Hybrid tensor networks offer a promising route to enhance the expressivity of classical tensor network methods by incorporating quantum states prepared on a quantum computer. Existing approaches are limited by the variational optimization of the quantum component of the tensor network. In this work, we introduce an alternative strategy that combines a non-parametric quantum state prepared through quantum annealing and a classical isometric tensor network. The latter is variationally optimized while the former is used as a fixed, boundary tensor resource in the form of classical shadows. We demonstrate the feasibility of this approach through extensive numerical simulations on the transverse-field Ising model, showing that the optimization procedure remains robust under statistical and hardware noise. Moreover, our results indicate that our newly proposed setup improves the accuracy of the obtained ground state approximation compared to the original quantum simulation, without increasing the depth of the applied quantum circuits. Therefore, this setup offers a practical route to scale variational quantum algorithms towards the quantum utility scale.
Hybrid tensor networks offer a promising route to enhance the expressivity of classical tensor network methods by incorporating quantum states prepared on a quantum computer. Existing approaches are limited by the variational optimization of the quantum component of the tensor network. In this work, we introduce an alternative strategy that combines a non-parametric quantum state prepared through quantum annealing and a classical isometric tensor network. The latter is variationally optimized while the former is used as a fixed, boundary tensor resource in the form of classical shadows. We demonstrate the feasibility of this approach through extensive numerical simulations on the transverse-field Ising model, showing that the optimization procedure remains robust under statistical and hardware noise. Moreover, our results indicate that our newly proposed setup improves the accuracy of the obtained ground state approximation compared to the original quantum simulation, without increasing the depth of the applied quantum circuits. Therefore, this setup offers a practical route to scale variational quantum algorithms towards the quantum utility scale.
Variational quantum algorithms are practical approaches to prepare ground states, but their potential for quantum advantage remains unclear. Here, we use differentiable 2D tensor networks (TN) to optimize parameterized quantum circuits that prepare the ground state of the transverse field Ising model (TFIM). Our method enables the preparation of states with high energy accuracy, even for large systems beyond 1D. We show that TN pre-optimization can mitigate the barren plateau issue by giving access to enhanced gradient zones that do not shrink exponentially with system size. We evaluate the classical simulation cost evaluating energies at these warm-starts, and identify regimes where quantum hardware offers better scaling than TN simulations.
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.
Near-term quantum devices generally suffer from shallow circuit depth and hence limited expressivity due to noise and decoherence. To address this, we propose tensor-network-assisted parametrized quantum circuits, which concatenate a classical tensor-network operator with a quantum circuit to effectively increase the circuit's expressivity without requiring a physically deeper circuit. We present a framework for tensor-network-assisted variational quantum algorithms that can solve quantum many-body problems using shallower quantum circuits. We demonstrate the efficiency of this approach by considering two examples of unitary matrix-product operators and unitary tree tensor networks, showing that they can both be implemented efficiently. Through numerical simulations, we show that the expressivity of these circuits is greatly enhanced with the assistance of tensor networks. We apply our method to two-dimensional Ising models and one-dimensional time-crystal Hamiltonian models with up to 16 qubits and demonstrate that our approach consistently outperforms conventional methods using shallow quantum circuits.
Computational methods are central to modern physics, yet many problems, especially those involving Quantum Many-Body Systems (QMBSs), remain challenging at scale. The core difficulty of simulating QMBSs is the exponential growth of the solution space with system size. This renders exact treatments intractable and pushes approximate methods to their limits when tackling two- and higher-dimensional systems and long-time dynamics. Quantum computers offer a quantum-native platform for representing and manipulating such systems. Recent algorithmic and technological advancements have enabled experiments on quantum processors in the quantum utility regime, a scale at which exact classical reproduction is no longer feasible. This represents the first step towards realizing quantum advantage, where quantum computers solve specific, practically relevant problems more efficiently than any classical approach. While advancing towards the goal of quantum advantage, useful computations on quantum processors are expected to be realised through hybrid quantum-classical algorithms, which combine quantum experiments and classical computational resources. A prominent example are variational quantum algorithms, where a Parametrized Quantum Circuit (PQC) prepares a trial quantum state whose parameters are iteratively updated by a classical optimisation routine to minimize a target cost function (e.g., the energy of a target Hamiltonian). However, the optimisation of such PQCs is affected by trainability issues, such as barren plateaus in the parameter landscape, which can render the optimisation increasingly difficult for larger system sizes. This motivates the development of novel hybrid approaches overcoming the limits of PQCs. In this thesis, we introduce a new class of approaches based on hybrid Tensor Networks (TNs). A hybrid TN is obtained by integrating quantum states prepared on a quantum processor within TNs -- one of the state-of-the-art approaches for simulating quantum systems. We first show how hybrid TNs can be used for ground-state searches within a variational setting. Then, we further extend the framework to non-variational hybrid TN ansätze for which the quantum tensors are prepared via quantum annealing and subsequently post-processed by classical TNs. In the context of quantum machine learning, which also relies on PQCs, we further investigate heuristic best practices that mitigate the same exponential concentration effects in kernel-based models for anomaly detection. Another class of hybrid algorithms are quantum error-mitigation workflows, where classical post-processing is used to reconstruct expectation values from noise-affected circuit executions. Such methods generally feature a trade-off between the bias and the variance: the price to pay for mitigating noise in a bias-free fashion is that the number of measurements usually grows exponentially with the depth of the circuit in order to achieve a fixed accuracy. In this thesis, we develop a low-resource error-mitigation strategy that reduces the measurement overhead in practical digital Hamiltonian-simulation workflows. We apply this method to Trotterized dynamics in a lattice-gauge-theory model, enabling the extraction of expectation values with improved accuracy at time-scales that allow the observation of post-collision scattering dynamics.
While recent breakthroughs have proven the ability of noisy intermediate-scale quantum (NISQ) devices to achieve quantum advantage in classically-intractable sampling tasks, the use of these devices for solving more practically relevant computational problems remains a challenge. Proposals for attaining practical quantum advantage typically involve parametrized quantum circuits (PQCs), whose parameters can be optimized to find solutions to diverse problems throughout quantum simulation and machine learning. However, training PQCs for real-world problems remains a significant practical challenge, largely due to the phenomenon of barren plateaus in the optimization landscapes of randomly-initialized quantum circuits. In this work, we introduce a scalable procedure for harnessing classical computing resources to provide pre-optimized initializations for PQCs, which we show significantly improves the trainability and performance of PQCs on a variety of problems. Given a specific optimization task, this method first utilizes tensor network (TN) simulations to identify a promising quantum state, which is then converted into gate parameters of a PQC by means of a high-performance decomposition procedure. We show that this learned initialization avoids barren plateaus, and effectively translates increases in classical resources to enhanced performance and speed in training quantum circuits. By demonstrating a means of boosting limited quantum resources using classical computers, our approach illustrates the promise of this synergy between quantum and quantum-inspired models in quantum computing, and opens up new avenues to harness the power of modern quantum hardware for realizing practical quantum advantage.
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.