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        "description": "The speedup concerns preparing a quantum state proportional to a linear-system solution, under sparse/access, conditioning and accuracy assumptions. It does not output every solution coordinate cheaply; state preparation and readout costs matter. Poor conditioning and arbitrary dense/classically supplied data can erase the claimed advantage. The original theorem is not a demonstrated general-purpose hardware speedup.",
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      "statement": "# HHL: a quantum state of a linear-system solution\n\nHarrow, Hassidim and Lloyd give a quantum algorithm that prepares a state proportional to a linear-system solution and enables selected observable estimates, with strong dimension scaling under explicit matrix-access, conditioning, preparation and precision assumptions.\n\nMany computations reduce to solving Ax=b. HHL changes what the solver returns: a quantum state whose amplitudes encode a normalized solution, rather than a printed list of numbers. It estimates eigenvalues, rotates an auxiliary system to weight each component by the inverse eigenvalue, then reverses the estimation and selects successful outcomes. For suitable sparse matrices with efficient access and a readily prepared input state, the dependence on matrix dimension can be logarithmic. The condition number, desired precision and measurement costs still matter. Reading every classical component removes that advantage; a hard-to-load input or poorly conditioned matrix can also dominate the cost. The 2009 revised paper includes detailed error analysis and complexity-theoretic arguments, not just a small example. This record covers the original quantum-state algorithm and its declared assumptions, without claiming that every classical linear system becomes exponentially faster or that a practical quantum computer was demonstrated.\n\n## Scientific scope\n\nThe speedup concerns preparing a quantum state proportional to a linear-system solution, under sparse/access, conditioning and accuracy assumptions. It does not output every solution coordinate cheaply; state preparation and readout costs matter. Poor conditioning and arbitrary dense/classically supplied data can erase the claimed advantage. The original theorem is not a demonstrated general-purpose hardware speedup.\n\n## People and systems\n\nAram W. Harrow; Avinatan Hassidim; Seth Lloyd\n\n\n\n## Original sources\n\n- [Harrow, Hassidim and Lloyd, Quantum algorithm for linear systems of equations, exactarXiv0811.3171v3,2009](https://arxiv.org/pdf/0811.3171v3)\n- [arXiv0811.3171v3 submissionhistory and completeauthor metadata](https://arxiv.org/abs/0811.3171v3)\n",
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