We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.
We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.
In optimization, one of the well-known classical algorithms is power iterations. Simply stated, the algorithm recovers the dominant eigenvector of some diagonalizable matrix. Since numerous optimization problems can be formulated as an eigenvalue/eigenvector search, this algorithm features wide applicability. Operationally, power iterations consist of performing repeated matrix-to-vector multiplications (or MatVec) followed by a renormilization step in order to converge to the dominant eigenvalue/eigenvector. However, classical realizations, including novel tensor network based approaches, necessitate an exponential scaling for the algorithm's run-time. In this paper, we propose a quantum realiziation to circumvent this pitfall. Our methodology involves casting low-rank representations; Matrix Product Operators (MPO) for matrices and Matrix Product States (MPS) for vectors, into quantum circuits. Specifically, we recover a unitary approximation by variationally minimizing the Frobenius distance between a target MPO and an MPO ansatz wherein the tensor cores are constrained to unitaries. Such an unitary MPO can easily be implemented as a quantum circuit with the addition of ancillary qubits. Thereafter, with appropriate initialization and post-selection on the ancillary space, we realize a single iteration of the classical algorithm. With our proposed methodology, power iterations can be realized entirely on a quantum computer via repeated, static circuit blocks; therefore, a run-time advantage can indeed be guaranteed. Moreover, by exploiting Riemannian optimization and cross-approximation techniques, our methodology becomes instance agnostic and thus allows one to address black-box optimization within the framework of quantum computing.
We adopt a two-dimensional tensor-network (TN) ansatz to simulate variational quantum algorithms on two-dimensional qubit architectures, demonstrating its capability to accurately simulate deep circuits through the Quantum Approximate Optimization Algorithm (QAOA) applied to Ising spin-glass problems on heavy-hexagonal and square lattices. For heavy-hexagonal problems with up to three-body interactions, parameters trained on small instances and transferred to systems an order of magnitude larger improve the sampled energy distribution only up to intermediate depths, indicating a fundamental limit of parameter concentration as a transfer strategy. By extending the training itself with TN simulations on larger system sizes, we avoid local minima and obtain lower-energy samples. Analyses of entanglement growth and importance sampling show that the simulation remains classically feasible with moderate bond dimension. We find that parameter concentration also persists on square lattices, albeit at substantially higher computational cost to perform reliable sampling. Overall, our TN framework not only provides an efficient and controlled framework for benchmarking variational quantum algorithms on two-dimensional lattices, but also serves as an effective surrogate model for training variational algorithms.
Quantum computing has long promised to revolutionize the way we solve complex problems. At the same time, tensor networks are widely used across various fields due to their computational efficiency and capacity to represent intricate systems. While both technologies can address similar problems, the primary aim of this thesis is not to compare them. Such comparison would be unfair, as quantum devices are still in an early stage, whereas tensor network algorithms represent the state-of-the-art in quantum simulation. Instead, we explore a potential synergy between these technologies, focusing on how two flagship algorithms from each paradigm, the Density Matrix Renormalization Group (DMRG) and quantum annealing, might collaborate in the future. Furthermore, a significant challenge in the DMRG algorithm is identifying an appropriate tensor network representation for the quantum system under study. The representation commonly used is called Matrix Product Operator (MPO), and it is notoriously difficult to obtain for certain systems. This thesis outlines an approach to this problem using finite automata, which we apply to construct the MPO for our case study. Finally, we present a practical application of this framework through the quadratic knapsack problem (QKP). Despite its apparent simplicity, the QKP is a fundamental problem in computer science with numerous practical applications. In addition to quantum annealing and the DMRG algorithm, we implement a dynamic programming approach to evaluate the quality of our results. Our results highlight the power of tensor networks and the potential of quantum annealing for solving optimization problems. Moreover, this thesis is designed to be self-explanatory, ensuring that readers with a solid mathematical background can fully understand the content without prior knowledge of quantum mechanics.
This is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algorithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the "quantics" representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross-Pitaevskii) within the "quantics" representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements.
Abstract Tensor networks (TNs) are a family of computational methods built on graph-structured factorizations of large tensors, which have long represented state-of-the-art methods for the approximate simulation of complex quantum systems on classical computers. The rapid pace of recent advancements in numerical computation, notably the rise of GPU and TPU hardware accelerators, have allowed TN algorithms to scale to even larger quantum simulation problems, and to be employed more broadly for solving machine learning tasks. The ‘quantum-inspired’ nature of TNs permits them to be mapped to parametrized quantum circuits (PQCs), a fact which has inspired recent proposals for enhancing the performance of TN algorithms using near-term quantum devices, as well as enabling joint quantum–classical training frameworks that benefit from the distinct strengths of TN and PQC models. However, the success of any such methods depends on efficient and accurate methods for approximating TN states using realistic quantum circuits, which remains an unresolved question. This work compares a range of novel and previously-developed algorithmic protocols for decomposing matrix product states (MPS) of arbitrary bond dimension into low-depth quantum circuits consisting of stacked linear layers of two-qubit unitaries. These protocols are formed from different combinations of a preexisting analytical decomposition method together with constrained optimization of circuit unitaries, with initialization by the former method helping to avoid poor-quality local minima in the latter optimization process. While all of these protocols have efficient classical runtimes, our experimental results reveal one particular protocol employing sequential growth and optimization of the quantum circuit to outperform all others, with even greater benefits in the setting of limited computational resources. Given these promising results, we expect our proposed decomposition protocol to form a useful ingredient within any joint application of TNs and PQCs, further unlocking the rich and complementary benefits of classical and quantum computation.
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.