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arXiv:2504.16718 · Physical review. B./Physical review. B

Simulating quantum circuits with tree tensor networks using density-matrix renormalization group algorithm

Aditya Dubey, Zeki Zeybek, Peter Schmelcher

Published Aug 28, 2025• 1 citation
arXivHTMLPDF

Abstract

Quantum computing offers the potential for computational abilities that can go beyond classical machines. However, they are still limited by several challenges such as noise, decoherence, and gate errors. As a result, efficient classical simulation of quantum circuits is vital not only for validating and benchmarking quantum hardware but also for gaining deeper insights into the behavior of quantum algorithms. A promising framework for classical simulation is provided by tensor networks. Recently, the density-matrix renormalization group (DMRG) algorithm was developed for simulating quantum circuits using matrix product states (MPS). Although MPS are efficient for representing quantum states with one-dimensional correlation structures, the fixed linear geometry restricts the expressive power of the MPS. In this work, we extend the DMRG algorithm for simulating quantum circuits to tree tensor networks (TTNs). The framework employs a variational compression scheme that optimizes the TTN to approximate the evolved quantum state. To benchmark the method, we simulate random circuits and the quantum approximate optimization algorithm (QAOA) with various two-qubit gate connectivities. For the random circuits, we devise treelike gate layouts that are suitable for TTN and show that TTN requires less memory than MPS for the simulations. For the QAOA circuits, a naive TTN construction that exploits graph structure significantly improves the simulation fidelities. Our findings show that the DMRG algorithm with TTNs provides a promising framework for simulating quantum circuits, particularly when gate connectivities exhibit clustering or a hierarchical structure.

