Develops a quantum algorithm for linear matrix differential equations with query complexity O~(ν L t / ε) that is nearly optimal and yields polynomial to exponential speedups for open quantum system simulation.
A shortcut to an optimal quantum linear system solver
11 Pith papers cite this work, alongside 1 external citations. Polarity classification is still indexing.
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abstract
Given a linear system of equations $A\boldsymbol{x}=\boldsymbol{b}$, quantum linear system solvers (QLSSs) approximately prepare a quantum state $|\boldsymbol{x}\rangle$ for which the amplitudes are proportional to the solution vector $\boldsymbol{x}$. Asymptotically optimal QLSSs have query complexity $O(\kappa \log(1/\varepsilon))$, where $\kappa$ is the condition number of $A$, and $\varepsilon$ is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm $\lVert\boldsymbol{x}\rVert$ is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is $(1+O(\varepsilon))\kappa \ln(2\sqrt{2}/\varepsilon)$. If the norm is unknown, our method allows it to be estimated up to a constant factor using $O(\log\log(\kappa))$ applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal $O(\kappa \log\log(\kappa)\log\log\log(\kappa)+\kappa\log(1/\varepsilon))$ total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that $O(\kappa)$ complexity can be achieved for norm estimation, yielding an optimal QLSS with $O(\kappa\log(1/\varepsilon))$ complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of $56\kappa+1.05\kappa \ln(1/\varepsilon)+o(\kappa)$ for the complexity of our optimal QLSS.
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Two quantum linear system solvers are presented with query complexity independent of the condition number, scaling instead with an effective condition number or a solution-norm ratio.
New SelectCopy architecture and qubit-constrained optimizations reduce QROM Toffoli cost from ~2N/λ to ~(1 + 1/b)N/λ while preserving the ability to trade dirty qubits for lower gate count.
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.
Probabilistic quantum algorithm prepares mixed states proportional to Lyapunov equation solutions and matrix inverses using oracles for input matrices and a deterministic stopping rule.
Using multi-product formulas in LCHS produces commutator-sensitive error bounds and better quadrature scaling than norm-based analyses for dissipative dynamics.
Quantum Koopman Algorithms define an observable-space quantum framework for simulating linear quantum and nonlinear classical dynamics with polylog gate costs in some cases.
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.
Adiabatic solver slightly outperforms shortcut when solution norm unknown; shortcut significantly better for non-Hermitian matrices when norm known.
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Efficient quantum algorithm for linear matrix differential equations and applications to open quantum systems
Develops a quantum algorithm for linear matrix differential equations with query complexity O~(ν L t / ε) that is nearly optimal and yields polynomial to exponential speedups for open quantum system simulation.
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Two quantum linear system solvers are presented with query complexity independent of the condition number, scaling instead with an effective condition number or a solution-norm ratio.
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New SelectCopy architecture and qubit-constrained optimizations reduce QROM Toffoli cost from ~2N/λ to ~(1 + 1/b)N/λ while preserving the ability to trade dirty qubits for lower gate count.
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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 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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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
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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.