A Quantum Computer Amenable Sparse Matrix Equation Solver
Bibliographic record
Abstract
Quantum computation offers a promising alternative to classical computing methods in many areas of numerical science, with algorithms that make use of the unique way in which quantum computers store and manipulate data often achieving dramatic improvements in performance over their classical counterparts. The potential efficiency of quantum computers is particularly important for numerical simulations, where the capabilities of classical computing systems are often insufficient for the analysis of real-world problems. In this work, we study problems involving the solution of matrix equations, for which there currently exists no efficient, general quantum procedure. We develop a generalization of the Harrow/Hassidim/Lloyd algorithm by providing an alternative unitary for eigenphase estimation. This unitary, which we have adopted from research in the area of quantum walks, has the advantage of being well defined for any arbitrary matrix equation, thereby allowing the solution procedure to be directly implemented on quantum hardware for any well-conditioned system. The procedure is most useful for sparse matrix equations, as it allows for the inverse of a matrix to be applied with $\mathcal{O}\left(N_{nz}\log\left(N\right)\right)$ complexity, where $N$ is the number of unknowns, and $N_{nz}$ is the total number of nonzero elements in the system matrix. This efficiency is independent of the matrix structure, and hence the quantum procedure can outperform classical methods for many common system types. We show this using the example of sparse approximate inverse (SPAI) preconditioning, which involves the application of matrix inverses for matrices with $N_{nz}=\mathcal{O}\left(N\right)$.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".