Here, we focus on the problem of minimizing complex classical cost functions associated with prototypical discrete neural networks, specifically the paradigmatic Hopfield model and binary perceptron. We show that the adiabatic time evolution of QA can be efficiently represented as a suitable Tensor Network. This representation allows for simple classical simulations, well-beyond small sizes amenable to exact diagonalization techniques. We show that the optimized state, expressed as a Matrix Product State (MPS), can be recast into a Quantum Circuit, whose depth scales only linearly with the system size and quadratically with the MPS bond dimension. This may represent a valuable starting point allowing for further circuit optimization on near-term quantum devices.
Here, we focus on the problem of minimizing complex classical cost functions associated with prototypical discrete neural networks, specifically the paradigmatic Hopfield model and binary perceptron. We show that the adiabatic time evolution of QA can be efficiently represented as a suitable Tensor Network. This representation allows for simple classical simulations, well-beyond small sizes amenable to exact diagonalization techniques. We show that the optimized state, expressed as a Matrix Product State (MPS), can be recast into a Quantum Circuit, whose depth scales only linearly with the system size and quadratically with the MPS bond dimension. This may represent a valuable starting point allowing for further circuit optimization on near-term quantum devices.
Quantum annealing and quantum approximate optimization algorithms hold a great potential to speed-up optimization problems. This could be game-changing for a plethora of applications. Yet, in order to hope to beat classical solvers, quantum circuits must scale up to sizes and performances much beyond current hardware. In that quest, intense experimental effort has been recently devoted to optimizations on 3-regular graphs, which are computationally hard but experimentally relatively amenable. However, even there, the amount and quality of quantum resources required for quantum solvers to outperform classical ones is unclear. Here, we show that quantum annealing for 3-regular graphs can be classically simulated even at scales of 1000 qubits and 5000000 two-qubit gates with all-to-all connectivity. To this end, we develop a graph tensor-network quantum annealer (GTQA) able of high-precision simulations of Trotterized circuits of near-adiabatic evolutions. Based on a recently proposed belief-propagation technique for tensor canonicalization, GTQA is equipped with re-gauging and truncation primitives that keep approximation errors small in spite of the circuits generating significant amounts of entanglement. As a result, even with a maximal bond dimension as low as 4, GTQA produces solutions competitive with those of state-of-the-art classical solvers. For non-degenerate instances, the unique solution can be read out from the final reduced single-qubit states. In contrast, for degenerate problems, such as MaxCut, we introduce an approximate measurement simulation algorithm for graph tensor-network states. On one hand, our findings showcase the potential of GTQA as a powerful quantum-inspired optimizer. On the other hand, they considerably raise the bar required for experimental demonstrations of quantum speed-ups in combinatorial optimizations.
Classical simulation of quantum circuits is an essential tool in quantum information science, but its applicability is constrained by the exponential growth of the Hilbert space and the entanglement structure of quantum states. In this work, we introduce Matrix Product Evolution (MPE), a tensor-train representation of quantum circuits constructed along the circuit depth rather than along the qubit index. Within this formulation, the simulation of a quantum circuit is modeled as the contraction of multiple MPE tensors. We develop an efficient contraction strategy based on a zip-up procedure to carry out this contraction in practice. We investigate the numerical behavior of this MPE-based contraction framework through simulations of random quantum circuits and the time evolution of a quantum many-body state. Our results characterize the growth of temporal bond dimensions, clarify how post-selection modifies the contraction cost and approximation accuracy, and identify regimes in which depth-oriented tensor-network contractions provide a useful complement to standard MPS-based simulation approaches.
The accurate simulation of quantum circuits remains a significant challenge due to the exponential scaling of computational resources with circuit size. This work explores the application of tensor network methods, specifically Matrix Product States (MPS), to provide a scalable approach for simulating quantum circuits. We demonstrate how tensor networks can efficiently represent and manipulate the states of quantum circuits, reducing the computational burden associated with traditional methods. The core claim of this research is that tensor networks offer a viable path to accurately simulating larger and more complex quantum systems. This research introduces a novel framework for symbolic simulation, leveraging the power of tensor network representations to overcome the limitations of conventional simulation techniques. The effectiveness of this approach is illustrated through theoretical analysis and conceptual examples.
The Thesis explores the application of Matrix Product States (MPS) in the domain of quantum many-body systems and quantum computing, show casing the versatility and effectiveness of MPS for addressing complex com putational challenges. The opening chapter introduces essential tools for Tensor Networks, addressing the curse of dimensionality and elucidating the main characteristics of Matrix Product States and Matrix Product Op erators. The discussion encompasses also important numerical techniques, including Density Matrix Renormalization Group and Time Evolving Block Decimation. Furthermore, the relation between MPS and quantum circuits is discussed. The subsequent Chapters delve into specific applications of MPS. In the second Chapter, we introduce a Tensor Network ansatz inspired by the backflow transformation for correlated systems. This extension of the MPS representation ensures an area law for entanglement in dimensions one or greater. We employ an optimization scheme combining DMRG and variational Monte Carlo algorithms for efficient ground-state search. Bench marking against spin models demonstrates high accuracy. The third Chapter explores quantum annealing for optimizing complex classical spin Hamil tonians, as the Hopfield model and the binary perceptron. We introduce an efficient Tensor Network representation for the adiabatic time evolution of quantum annealing, enabling scalable classical simulations. The use of MPS in mitigating Trotter errors and mapping to quantum circuits is also explored. In the fourth Chapter we present a novel method for evaluating the amount of nonstabilizerness (also known as quantum magic) contained in an MPS. We overcome the exponentially hard evaluation of this quan tity by employing a simple perfect sampling technique of Pauli string. Our innovative MPS approach enables efficient computation. Benchmarked on magic states and the quantum Ising chain ground state, this method offers easy access to the non-equilibrium dynamics of nonstabilizerness following a quantum quench. The Appendices provide additional insights, including: an introduction to variational Monte Carlo, technical details on MPS simula tions for the quantum annealing and a comprehensive overview on stabilizer formalism and nonstabilizerness.
Quantum annealing is a promising paradigm for building practical quantum computers. Compared to other approaches, quantum annealing technology has been scaled up to a larger number of qubits. On the other hand, deep learning has been profoundly successful in pushing the boundaries of AI. It is thus natural to investigate potentially game changing technologies such as quantum annealers to augment the capabilities of deep learning. In this work, we explore binary neural networks, which are lightweight yet powerful models typically intended for resource constrained devices. Departing from current training regimes for binary networks that smooth/approximate the activation functions to make the network differentiable, we devise a quadratic unconstrained binary optimization formulation for the training problem. While the problem is intractable, i.e., the cost to estimate the binary weights scales exponentially with network size, we show how the problem can be optimized directly on a quantum annealer, thereby opening up to the potential gains of quantum computing. We experimentally validated our formulation via simulation and testing on an actual quantum annealer (D-Wave Advantage), the latter to the extent allowable by the capacity of current technology.
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.