### Efficient Tensor Network Simulation of IBM’s Eagle-Kicked Ising Experiment

The research paper focuses on the classical simulation of a kicked Ising quantum system on the heavy hexagon lattice. It compares the accuracy and efficiency of this classical simulation with a recent quantum processor simulation. The authors employ a tensor network approach that reflects the lattice’s geometry and is contracted using belief propagation, resulting in significantly more accurate and precise results than those obtained from the quantum processor and other classical methods. They quantify the treelike correlations of the wave function to explain the accuracy of their belief propagation-based approach and demonstrate its ability to simulate the system to long times in the thermodynamic limit, equivalent to a quantum computer with an infinite number of qubits. The paper suggests that their tensor network approach has broader applications for simulating the dynamics of quantum systems with treelike correlations.

The introduction of the research paper discusses the inherent noise and errors in quantum computers and the potential use of error-correcting codes to address these issues. The paper emphasizes the practical advantage of noisy quantum technologies over classical computers, despite the current infeasibility of quantum error correction. It also explores the impact of noise on the power and simulation of noisy quantum devices. The authors demonstrate the effectiveness of belief propagation tensor network approach for solving quantum dynamics problems and simulating systems with locally treelike correlations and limited entanglement. They provide evidence of achieving accurate results for a 127-qubit problem, surpassing classical tensor network approaches and error-mitigated quantum computers. The research anticipates the applicability of their methodology to problems involving treelike correlations and limited entanglement, serving as a benchmark for many-body dynamics problems.

Model and ansatz

The research paper focuses on the dynamics of the Trotterized kicked transverse-field Ising model represented by a unitary involving Pauli operators Z and X, where v and v denote neighboring qubits on the “heavy hex” lattice. This lattice is a hexagonal lattice decorated with additional qubits along the edges. The dynamics of the model were simulated on the IBM Eagle quantum processor, which consists of a lattice of 6 × 3 heavy hexagons plus two additional qubits. To simulate the system on a classical computer, the authors use a tensor network approach that respects the qubit connectivity of the heavy hex lattice and fix a maximum amount of entanglement by limiting the bond dimension of the network. The tensor network state (TNS) is evolved by applying gates under the belief propagation (BP) approximation, and expectation values are extracted using belief propagation. The BP method is fully controlled on trees but incurs a potentially small but uncontrolled approximation when there are loops in the network; however, the error from this approximation is generally smaller for larger loops. The research demonstrates that even for a large lattice and significant circuit depths, the correlations in this model persist.

Results

The research paper focuses on the application of tensor network state (TNS) and gate-evolution methods to simulate the dynamics of heavy hexagon lattices. The study investigates the accuracy of the belief propagation (BP) approximation in these simulations and its dependence on system size, Trotter steps, and bond dimension. The paper demonstrates that the BP-approximated TNS method achieves high accuracy in simulating the dynamics of the heavy hexagon lattice, as evidenced by the agreement with exact simulation results and quantum processor experimental data. It is observed that the accuracy of the BP approximation improves with increasing system size and number of heavy hexagons. The paper also presents evidence of the high accuracy of the BP-approximated TNS method throughout the phase diagram, showing close agreement with exact results. Comparisons with an independent matrix product state (MPS) approach further support the accuracy of the BP-approximated TNS method, as the two methods agree closely when the error in the MPS method is small. The paper highlights the remarkable accuracy achieved by the BP-approximated TNS method in simulating the dynamics of heavy hexagon lattices and provides insights into the factors influencing its accuracy, such as system size, Trotter steps, and bond dimension.

Dynamics of the infinite heavy hex lattice

The research paper focuses on the application of tensor network methods to simulate dynamics of the infinite heavy hex lattice. It presents results on the dynamics of the kicked transverse-field Ising model on an infinite heavy hexagon lattice, approximating the dynamics and taking expectation values using the belief propagation (BP) approximation. The paper illustrates the close agreement between the magnetization of the infinite and finite heavy hex lattices, indicating minimal boundary effects in the finite system. The time-dependence of the bipartite entanglement entropy per edge is analyzed, showing a sharp linear growth at short times before slowing down and potentially saturating over a large time scale. The study suggests that the time-to-decay scales exponentially with the inverse of a parameter θh, and it concludes that accurate simulation of the infinite quantum processor is possible up to large circuit depths for smaller values of θh. For larger values of θh, the entanglement grows too quickly to accurately determine the entanglement entropy in the system beyond approximately 25 Trotter steps with current resources. The paper provides evidence of the accuracy of the BP approximation for the finite system and implies that the presented results for the infinite lattice are highly accurate based on this evidence.

