Explicit quantum-circuit simulation of nonlinear 1D fluid via second-order Carleman-linearized Boltzmann equation and QSVD Taylor ODE solver, with logarithmic scaling analysis.
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Quantum linear system solvers: A survey of algorithms and applications
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abstract
Solving linear systems of equations plays a fundamental role in numerous computational problems from different fields of science. The widespread use of numerical methods to solve these systems motivates investigating the feasibility of solving linear systems problems using quantum computers. In this work, we provide a survey of the main advances in quantum linear systems algorithms, together with some applications. We summarize and analyze the main ideas behind some of the algorithms for the quantum linear systems problem in the literature. The analysis begins by examining the Harrow-Hassidim-Lloyd (HHL) solver. We note its limitations and reliance on computationally expensive quantum methods, then highlight subsequent research efforts which aimed to address these limitations and optimize runtime efficiency and precision via various paradigms. We focus in particular on the post-HHL enhancements which have paved the way towards optimal lower bounds with respect to error tolerance and condition number. By doing so, we propose a taxonomy that categorizes these studies. Furthermore, by contextualizing these developments within the broader landscape of quantum computing, we explore the foundational work that have inspired and informed their development, as well as subsequent refinements. Finally, we discuss the potential applications of these algorithms in differential equations, quantum machine learning, and many-body physics.
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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.
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.
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.
QLS applied to single- and multi-reference linearized coupled cluster shows polylog κ and sublinear sparsity on model systems, supporting prospects of exponential advantage over conjugate gradient.
Tangent-fermion Dirac operator admits efficient LCU block-encoding via local operator pencil formulation, with lattice-size independent resources.
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.
Presents LCNU-plus-embedding data loading for any polynomial Carleman-linearized autonomous system and applies it to the 3D LBE, yielding Ns ~ O(α²Q²) terms and explicit T-gate resource estimates for two solvers.
Constrained optimal polynomials (CUP and CAP) reduce quantum linear system solver errors under noise by jointly optimizing approximation accuracy and block-encoding normalization, outperforming standard QSVT and Chebyshev-iteration solvers in numerical benchmarks.
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.
Demonstration of quantum circuit implementation for 2D obstacle flow via Carleman-linearized LBM solved with QSVT, achieving logarithmic qubit and gate scaling with lattice points.
QAFE² encodes all macroscopic RVE problems into orthogonal quantum subspaces and applies one fixed-point circuit, claiming O(M log^c M + log^c N) complexity against classical O(M N^c).
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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Explicit Quantum Circuit Simulation of Nonlinear 1-Dimensional Fluid with Carleman-linearized Boltzmann Method
Explicit quantum-circuit simulation of nonlinear 1D fluid via second-order Carleman-linearized Boltzmann equation and QSVD Taylor ODE solver, with logarithmic scaling analysis.
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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.
-
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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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 linear solvers for quantum chemistry: prospects of exponential quantum advantage
QLS applied to single- and multi-reference linearized coupled cluster shows polylog κ and sublinear sparsity on model systems, supporting prospects of exponential advantage over conjugate gradient.
-
Quantum circuit decomposition of the tangent-fermion Dirac operator
Tangent-fermion Dirac operator admits efficient LCU block-encoding via local operator pencil formulation, with lattice-size independent resources.
-
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.
-
Quantum Data Loading for Carleman Linearized Systems: Application to the Lattice-Boltzmann Equation
Presents LCNU-plus-embedding data loading for any polynomial Carleman-linearized autonomous system and applies it to the 3D LBE, yielding Ns ~ O(α²Q²) terms and explicit T-gate resource estimates for two solvers.
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Nonisothermal global-pressure exactness in fractured multiphase flow with aperture feedback
Constrained optimal polynomials (CUP and CAP) reduce quantum linear system solver errors under noise by jointly optimizing approximation accuracy and block-encoding normalization, outperforming standard QSVT and Chebyshev-iteration solvers in numerical benchmarks.
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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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A Demonstration of Quantum Circuit Implementation for Obstacle Flow Using Carleman-Linearized Lattice Boltzmann Method
Demonstration of quantum circuit implementation for 2D obstacle flow via Carleman-linearized LBM solved with QSVT, achieving logarithmic qubit and gate scaling with lattice points.
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QAFE$^2$: Quantum accelerated multiscale finite element analysis
QAFE² encodes all macroscopic RVE problems into orthogonal quantum subspaces and applies one fixed-point circuit, claiming O(M log^c M + log^c N) complexity against classical O(M N^c).
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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.