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algorithms/qsvt/qsvt_matrix_inversion/qsvt_matrix_inversion.ipynb

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"> **Quantum Matrix Inversion (or Quantum Linear Solver) via QSVT** provides a framework for solving linear systems exponentially faster than known classical methods under certain conditions [\\[1\\]](#ref-grand). The algorithm employs a polynomial transformation of singular values to block-encode the inverse of a matrix, which can then be applied on a prepared quantum state to solve a linear equation. Given an efficient routine to embed the classical matrix as a quantum function (block-encoding), this algorithm gives a clean and optimal way to implement matrix inversion compared to other quantum methods. Quantum linear solvers based on block-encoding have many applications, e.g., in [Plasma Physics](https://github.com/Classiq/classiq-library/blob/main/applications/plasma/vlasov_ampere/vlasov_ampere.ipynb) and [Fluid Dynamics](https://github.com/Classiq/classiq-library/blob/main/applications/cfd/linear_qls_for_hybrid_solvers/qls_qsvt.ipynb).\n",
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"> **Quantum Matrix Inversion (or Quantum Linear Solver) via QSVT** provides a framework for solving linear systems exponentially faster than known classical methods under certain conditions [\\[1\\]](#ref-grand). The algorithm employs a polynomial transformation of singular values to block-encode the inverse of a matrix, which can then be applied on a prepared quantum state to solve a linear equation. Given an efficient routine to embed the classical matrix as a quantum function (block-encoding), this algorithm gives a clean and optimal way to implement matrix inversion compared to other quantum methods. Quantum linear solvers based on block-encoding have many applications, e.g., in [Plasma Physics](https://github.com/Classiq/classiq-library/blob/main/applications/plasma/vlasov_ampere/vlasov_ampere.ipynb) and [Fluid Dynamics](https://github.com/Classiq/classiq-library/blob/main/applications/cfd/qls_for_hybrid_solvers/qls_qsvt.ipynb).\n",
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"> - **Input:** An $N \\times N$ matrix $A$ with condition number $\\kappa$ (the ratio between the maximal and minimal singular values), and a vector $|b\\rangle$ prepared as a quantum state.\n",
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"> - **Promise:** The matrix $A$ is well-conditioned (invertible, with bounded spectrum) and can be efficiently block-encoded in a unitary $U_A$ (see technical note at the end of this demo).\n",

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