Evaluating the entanglement spectrum is essential for characterizing exotic quantum phases such as quantum criticality and topological order. However, for large quantum many-body systems, this task is hindered by the exponential measurement complexity of standard tomographic techniques. To address this challenge, we introduce a hybrid quantum-classical variational framework for partial singular value decomposition of bipartite states, built on the canonical form of matrix product states. We employ a deflation-based optimization approach to sequentially extract dominant and subdominant Schmidt components of target states. Because hardware noise and finite circuit depth can compromise the mutual orthogonality of these extracted vectors, we propose an improved deflation algorithm incorporating explicit classical orthogonality correction. This classical post-processing acts as an error-filtering mechanism, enabling shallow and suboptimal quantum circuits. As a result, numerical accuracy is decoupled from quantum circuit optimization, mitigating optimization difficulties caused by barren plateaus and hardware noise. Furthermore, shallow ansatzes enable a concurrent execution strategy. Overlap matrices are evaluated by classical tensor network contractions, while cross terms between the target state and the extracted vectors are computed using an auxiliary reference state. This concurrent hybrid design improves computational throughput and bypasses the overhead of controlled target-state preparation. Numerical benchmarks on the ground states of one- and two-dimensional Heisenberg models demonstrate improved accuracy and numerical stability. By mitigating hurdles of circuit depth, optimization hardness, and measurement complexity, our framework provides a robust pathway for large-scale entanglement spectrum estimation on advanced near-term quantum devices.
Evaluating the entanglement spectrum is essential for characterizing exotic quantum phases such as quantum criticality and topological order. However, for large quantum many-body systems, this task is hindered by the exponential measurement complexity of standard tomographic techniques. To address this challenge, we introduce a hybrid quantum-classical variational framework for partial singular value decomposition of bipartite states, built on the canonical form of matrix product states. We employ a deflation-based optimization approach to sequentially extract dominant and subdominant Schmidt components of target states. Because hardware noise and finite circuit depth can compromise the mutual orthogonality of these extracted vectors, we propose an improved deflation algorithm incorporating explicit classical orthogonality correction. This classical post-processing acts as an error-filtering mechanism, enabling shallow and suboptimal quantum circuits. As a result, numerical accuracy is decoupled from quantum circuit optimization, mitigating optimization difficulties caused by barren plateaus and hardware noise. Furthermore, shallow ansatzes enable a concurrent execution strategy. Overlap matrices are evaluated by classical tensor network contractions, while cross terms between the target state and the extracted vectors are computed using an auxiliary reference state. This concurrent hybrid design improves computational throughput and bypasses the overhead of controlled target-state preparation. Numerical benchmarks on the ground states of one- and two-dimensional Heisenberg models demonstrate improved accuracy and numerical stability. By mitigating hurdles of circuit depth, optimization hardness, and measurement complexity, our framework provides a robust pathway for large-scale entanglement spectrum estimation on advanced near-term quantum devices.
Abstract Schmidt decomposition of a vector can be understood as writing the singular value decomposition (SVD) in vector form. A vector can be written as a linear combination of tensor product of two dimensional vectors by recursively applying Schmidt decompositions via SVD to all subsystems. Given a vector expressed as a linear combination of tensor products, using only the k principal terms yields a k-rank approximation of the vector. Therefore, writing a vector in this reduced form allows to retain most important parts of the vector while removing small noises from it, analogous to SVD-based denoising. In this paper, we show that quantum circuits designed based on a value k (determined from the tensor network decomposition of the mean vector of the training sample) can approximate the reduced-form representations of entire datasets. We then employ this circuit ansatz with a classical neural network head to construct a hybrid machine learning model. Since the output of the quantum circuit for an 2n dimensional vector is an n dimensional probability vector, this provides an exponential compression of the input and potentially can reduce the number of learnable parameters for training large-scale models. We use datasets provided in the Python scikit-learn module for the experiments. The results confirm the quantum circuit is able to compress data successfully to provide effective k-rank approximations to the classical processing component.
