In this report, we propose a novel quantum diagonalization algorithm based on the optimization of variational quantum circuits. Diagonalizing a quantum state is a fundamental yet computationally challenging task in quantum information science, especially as the system size increases. To address this challenge, we reformulate the problem as a variational optimization process, where parameterized quantum circuits are trained to transform the input state into a diagonal form. To guide the optimization, we develop two objective functions based on measurement outcomes in the computational basis. The first objective function utilizes global computational basis probabilities, with the optimized value directly yielding the purity of the input state. The second objective function, designed for enhanced experimental feasibility, is constructed solely from single-qubit probabilities. It admits an elegant and compact analytical form that significantly reduces the exponential measurement complexity, while still effectively driving the state toward a diagonal representation. Through numerical simulations and analytical insights, we demonstrate that our variational optimization framework successfully produces the diagonal form of an input quantum state, offering a scalable and practical solution for quantum state diagonalization.
In this report, we propose a novel quantum diagonalization algorithm based on the optimization of variational quantum circuits. Diagonalizing a quantum state is a fundamental yet computationally challenging task in quantum information science, especially as the system size increases. To address this challenge, we reformulate the problem as a variational optimization process, where parameterized quantum circuits are trained to transform the input state into a diagonal form. To guide the optimization, we develop two objective functions based on measurement outcomes in the computational basis. The first objective function utilizes global computational basis probabilities, with the optimized value directly yielding the purity of the input state. The second objective function, designed for enhanced experimental feasibility, is constructed solely from single-qubit probabilities. It admits an elegant and compact analytical form that significantly reduces the exponential measurement complexity, while still effectively driving the state toward a diagonal representation. Through numerical simulations and analytical insights, we demonstrate that our variational optimization framework successfully produces the diagonal form of an input quantum state, offering a scalable and practical solution for quantum state diagonalization.
Abstract Despite extensive research efforts, few quantum algorithms for classical optimization demonstrate realizable quantum advantage. The utility of many quantum algorithms is limited by high requisite circuit depth and nonconvex optimization landscapes. We tackle these challenges by introducing a new variational quantum algorithm that utilizes multi-basis graph encodings and nonlinear activation functions. Our technique results in increased optimization performance, a factor of two increase in effective quantum resources, and a quadratic reduction in measurement complexity. While the classical simulation of many qubits with traditional quantum formalism is impossible due to its exponential scaling, we mitigate this limitation with exact circuit representations using factorized tensor rings. In particular, the shallow circuits permitted by our technique, combined with efficient factorized tensor-based simulation, enable us to successfully optimize the MaxCut of the nonlocal 512-vertex DIMACS library graphs on a single GPU. By improving the performance of quantum optimization algorithms while requiring fewer quantum resources and utilizing shallower, more error-resistant circuits, we offer tangible progress for variational quantum optimization.
Variational quantum algorithms have promising applications in noisy intermediate-scale quantum (NISQ) devices. These algorithms rely on a classical optimization outer loop that minimizes a parameterized quantum circuit function. The optimization in variational quantum eigensolver (VQE) is NP-hard, meaning that finding the optimal solution is infeasible in the worst-case scenario. One way to address this challenge is through parallel optimization of parameters using multiple-parameterized quantum circuits. However, this approach is unsuitable for cloud-based quantum processing unit utilization due to the increased number of quantum circuit executions. Although NISQ devices have limitations in terms of gate depth, their size has been growing in recent years. Therefore, implementing multiple-parameterized quantum circuits in NISQ devices can suppress the increase in the number of executions. In this study, we propose a parallel-VQE, which leverages the parallel execution of parameterized quantum circuits to perform parallel parameter optimization in VQE, achieving convergence to solutions closer to the ground state. We validate the effectiveness of parallel-VQE in solving the random weighted max-cut problem using numerical simulations and a real quantum device. We present the results of running up to six circuits in parallel (120 qubits) and demonstrate the advantages of using multiple units to improve computational accuracy. This study provides a potential method for solving eigenvalue problems and combinatorial optimization problems for future quantum devices.
Variational quantum algorithms (VQAs) have emerged as the leading strategy for using these noisy devices to achieve an advantage over their classical counterparts.VQAs use a hybrid quantum-classical framework, in which one employs the quantum computer to prepare a parameterized quantum state and outsource the optimization to a classical subroutine. VQAs have been proposed for applications in various fields, including physics, quantum chemistry, machine learning, and optimization. Nevertheless, many challenges must be addressed before we can use them for practical tasks. In this dissertation, we propose strategies to overcome some challenges in implementing VQAs. We first look at the issue of trainability for VQAs and propose three methods for optimizing them.The first method is based on natural evolutionary strategies (NES) and can alleviate the issue of vanishing gradients for randomly initialized parameterized quantum circuits (PQCs). The second method is based on the notion of information flow in PQCs and uses a mutual information-based distance metric to define paths for stochastic optimization of VQAs. Finally, we develop gradient evaluation procedures for optimizing objectives with the unitary coupled-cluster ansatz. Next, we study the issue of expressibility of PQCs used in variational algorithms.First, using the theory of quantum optimal control we define a notion of reachability of general ans\"atze used in VQAs. Next, using a variational autoencoder (VAE) coupled with a regression model, we try to gain a better understanding of the parameterization of PQCs. Finally, we attempt to remedy the inefficient usage of quantum resources for storing quantum data using evolutionary strategies to design quantum circuits for compressing quantum data. We present numerical evidence that the work presented here can increase trainability, efficiency, and understanding of VQAs.We believe our work is of interest to the quantum computing community as we continue to push the limits of NISQ devices for solving practical problems.
