Solving large-scale linear systems problems is a central task in scientific and industrial computing. Classical iterative solvers face increasing difficulty as the number of unknowns becomes large, while fully quantum linear solvers require fault-tolerant resources that remain far beyond near-term feasibility. Here we propose a quantum-accelerated conjugate gradient (QACG) framework in which a fault-tolerant quantum algorithm is used exclusively to construct a spectrally informed initial guess for a classical conjugate gradient (CG) solver. Rather than replacing classical kernels, the quantum subroutine functions as a cooperative accelerator that selectively suppresses low-energy spectral components responsible for slow classical convergence. We analyze the total runtime and resource requirements of this integrated quantum-HPC platform for the 3D Poisson equation. A central feature of QACG is a controllable decomposition of the condition number between the quantum and the classical solver, enabling flexible allocation of computational effort across quantum and classical resources. Under explicit architectural assumptions, we identify regimes in which this cooperative strategy yields a runtime advantage over purely classical approaches while requiring substantially fewer quantum resources than end-to-end quantum linear solvers. These results illustrate a concrete pathway toward the scientific and industrial use of early-stage fault-tolerant quantum computing and point to a scalable hybrid paradigm in which quantum devices act as accelerators within high-performance computing workflows rather than as standalone replacements.
Solving large-scale linear systems problems is a central task in scientific and industrial computing. Classical iterative solvers face increasing difficulty as the number of unknowns becomes large, while fully quantum linear solvers require fault-tolerant resources that remain far beyond near-term feasibility. Here we propose a quantum-accelerated conjugate gradient (QACG) framework in which a fault-tolerant quantum algorithm is used exclusively to construct a spectrally informed initial guess for a classical conjugate gradient (CG) solver. Rather than replacing classical kernels, the quantum subroutine functions as a cooperative accelerator that selectively suppresses low-energy spectral components responsible for slow classical convergence. We analyze the total runtime and resource requirements of this integrated quantum-HPC platform for the 3D Poisson equation. A central feature of QACG is a controllable decomposition of the condition number between the quantum and the classical solver, enabling flexible allocation of computational effort across quantum and classical resources. Under explicit architectural assumptions, we identify regimes in which this cooperative strategy yields a runtime advantage over purely classical approaches while requiring substantially fewer quantum resources than end-to-end quantum linear solvers. These results illustrate a concrete pathway toward the scientific and industrial use of early-stage fault-tolerant quantum computing and point to a scalable hybrid paradigm in which quantum devices act as accelerators within high-performance computing workflows rather than as standalone replacements.
High-Performance Computing (HPC) is today the main tool for large-scale scientific computing, but performance gains are increasingly limited by architectural and energy constraints. At the same time, quantum computing is an emerging paradigm, still under development, but already promising for accelerating certain computational kernels. A central challenge is therefore to understand how to efficiently integrate quantum processors within existing HPC infrastructures, rather than treating them as isolated accelerators. The objective of this thesis is to explore hybrid quantum-classical computing, both from the algorithmic perspective and in terms of programming models. We investigate how to coherently combine classical and quantum resources within hybrid workflows, leveraging the respective strengths of each paradigm. As a primary use case, we focus on solving linear systems of equations, a fundamental building block of scientific computing and HPC. First, we address the challenge of encoding classical data into quantum representations, which remains a major bottleneck for hybrid algorithms. We propose an improved preprocessing method for matrix encoding into unitaries using a Pauli decomposition, reducing the associated computational cost. Next, we introduce a hybrid solver based on the Quantum Singular Value Transformation (QSVT). Inspired by classical numerical linear algebra, we propose a mixed-precision iterative refinement approach, combining low-precision quantum solves with high-precision classical corrections. A detailed analysis of convergence, errors, and computational cost shows that hybridization can reduce the quantum resources required to reach a given precision.Finally, we address programming and implementation aspects through the design of a hybrid quantum-classical programming framework, Q-Pragma, aimed at facilitating the integration of quantum acceleration into existing HPC environments. Overall, this work contributes to the development of more practical and usable hybrid quantum-HPC codes, bridging the gap between theoretical algorithms and real-world high-performance applications.
The advent of fault-tolerant quantum computing (FTQC) promises to tackle classically intractable problems. A key milestone is solving the Navier-Stokes equations (NSE), which has remained formidable for quantum algorithms due to their high input-output overhead and nonlinearity. Here, we establish a full-stack framework that charts a practical pathway to a quantum advantage for large-scale NSE simulation. Our approach integrates a spectral-based input/output algorithm, an explicit and synthesized quantum circuit, and a refined error-correction protocol. The algorithm achieves an end-to-end exponential speedup in asymptotic complexity, meeting the lower bound for general quantum linear system solvers. Through symmetry-based circuit synthesis and optimized error correction, we reduce the required logical and physical resources by two orders of magnitude. Our concrete resource analysis demonstrates that solving NSE on a 280-grid is feasible with 8.71 million physical qubits (at an error rate of 5×10−4) in 42.6 days -- outperforming a state-of-the-art supercomputer, which would require over a century. This work bridges the gap between theoretical quantum speedup and the practical deployment of high-performance scientific computing.
