spin systems using tensor-network representations of a cluster expansion. We benchmark this cluster tensor network operator (cluster TNO) for one-dimensional systems, and show that the approximation works well for large real- or imaginary-time steps. We use this formalism for representing the thermal density operator of a two-dimensional quantum spin system at a certain temperature as a single cluster TNO, which we can then contract by standard contraction methods for two-dimensional tensor networks. We apply this approach to the thermal phase transition of the transverse-field Ising model on the square lattice, and we find through a scaling analysis that the cluster-TNO approximation gives rise to a continuous phase transition in the correct universality class; by increasing the order of the cluster expansion we find good values of the critical point up to surprisingly low temperatures.

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.

Entanglement at the interplay between single- and many-bodyness

The tensor network representation of the ground state of a Bethe chain is analytically obtained and studied in relation to its entanglement distribution. Block entanglement displays a maximum at the interplay between single- and many-bodyness. In systems of two fermions, tensor networks describing states with substantial many-body entropy cannot be written as a sequence of next-neighbor unitaries applied on an uncorrelated state, but need four-next-neighbor unitaries in addition. This differs from the idea that ground states can be obtained as a sequence of next-neighbor operations applied on an initial tensor network lacking many-body correlations. The work uncovers the transcendence of the notion of many-bodyness in the implementation of protocols based on matrix product states.

Efficient separation of quantum from classical correlations for mixed states with a fixed charge

Entanglement is the key resource for quantum technologies and is at the root of exciting many-body phenomena. However, quantifying the entanglement between two parts of a real-world quantum system is challenging when it interacts with its environment, as the latter mixes cross-boundary classical with quantum correlations. Here, we efficiently quantify quantum correlations in such realistic open systems using the operator space entanglement spectrum of a mixed state. If the system possesses a fixed charge, we show that a subset of the spectral values encode coherence between different cross-boundary charge configurations. The sum over these values, which we call "configuration coherence", can be used as a quantifier for cross-boundary coherence. Crucially, we prove that for purity non-increasing maps, e.g., Lindblad-type evolutions with Hermitian jump operators, the configuration coherence is an entanglement measure. Moreover, it can be efficiently computed using a tensor network representation of the state&amp;apos;s density matrix. We showcase the configuration coherence for spinless particles moving on a chain in presence of dephasing. Our approach can quantify coherence and entanglement in a broad range of systems and motivates efficient entanglement detection.

Preparing valence-bond-solid states on noisy intermediate-scale quantum computers

Quantum state preparation is a key step in all digital quantum simulation algorithms. Here we propose methods to initialize on a gate-based quantum computer a general class of quantum spin wave functions, the so-called valence-bond-solid (VBS) states, that are important for two reasons. First, VBS states are the exact ground states of a class of interacting quantum spin models introduced by Affleck, Kennedy, Lieb, and Tasaki (AKLT). Second, the two-dimensional VBS states are universal resource states for measurement-based quantum computing. We find that schemes to prepare VBS states based on their tensor-network representations yield quantum circuits that are too deep to be within reach of noisy intermediate-scale quantum (NISQ) computers. We then apply the general nondeterministic method herein proposed to the preparation of the spin-1 and spin-$3/2$ VBS states, the ground states of the AKLT models defined in one dimension and in the honeycomb lattice, respectively. Shallow quantum circuits of depth independent of the lattice size are explicitly derived for both cases, making use of optimization schemes that outperform standard basis gate decomposition methods. The probabilistic nature of the proposed routine translates into an average number of repetitions to successfully prepare the VBS state that scales exponentially with the number of lattice sites $N$. However, two strategies to quadratically reduce this repetition overhead for any bipartite lattice are devised. Our approach should permit to use NISQ processors to explore the AKLT model and variants thereof, outperforming conventional numerical methods in the near future.

Projected d -wave superconducting state: A fermionic projected entangled pair state study

We investigate the physics of projected $d$-wave pairing states using their fermionic projected entangled pair state (fPEPS) representation. First, we approximate a $d$-wave Bardeen-Cooper-Schrieffer state using the Gaussian fPEPS. Next, we translate the resulting state into fPEPS tensors and implement the Gutzwiller projection which removes double occupancy by modifying the local tensor elements. The tensor network representation of the projected $d$-wave pairing state allows us to evaluate physical quantities in the thermodynamic limit without employing the Gutzwiller approximation. Despite having very few variational parameters, such physically motivated tensor network states are shown to exhibit competitive energies for the doped $t\text{\ensuremath{-}}J$ model. We expect that such construction offers useful initial states and guidance for variational tensor network calculations.