State ansatz

The research paper proposes an ansatz for the wave function of the system using a tensor network that reflects the qubit connectivity of the processor. The physical properties of the tensor network are invariant under a gauge symmetry, which is utilized to keep the network in the Vidal gauge. This gauge involves positive, diagonal bond tensors and on-site tensors with specific isometric properties. The TNS of the system after applications of U(θ h ) is denoted as |ψ(θ h , n). Single-site X rotations in U(θ h ) can be applied exactly, while two-site gates are applied approximately using a simple update procedure involving truncating internal indices to a prescribed maximum bond dimension χ. The tensor network is regauged using belief propagation after a single Trotter step to improve the accuracy of the simple update procedure. Belief propagation efficiently “regauges” the tensor network by performing message passing over the network. The results presented could have been achieved with other known gauging methods, highlighting the effectiveness of the proposed approach.

Measuring expectation values

The section measures single-site expectation values of the gauged state |ψ(θ h , n) using a rank-one approximation for the environments of local regions of the network. The study demonstrates that the network is locally treelike, allowing for the approximation method. It discusses the measurement of higher-weight observables in the “regime of strong entanglement” by exploiting the Clifford properties of the circuit at θ h = π/2. The extended time evolution method is used to measure all higher-weight observables. It notes that the bond dimension needed to evolve the system by n Trotter steps and measure a string can be 2^n+n, and highlights that accuracy can be achieved with relatively small values of the bond dimension. The study also emphasizes that higher-weight observables could be measured with other tensor network methods, such as the boundary MPS method, which was used to approximately compute the edge environment and showed close agreement with boundary MPS when measuring single-site observables.

The conclusion of the research paper highlights the application of tensor networks in accurately simulating the dynamics of a 127-qubit kicked Ising model on a heavy hexagon lattice. The method, based on belief propagation approximation, demonstrates a computational scaling of O(Lχ 4 ), where L is the number of qubits and χ is the bond dimension. The study shows that the assumption of locally treelike lattice is increasingly valid with increasing system size, making the gate-evolution method highly accurate and revealing loop-free behavior in the model’s dynamics. The research also indicates the potential of the belief-propagation-based method for efficient tensor network simulations of various dynamics problems, especially in higher dimensions. The paper highlights the importance of lattice topology and the flexibility of tensor network methods in benchmarking new quantum processor designs. The authors emphasize the potential applicability of their approach in classical simulation of quantum many-body systems and the ongoing development of software to facilitate the rapid testing and deployment of tensor network methods on arbitrary graphs. The study opens new directions for utilizing tensor network approaches to benchmark quantum processor designs and delineate systems challenging for classical computing techniques.

*Now let us see how the authors handled the following topics:*

METHODS

The researchers in the study proposed a tensor network ansatz for representing the state of the system, which includes local tensors v on the network sites and bond tensors e on the edges. They utilized the “Vidal” gauge, characterized by isometric constraints on specific groupings of local tensors. For every edge connecting vertices, they incorporated the isometric tensor condition, crucial for accuracy in applying two-site gates and calculating expectation values. The paper then outlines the methods used for these operations and describes the use of belief propagation to maintain the Vidal gauge during simulations. The researchers stressed that working in the Vidal gauge is not mandatory, as the same operations and results can be obtained in an arbitrary gauge using message tensors generated from performing belief propagation on the tensor network state. They named their tensor network state a “BP-approximated TNS” and offered to discuss this further in the subsequent sections.

The simple update procedure

The research paper discusses the application of two-site gates to the tensor network state using the simple update procedure. The procedure is illustrated in a diagram and is further optimized for efficiency using the “reduced tensor” variant. The simple update procedure can be implemented on a tensor network state in any gauge by utilizing fixed point message tensors obtained from belief propagation on the TNS. The update involves modifying the message tensor on the edge where the gate is applied with the bond matrix returned from the singular value decomposition (SVD) procedure. This approach bears resemblance to prior work where message tensors were employed for energy optimization.