Fault tolerance is an indispensable prerequisite for constructing large-scale universal quantum computers. Drawing philosophies from classical computer architecture, this paper presents a hardware-agnostic three-layer high-level architectural framework for generic fault-tolerant quantum computation. Guided by the real execution workflows of fault-tolerant quantum algorithms, the proposed model is decoupled from specific physical qubit hardware platforms and quantum error correction codes, serving as a universal abstract standard rather than a platform-specific implementation scheme. Special attention is devoted to the intermediate Fault-Tolerance Layer, which serves as the architectural bridge between application-level logical programs and hardware-level execution. We systematically characterize its five internal components, the interfaces and data exchanged among them, and the execution, correction, and adaptation paths that together enable logical synthesis, fault-tolerant resources management, decoding, and runtime fault-tolerant control. An end-to-end example is further provided to illustrate the full-stack operating pipeline of fault-tolerant quantum algorithms under this framework. Given the increasing emphasis on modular, heterogeneous, and cross-layer fault-tolerant quantum systems, our architecture provides a unified foundational model for organizing such designs.
We investigate the feasibility of early fault-tolerant quantum algorithms focusing on ground-state energy estimation problems. In particular, we examine the computation of the cumulative distribution function (CDF) of the spectral measure of a Hamiltonian and the identification of its discontinuities. Scaling these methods to larger system sizes reveals three key challenges: the smoothness of the CDF for large supports, the lack of tight lower bounds on the overlap with the true ground state, and the difficulty of preparing high-quality initial states. To address these challenges, we propose a signal processing approach to find these estimates automatically, in the regime where the quality of the initial state is unknown. Rather than aiming for exact ground-state energy, we advocate for improving classical estimates by targeting the low-energy support of the initial state. Additionally, we provide quantitative resource estimates, demonstrating a constant factor improvement in the number of samples required to detect a specified change in CDF. Our numerical experiments, conducted on a 26-qubit fully connected Heisenberg model, leverage a truncated density-matrix renormalization group (DMRG) initial state with a low bond dimension. The results show that the predictions from the quantum algorithm align closely with the DMRG-converged energies at larger bond dimensions while requiring several orders of magnitude fewer samples than theoretical estimates suggest. These findings underscore that CDF-based quantum algorithms are a practical and resource-efficient alternative to quantum phase estimation, particularly in resource-constrained scenarios.
While quantum computers promise to solve classically intractable problems, identifying the point at which fault-tolerant quantum computation outperforms the best classical algorithms for practical applications remains an outstanding challenge. Here we establish a concrete quantum-classical crossover for quantum many-body dynamics under realistic hardware conditions. We introduce a scalable fault-tolerant framework that combines coherent observable estimation with a space-time-efficient implementation of non-Clifford rotations, suppressing the residual logical errors that limit existing partially fault-tolerant approaches. A benchmark against state-of-the-art tensor-network and variational Monte Carlo algorithms reveals a concrete crossover for mixed-field Ising dynamics at modest system sizes. For a physical error rate of p=10−3, fault-tolerant simulation requires approximately 2 hours and 3.7×105 physical qubits for a 100-site 1D system, whereas tensor network approaches would require about 100 years. For 2D models, where rapid entanglement growth limits the classical evolution time, we project quantum runtimes within minutes. A physical error rate of p=10−4 leads to at least an order of magnitude reduction in qubit count (3.1×104 physical qubits) and runtime (minutes for 1D and seconds for 2D). The reduction in quantum runtime arises from our improved rotation-state injection and co-design of quantum error correction and observable-estimation protocols, which jointly suppress logical-error accumulation and reduce sampling overhead. Our results establish a scalable route towards practical quantum advantage and identify quantitative engineering targets for future fault-tolerant architectures.
As industrial models and designs grow increasingly complex, the demand for optimal control of large-scale dynamical systems has significantly increased. However, traditional methods for optimal control incur significant overhead as problem dimensions grow. In this paper, we introduce an end-to-end quantum algorithm for linear-quadratic control with provable speedups. Our algorithm, based on a policy gradient method, incorporates a novel quantum subroutine for solving the matrix Lyapunov equation. Specifically, we build a quantum-assisted differentiable simulator for efficient gradient estimation that is more accurate and robust than classical methods relying on stochastic approximation. Compared to the classical approaches, our method achieves a super-quadratic speedup. To the best of our knowledge, this is the first end-to-end quantum application to linear control problems with provable quantum advantage.