An efficient representation of a quantum circuit is of great importance in achieving a quantum advantage on current noisy intermediate scale quantum (NISQ) devices and the classical simulation of quantum manybody systems.The quantum circuits are playing the key ingredient in the performance of variational quantum algorithms and quantum dynamics in problems of physics and chemistry.In this paper, we study the role of the network structure of a quantum circuit in its performance.We discuss the variational optimization of quantum circuit (a unitary tensor-network circuit) with different network structures.The ansatz is performed based on a generalization of well-developed multiscale entanglement renormalization algorithm and also the conjugategradient method with an effective line search.We present the benchmarking calculations for different network structures by studying the Heisenberg model in a strongly disordered magnetic field and a tensor-network QR decomposition.Our work can contribute to achieve the most out of NISQ hardware and to classically develop isometric tensor network states.
An efficient representation of a quantum circuit is of great importance in achieving a quantum advantage on current noisy intermediate scale quantum (NISQ) devices and the classical simulation of quantum manybody systems.The quantum circuits are playing the key ingredient in the performance of variational quantum algorithms and quantum dynamics in problems of physics and chemistry.In this paper, we study the role of the network structure of a quantum circuit in its performance.We discuss the variational optimization of quantum circuit (a unitary tensor-network circuit) with different network structures.The ansatz is performed based on a generalization of well-developed multiscale entanglement renormalization algorithm and also the conjugategradient method with an effective line search.We present the benchmarking calculations for different network structures by studying the Heisenberg model in a strongly disordered magnetic field and a tensor-network QR decomposition.Our work can contribute to achieve the most out of NISQ hardware and to classically develop isometric tensor network states.
Despite the advantage quantum computers are expected to deliver when performing simulations compared to their classical counterparts, the current noisy intermediate-scale quantum (NISQ) devices remain limited in their capabilities. The training of parameterized quantum circuits (PQCs) remains a significant practical challenge, exacerbated by the requirement of shallow circuit depth necessary for their hardware implementation. Hybrid methods employing classical computers alongside quantum devices, such as the Variational Quantum Eigensolver (VQE), have proven useful for analyzing the capabilities of NISQ devices to solve relevant optimization problems. Still, in the simulation of complex structures involving the many-body problem in quantum mechanics, major issues remain about the representation of the system and obtaining results which clearly outperform classical computational devices. In this research contribution we propose the use of parameterized Tensor Networks (TNs) to attempt an improved performance of the VQE algorithm. A joint approach is presented where the Hamiltonian of a system is encapsulated into a Matrix Product Operator (MPO) within a parameterized unitary TN hereby splitting up the optimization task between the TN and the VQE. We show that the hybrid TN-VQE implementation improves the convergence of the algorithm in comparison to optimizing randomly-initialized quantum circuits via VQE.
We characterize the variational power of quantum circuit tensor networks in the representation of physical many-body ground states.Such tensor networks are formed by replacing the dense block unitaries and isometries in standard tensor networks by local quantum circuits.We explore both quantum circuit matrix product states and the quantum circuit multiscale entanglement renormalization Ansatz, and introduce an adaptive method to optimize the resulting circuits to high fidelity with more than 10 4 parameters.We benchmark their expressiveness against standard tensor networks, as well as other common circuit architectures, for the 1D and 2D Heisenberg and 1D Fermi-Hubbard models.We find quantum circuit tensor networks to be substantially more expressive than other quantum circuits for these problems, and that they can even be more compact than standard tensor networks.Extrapolating to circuit depths which can no longer be emulated classically, this suggests a region of advantage in quantum expressiveness in the representation of physical ground states.
Several tensor networks are built of isometric tensors, i.e. tensors satisfying W\dagger W = \mathbb{1} W † W = 1 . Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.
We propose a hybrid quantum-classical algorithm for approximating the ground state of two-dimensional quantum systems using an isometric tensor network ansatz, which maps naturally to quantum circuits. Inspired by the density matrix renormalization group, we optimize tensors sequentially by diagonalizing a series of effective Hamiltonians. These are constructed using a tomography-inspired method on a qubit subset whose size depends only on the bond dimension. Our approach leverages quantum computers to enable accurate solutions without relying on approximate contractions, circumventing the exponential complexity faced by classical techniques. We demonstrate our method on the two-dimensional (2D) transverse-field Ising model, achieving ground-state optimization on up to 25 qubits with modest quantum overhead -- significantly less than standard solutions based on variational quantum eigensolvers. Overall, our results offer a path towards scalable variational quantum algorithms in both noisy and fault-tolerant regimes.
Variational quantum algorithms (VQAs) have emerged as a promising candidate for realizing quantum advantage in machine learning and optimization tasks on noisy intermediate-scale quantum (NISQ) devices. Despite significant theoretical promise, the practical impact of VQAs is constrained by the exponential growth in computational resources required for classical simulation, the onset of barren plateaus in parameterized landscapes, and the inherent difficulties of scaling classical optimization routines to circuits with many qubits and layers. Addressing these intertwined challenges is essential for both advancing quantum algorithm development and extracting actionable physical insights from modern quantum hardware.This dissertation develops, analyzes, and validates an efficient framework based on approximate tensor ring decompositions for the classical simulation, training, and benchmarking of variational quantum circuits and eigensolvers. The first component of this work formalizes the representation of quantum states and gates within a cyclic tensor ring structure, enabling the compression of parameterized quantum circuits to linear scaling in both qubit number and circuit depth, while preserving essential quantum correlations and entanglement structure. Gate operations are efficiently approximated via truncated singular value decomposition, and novel algorithms for forward propagation and gradient-based training in the tensor ring domain are developed. These methodological advances are systematically benchmarked on standard supervised learning tasks, including Iris and MNIST datasets, and compared to conventional statevector and matrix product state baselines, resulting in competitive accuracy and orders-of-magnitude reductions in simulation cost. Extending this framework, the dissertation also introduces a scalable method for gradient evaluation in variational quantum eigensolvers (VQE) using the approximate tensor ring representation. This approach enables the simulation and training of large circuits for combinatorial optimization problems, supporting practical qubit counts and circuit depths inaccessible to traditional simulation methods. Detailed experiments on the Maximum-Cut optimization problem demonstrate both the computational efficiency and robustness of the tensor ring framework in nontrivial optimization landscapes.Collectively, the results establish approximate tensor ring decomposition as a versatile and powerful framework for advancing quantum machine learning and discrete optimization, supporting both pre-deployment benchmarking on classical hardware and algorithm design for next-generation quantum devices. This work provides a foundation for the further exploration of scalable quantum-classical hybrid algorithms, with implications for quantum simulation, optimization, and beyond.
The variational quantum eigensolver (VQE) combines classical and quantum resources in order simulate classically intractable quantum states. Amongst other variables, successful VQE depends on the choice of variational ansatz for a problem Hamiltonian. We give a detailed description of a quantum circuit version of the 2D isometric tensor network (isoTNS) ansatz which we call qisoTNS. We benchmark the performance of qisoTNS on two different 2D spin 1/2 Hamiltonians. We find that the ansatz has several advantages. It is qubit efficient with the number of qubits allowing for access to some exponentially large bond-dimension tensors at polynomial quantum cost. In addition, the ansatz is robust to the barren plateau problem due emergent layerwise training. We further explore the effect of noise on the efficacy of the ansatz. Overall, we find that qisoTNS is a suitable variational ansatz for 2D Hamiltonians with local interactions.