Quantum computing has emerged as one of the most influential paradigms capable of transforming computational processes that struggle under classical limitations. In recent years, hybrid quantum–classical algorithms have gained significant attention for their ability to leverage the strengths of both architectures, especially within the Noisy Intermediate-Scale Quantum (NISQ) era. This research paper explores a quantum-enhanced optimization framework designed to improve machine learning model performance through parameterized quantum circuits (PQCs). The goal of this study is to provide a practical, scalable approach that can be integrated into existing machine learning workflows without requiring fully developed fault-tolerant quantum hardware. The proposed framework uses classical preprocessing to encode data into quantum states, followed by quantum transformations that explore richer feature spaces due to quantum superposition and entanglement. A classical optimizer then updates the quantum circuit parameters based on measurement results, forming an end-to-end feedback loop. This hybrid model aims to demonstrate improved optimization efficiency, reduction in loss-function plateaus, and better generalization for specific ML tasks involving small to medium-sized datasets. This paper provides a comprehensive investigation of methodology, literature background, and experimental reasoning with statistical analysis, along with conceptual diagrams illustrating system architecture and circuit layouts. The outcomes further strengthen the viability of hybrid quantum–classical computation as a promising direction for advanced machine learning systems.
Abstract Variational hybrid quantum-classical algorithms are promising candidates for near-term implementation on quantum computers. In these algorithms, a quantum computer evaluates the cost of a gate sequence (with speedup over classical cost evaluation), and a classical computer uses this information to adjust the parameters of the gate sequence. Here we present such an algorithm for quantum state diagonalization. State diagonalization has applications in condensed matter physics (e.g., entanglement spectroscopy) as well as in machine learning (e.g., principal component analysis). For a quantum state ρ and gate sequence U, our cost function quantifies how far UρU† U ρ U † is from being diagonal. We introduce short-depth quantum circuits to quantify our cost. Minimizing this cost returns a gate sequence that approximately diagonalizes ρ. One can then read out approximations of the largest eigenvalues, and the associated eigenvectors, of ρ. As a proof-of-principle, we implement our algorithm on Rigetti’s quantum computer to diagonalize one-qubit states and on a simulator to find the entanglement spectrum of the Heisenberg model ground state.
Many fundamental problems in quantum information theory can be formulated as optimization tasks over quantum states and measurements. While the convex structure of quantum theory often enables powerful semidefinite programming formulations, a major challenge persists: the exponential growth of Hilbert space dimension with system size. This dissertation discusses techniques to address these and related obstacles across several core problems in quantum information. We first study optimization over separable quantum states. Such problems are typically NP-hard and standard semidefinite programming hierarchies suffer from the curse of dimensionality. We introduce a hybrid algorithm that leverages a quantum co-processor to reduce the effective dimension of the problem. While several existing approaches provide only a numerical bound on the optimal value, our method is guaranteed to return an explicit feasible solution: a valid separable state, even in instances where it is not optimal. We apply this framework to approximate the minimum separable energy of many-body Hamiltonians, define an entanglement measure for their ground spaces, and demonstrate numerical scalability to systems of up to 28 qubits. We next consider the general Bayes framework, as introduced by Helstrom, for state discrimination when, instead of a classical description of the candidate states, one has access to their source code: the quantum circuit that prepares them. We show that the typical semidefinite program for the discrimination problem can be reformulated in terms of the Gram matrix of these states, reducing the variable dimensions from dL to NL, where d is the Hilbert space dimension, N is the number of candidate states, and L is the number of possible guesses. Importantly, we further introduce a quantum pre-processing procedure which efficiently constructs the reduced problem from the source code, enabling our method to operate directly on quantum data. We consider two applications. First, we characterize the optimal identifications for quantum changepoint problems under several reward structures whose associated discrimination problems were previously computationally inaccessible. Second, we consider a quantum error classification problem and show how our reduction makes it tractable for systems of hundreds of qubits. We also analyze structural measurement strategies. As a counterpart to the well-known pretty good measurement for minimum-error state discrimination, we introduce the pretty bad measurement for the exclusion variant, in which the goal is to identify a state that was not sent, rather than to identify the one that was, and prove that both are guaranteed to perform at least as well as blind guessing in their respective objectives. Finally, we consider evaluating Boolean functions under severe memory constraints and prove that a simple local "greedy" measurement strategy is globally optimal precisely for affine Boolean functions, while more generally achieving at least the square of the optimal global success probability. Together, these results demonstrate that high-dimensional quantum optimization problems often admit hidden structure enabling dimension reduction, hybrid computation, and provable approximation guarantees.