Quantum algorithms achieve polylog(N) complexity for high-dimensional linear SDEs by amplitude-encoding the solution and noise via Dyson series or Euler-Maruyama approximations plus quantum linear systems solvers.
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Quantum Linear System Solvers: A Survey of Algorithms and Applications
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Constrained Uniform Polynomial (CUP) and Constrained Adaptive Polynomial (CAP) solvers achieve lower error than standard QSVT and Chebyshev methods in noise-limited regimes by optimizing accuracy versus block-encoding normalization under uniform or moment-based spectral models.
A polylog-sized quantum computer achieves exponential advantage over classical machines in classification and dimension reduction of massive classical data using quantum oracle sketching combined with classical shadows.
QAFE² uses quantum parallelism to evaluate every RVE problem at all quadrature points simultaneously, delivering polylog complexity in microscopic mesh size and exponential speedup over classical solvers.
A modular block-encoding framework for finite-difference Laplacians supporting arbitrary combinations of Dirichlet, periodic, and Neumann boundary conditions across dimensions.
Probabilistic quantum algorithm prepares mixed states proportional to Lyapunov equation solutions and matrix inverses using oracles for input matrices and a deterministic stopping rule.
Pivot-shifted Carleman linearization with Lyapunov transform enables logarithmic truncation and removes initial-condition lower bounds for quantum simulation of a broader class of nonlinear ODEs.
A matrix decomposition into linear combinations of non-unitaries produces an LCU for any Carleman-linearized polynomial system and yields an O(α² Q²) term count for the 3D lattice Boltzmann equation independent of spatial or temporal grid points.
The Eclipse Qrisp BlockEncoding interface provides high-level programming abstractions for block-encodings, enabling easier implementation of quantum algorithms such as QSVT, matrix inversion, and Hamiltonian simulation.
Quantum spectral method solves non-periodic Dirichlet boundary value problems with polylogarithmic complexity by extending Fourier discretization with domain doubling, antisymmetric reflection, and quantum sine transform.
qANM applies high-order perturbation via Taylor series to convert nonlinear systems to linear equations solved by variational quantum linear solver and quantum Jacobi method, with simulator validation and 98% accuracy on a noisy superconducting processor.
Unitaria is a new open-source Python library that provides a high-level, composable interface for block encodings in quantum computing, enabling automatic circuit generation and classical simulation-based verification.
Hybrid quantum interior point methods for linear programming have no practical runtime advantage over classical solvers like HiGHS on realistic instances because their quantum lower bounds already exceed classical performance under optimistic assumptions.
Adiabatic solver slightly outperforms shortcut when solution norm unknown; shortcut significantly better for non-Hermitian matrices when norm known.
citing papers explorer
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Quantum algorithm for solving high-dimensional linear stochastic differential equations via amplitude encoding of the noise term
Quantum algorithms achieve polylog(N) complexity for high-dimensional linear SDEs by amplitude-encoding the solution and noise via Dyson series or Euler-Maruyama approximations plus quantum linear systems solvers.
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Constrained Optimal Polynomials for Quantum Linear System Solvers
Constrained Uniform Polynomial (CUP) and Constrained Adaptive Polynomial (CAP) solvers achieve lower error than standard QSVT and Chebyshev methods in noise-limited regimes by optimizing accuracy versus block-encoding normalization under uniform or moment-based spectral models.
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Exponential quantum advantage in processing massive classical data
A polylog-sized quantum computer achieves exponential advantage over classical machines in classification and dimension reduction of massive classical data using quantum oracle sketching combined with classical shadows.
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QAFE$^2$: Quantum Accelerated Multiscale Finite Element Analysis
QAFE² uses quantum parallelism to evaluate every RVE problem at all quadrature points simultaneously, delivering polylog complexity in microscopic mesh size and exponential speedup over classical solvers.
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Explicit Block Encodings of Discrete Laplacians with Mixed Boundary Conditions
A modular block-encoding framework for finite-difference Laplacians supporting arbitrary combinations of Dirichlet, periodic, and Neumann boundary conditions across dimensions.
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Probabilistic quantum algorithm for Lyapunov equations and matrix inversion
Probabilistic quantum algorithm prepares mixed states proportional to Lyapunov equation solutions and matrix inverses using oracles for input matrices and a deterministic stopping rule.
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Quantum Algorithms for Nonlinear Differential Equations via Pivot-Shifted Carleman Linearization
Pivot-shifted Carleman linearization with Lyapunov transform enables logarithmic truncation and removes initial-condition lower bounds for quantum simulation of a broader class of nonlinear ODEs.
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Quantum Data Loading for Carleman Linearized Systems: Application to the Lattice-Boltzmann Equation
A matrix decomposition into linear combinations of non-unitaries produces an LCU for any Carleman-linearized polynomial system and yields an O(α² Q²) term count for the 3D lattice Boltzmann equation independent of spatial or temporal grid points.
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Block-encodings as programming abstractions: The Eclipse Qrisp BlockEncoding Interface
The Eclipse Qrisp BlockEncoding interface provides high-level programming abstractions for block-encodings, enabling easier implementation of quantum algorithms such as QSVT, matrix inversion, and Hamiltonian simulation.
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A Quantum Spectral Method for Non-Periodic Boundary Value Problems
Quantum spectral method solves non-periodic Dirichlet boundary value problems with polylogarithmic complexity by extending Fourier discretization with domain doubling, antisymmetric reflection, and quantum sine transform.
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A quantum nonlinear solver based on the asymptotic numerical method
qANM applies high-order perturbation via Taylor series to convert nonlinear systems to linear equations solved by variational quantum linear solver and quantum Jacobi method, with simulator validation and 98% accuracy on a noisy superconducting processor.
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Unitaria: Quantum Linear Algebra via Block Encodings
Unitaria is a new open-source Python library that provides a high-level, composable interface for block encodings in quantum computing, enabling automatic circuit generation and classical simulation-based verification.
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Practical lower bounds for hybrid quantum interior point methods in linear programming
Hybrid quantum interior point methods for linear programming have no practical runtime advantage over classical solvers like HiGHS on realistic instances because their quantum lower bounds already exceed classical performance under optimistic assumptions.
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Constant Factor Analysis of Optimal Quantum Linear Solvers in Practice
Adiabatic solver slightly outperforms shortcut when solution norm unknown; shortcut significantly better for non-Hermitian matrices when norm known.