DOI
10.1103/64hd-q4z5

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Quantum computing offers the potential for computational abilities that can go beyond classical machines. However, they are still limited by several challenges such as noise, decoherence, and gate errors. As a result, efficient classical simulation of quantum circuits is vital not only for validating and benchmarking quantum hardware but also for gaining deeper insights into the behavior of quantum algorithms. A promising framework for classical simulation is provided by tensor networks. Recently, the density-matrix renormalization group (DMRG) algorithm was developed for simulating quantum circuits using matrix product states (MPS). Although MPS are efficient for representing quantum states with one-dimensional correlation structures, the fixed linear geometry restricts the expressive power of the MPS. In this work, we extend the DMRG algorithm for simulating quantum circuits to tree tensor networks (TTNs). The framework employs a variational compression scheme that optimizes the TTN to approximate the evolved quantum state. To benchmark the method, we simulate random circuits and the quantum approximate optimization algorithm (QAOA) with various two-qubit gate connectivities. For the random circuits, we devise treelike gate layouts that are suitable for TTN and show that TTN requires less memory than MPS for the simulations. For the QAOA circuits, a naive TTN construction that exploits graph structure significantly improves the simulation fidelities. Our findings show that the DMRG algorithm with TTNs provides a promising framework for simulating quantum circuits, particularly when gate connectivities exhibit clustering or a hierarchical structure.
Variational Quantum Algorithms (VQAs) are being highlighted as key quantum algorithms for demonstrating quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices, which are limited to executing shallow quantum circuits because of noise. However, the barren plateau problem, where the gradient of the loss function becomes exponentially small with system size, hinders this goal. Recent studies suggest that embedding tensor networks into quantum circuits and initializing the parameters can avoid the barren plateau. Yet, embedding tensor networks into quantum circuits is generally difficult, and methods have been limited to the simplest structure, Matrix Product States (MPSs). This study proposes a method to embed Tree Tensor Networks (TTNs), characterized by their hierarchical structure, into shallow quantum circuits. TTNs are suitable for representing two-dimensional systems and systems with long-range correlations, which MPSs are inadequate for representing. Our numerical results show that embedding TTNs provides better initial quantum circuits than MPS. Additionally, our method has a practical computational complexity, making it applicable to a wide range of TTNs. This study is expected to extend the application of VQAs to two-dimensional systems and those with long-range correlations, which have been challenging to utilize.
We develop and analyze a method for simulating quantum circuits on classical computers by representing quantum states as rooted tree tensor networks. Our algorithm first determines a suitable, fixed tree structure adapted to the expected entanglement generated by the quantum circuit. The gates are sequentially applied to the tree by absorbing single-qubit gates into leaf nodes, and splitting two-qubit gates via singular value decomposition and threading the resulting virtual bond through the tree. We theoretically analyze the applicability of the method as well as its computational cost and memory requirements, and identify advantageous scenarios in terms of required bond dimensions as compared to a matrix product state representation. The study is complemented by numerical experiments for different quantum circuit layouts up to 37 qubits.
We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.
Tensor networks provide a powerful and elegant approach to quantum manybody simulation. The simplest example is the density matrix renormalisation group (DMRG), which is based on the variational update of a matrix product state (MPS). It has proved to be the most accurate approach for the numerical study of strongly correlated one dimensional systems. We use DMRG to study the one dimensional disordered Bose-Hubbard model at fillings N=L = 1=2, 1 and 2 and show that the whole phase diagram for each can be successfully obtained by analysing entanglement properties alone. We �nd that the average entanglement is insufficient to accurately locate all of the phases, however using the standard error on the mean we are able to construct a phase diagram that is consistent with previous studies. It has recently been shown that there is a connection between the geometry of tensor networks and the entanglement and correlation properties that it can encode, which is a generalisation of the so called area law for entanglement entropy. This suggests that whilst gapped quantum systems can be accurately modeled using an MPS, a tensor network with a holographic geometry is natural to capture the logarithmic entanglement scaling and power law decaying correlation functions of critical systems. We create an algorithm for the disordered Heisenberg Hamiltonian that self assembles a tensor network based on the disorder in the couplings. The geometry created is that of a disordered tree tensor network (TTN) that when averaged has the holographic properties characteristic of critical systems. We continue the analysis of holographic tensor network geometry by considering the average length of leaf-to-leaf paths in various tree graphs, which is related to two-point correlation functions in tensor networks. For regular, complete trees we analytically calculate the average path length and all statistical moments, and generalise it for any splitting number. We then turn to the Catalan trees, which is the set of unique binary trees with n vertices, as it has a similar geometry to the disordered TTNs. We calculate the average depth of a leaf and show that it is equal to the average path length. We compare these analytic results with the structures found in the TTN and randomly constructed trees to show that the renormalisation involved in the TTN algorithm is crucial in the selection of the tree structure.
We develop a density-matrix renormalization group (DMRG) algorithm for the simulation of quantum circuits.This algorithm can be seen as the extension of the time-dependent DMRG from the usual situation of Hermitian Hamiltonian matrices to quantum circuits defined by unitary matrices.For small circuit depths, the technique is exact and equivalent to other matrix product state-based techniques.For larger depths, it becomes approximate in exchange for an exponential speed up in computational time.Like an actual quantum computer, the quality of the DMRG results is characterized by a finite fidelity.However, unlike a quantum computer, the fidelity depends strongly on the quantum circuit considered.For the most difficult possible circuit for this technique, the so-called "quantum supremacy" benchmark of Google LLC [Arute et al., Nature 574, 505 (2019)], we find that the DMRG algorithm can generate bit strings of the same quality as the seminal Google experiment on a single computing core.For a more structured circuit used for combinatorial optimization (quantum approximate optimization algorithm), we find a drastic improvement of the DMRG results with error rates dropping by a factor of 100 compared with random quantum circuits.Our results suggest that the current bottleneck of quantum computers is their fidelities rather than the number of qubits.
We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the density-matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topology of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following [Y. Shi, L. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006)], we treat a treelike network ansatz with arbitrary coordination number zzz, where the z=2z=2z=2 case corresponds to the one-dimensional (1D) scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addition, we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1D interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calculations carried out on small quantum chemical systems support our approach.