Γv

The section discusses the approximation of the bond tensors in a network by treating the environment as a tensor product of environments coming from each of the neighbors of v. This approximation is equivalent to computing the expectation value in an arbitrary gauge by using the fixed-point message tensors of belief propagation as approximations of the environments, provided that the tensors of the network satisfy the Vidal gauge conditions. For locally treelike networks, this approximation can provide very good approximations for local observables. The study provides evidence that this approximation holds well for the model and lattice studied. The section also outlines how belief propagation is used to maintain the Vidal gauge during simulations, which helps to ensure accuracy in the update procedure and when taking expectation values.

Belief propagation on a tensor network state

The paper introduces belief propagation (BP) as a method for maintaining the gauge properties of a tensor network state (TNS). BP is used to approximate the marginals of probability distributions of graphical models and has gained interest in the context of contracting tensor networks. To perform BP on a TNS in the Vidal gauge, the square roots of the bond tensors are absorbed onto the v tensors, forming a closed network representing the square norm of the TNS. A series of “message tensors” over the norm network is defined, and self-consistent equations are established for the message tensors. These equations can be iterated to converge the messages, providing a rank-one approximation of the exact environment for a given tensor. BP message tensors can also be used to define a gauge transformation, bringing a TNS into the Vidal gauge and ensuring the satisfaction of specific conditions. The paper discusses the use of BP gauging after every Trotter step in simulations, emphasizing its importance for maintaining accuracy, particularly for longer depth circuits. Gauging the state before taking expectation values is also highlighted as crucial for accuracy, even for shorter circuits. The findings suggest that gauging after every Trotter step and before taking expectation values is essential for preserving the gauge properties and accuracy of the TNS in simulations.

Estimating the error of BP for general tensor networks

The section discusses the estimation of the error of belief propagation (BP) for general tensor networks. The paper focuses on quantifying the error that BP makes when approximating contractions of the network by computing the “edge environment” associated with cutting the norm network along a given edge. The edge environment is used to define a measure of the error of BP, assuming a rank-one edge environment along every edge of the network. The error from BP along an edge is estimated using the “index of separability,” which depends on the singular values of the network. The study finds that the choice of edge is unimportant and does not qualitatively change the results. Computing the BP error is costly due to the need to contract the full norm network. The paper also discusses the exponential decrease in error with the smallest loop size of the lattice and the scaling of the BP error for a lattice with gapped loop correlations. The findings of the study provide insights into the error estimation of BP for general tensor networks and its dependence on network characteristics such as edge environment and loop size.

Computing resources and software packages

The research paper utilizes the ITensorNetworks.jl package, a Julia package for manipulating tensor network states. It is built on ITensors.jl, which provides basic tensor operations. The ITensorNetworks.jl package includes code for belief propagation, gauging, and simple update procedures for arbitrary tensor network states. The paper provides an example script for simulating the model using a BP-approximated TNS approach. The paper also discusses the formation of edge environments from the norm network of a tensor network state, employing singular value decomposition on a single matrix obtained after contracting the network’s indices. The work references a study using dissipative mean-field theory to simulate a 127-qubit system out to short-circuit depths, obtaining relatively accurate results with some noticeable deviations from the true result. The paper highlights that their BP-evolved TNS method can compute to an accuracy of approximately 10^-14.

MPS CALCULATIONS

The paper presents MPS calculations for benchmarking in the unverifiable regime, specifically focusing on the 127-qubit kicked Ising model. An alternative ordering of the sites of the heavy hex lattice was implemented, resulting in slightly lower truncation errors when simulating the model at fixed MPS bond dimension. The two-site term in the propagator U(θ h ) is decomposed into a product of at most five commuting matrix product operators (MPOs) of bond dimension 2, and the MPS is evolved by successively applying these MPOs with truncation performed at each step. The paper compares various approaches to simulating the kicked transverse-field Ising model and discusses the use of light-cone depth reduction (LCDR) for improved efficiency. The error of the simulation is approximated by calculating the sum of the singular values discarded during the application of a single MPO. Additionally, the paper outlines the approximation of the edge environment of a tensor network state (TNS) using a boundary MPS-style scheme and analyzes the dynamics of the single-site magnetization of the kicked Ising model on heavy hex lattices of varying sizes. The results demonstrate the feasibility and accuracy of the MPS approach in simulating quantum systems, with implications for quantum computation and simulation.

Hopefully, I will be able to impart some insights into Tensor Networks Simulation and why it is important, as depicted in the paper.

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