This research paper explores the application of singular value decomposition (SVD) in quantum image processing (QIP), specifically focusing on the computation of eigenvalues using variational quantum algorithms. SVD is a powerful mathematical tool in image processing, used for tasks such as image compression, noise reduction, and feature extraction. In this study, we propose quantum variational singular value decomposition (QVSVD) based on the variational quantum deflation (VQD) algorithm to determine the eigenvalues that contributes in calculating the singular values of the image matrix and eventually doing the SVD. This facilitates the extraction of eigenvalues with exponential speedup on real hardware compared to classical methods. We detail the implementation of this quantum algorithm within the framework of QIP, highlighting its advantages in terms of computational efficiency and accuracy on a Fault-Tolerant Quantum Computing (FTQC). Furthermore, we present a comparative analysis of the quantum and the classical method of SVD, demonstrating the accuracy of the image data. The simulation results validate the theoretical advantages of using quantum algorithms for SVD in image processing. This work not only underscores the potential of quantum computing in enhancing image processing techniques but also sets the stage for future research in the field, exploring more complex image processing tasks and other quantum algorithms. Our findings suggest that quantum image processing can offer unprecedented capabilities, paving the way for advancements in various applications such as medical imaging, remote sensing, and multimedia processing.
Preparing arbitrary quantum states requires exponential resources. Matrix Product States (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50% and CNOT counts by 33%. We evaluate the full pipeline on datasets of varying complexity across systems of 19-50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.
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.
Hybrid quantum classical algorithms offer a promising approach for exploiting Noisy Intermediate Scale Quantum devices; however, their practical deployment remains limited by excessive circuit depth, hardware noise, and inefficient utilization of quantum resources. This paper presents a Hybrid Variational Optimization (HVO) framework that reduces quantum algorithmic complexity by integrating adaptive circuit pruning with classical feedback-driven optimization and a unified Quantum Complexity Metric (QCM). Unlike prior adaptive variational approaches such as RE-ADAPT-VQA, the proposed framework explicitly optimizes the trade-off between gate count, fidelity, and qubit resources while remaining modular and hardware-agnostic. To ensure a fair and transparent evaluation, the proposed method is compared against a well-tuned shallow Variational Quantum Eigensolver (VQE), QAOA, and RE-ADAPT-VQA. Experimental validation on IBM Quantum simulators and real NISQ hardware demonstrates up to a 35% reduction in circuit complexity, a 30% reduction in execution time, and improved fidelity of 0.94. Furthermore, performance evaluation using accuracy, precision, recall, and F1-score confirms the robustness and reliability of the proposed framework. These results demonstrate that HVO provides a scalable and noise-resilient solution for near term quantum computing applications.
The rapid advancement of quantum-inspired classical algorithms, particularly Quantics Tensor Trains (QTT), has demonstrated that exponentially large vectors can be manipulated efficiently via Matrix Product States (MPS). Conventionally, the compression of these tensor networks relies exclusively on Singular Value Decomposition (SVD)-based truncation. While SVD is optimal for minimizing the Frobenius norm error, it remains structurally blind to exact algebraic correlations, such as reversible logic gates or Clifford symmetries, that do not require numerical approximation to factor out. In this work, we propose a hybrid compression protocol that integrates the topological rigor of ZX-Calculus with the numerical power of SVD. We construct an explicit isomorphism between Rank-3 MPS tensors and ZX-diagrams, allowing us to subject QTT representations to formal diagrammatic rewriting rules prior to numerical truncation. We establish that this Topological Preconditioning can algebraically collapse the effective bond dimension (χ) and T-count of the underlying tensor network without information loss. By benchmarking the algorithmic implementation against hardware-accelerated SVD on discretized functions and stabilizer states, we demonstrate that ZX-driven algebraic erasure fundamentally bypasses standard [Formula: see text] bottlenecks, yielding up to a computational speedup. Finally, we formalize this advantage by defining the structural complexity class [Formula: see text], proving that for systems characterized by algebraic symmetry, this work reflects our ongoing effort to deepen understanding of Categorical Quantum Mechanics, aiming to contribute to the optimization of high-dimensional linear algebra.