Quantum algorithm for the gradient of a logarithm-determinant
Bibliographic record
Abstract
The logarithm-determinant is an widely-present operation in many areas of physics and computer science. Derivatives of the logarithm-determinant compute physically relevant quantities in statistical physics models, quantum field theories, as well as the inverses of matrices. A multi-variable version of the quantum gradient algorithm is developed here to evaluate the derivative of the logarithm-determinant. From this, the inverse of a sparse-rank input operator may be determined efficiently. Measuring an expectation value of the quantum state--instead of all $N^2$ elements of the input operator--can be accomplished in $O(k/\varepsilon^2)$ time in the idealized case for $k$ relevant eigenvectors of the input matrix with precision $\varepsilon$. A practical implementation of the required operator will likely need $\log_2N$ overhead, giving an overall complexity of $O((k\log_2 N)/\varepsilon^2)$. The method applies widely and converges super-linearly in $k$ when the condition number is high. The best classical method we are aware of scales as $N$. Given the same resource assumptions as other algorithms, such that an equal superposition of eigenvectors is available efficiently, the algorithm is evaluated in the practical case as $O(\log_2 N/\varepsilon^2)$. The output is given in $O(1)$ queries of oracle, which is given explicitly here and only relies on time-evolution operators that can be implemented with arbitrarily small error. The algorithm is envisioned for fully error-corrected quantum computers but may be implementable on near-term machines. We discuss how this algorithm can be used for kernel-based quantum machine-learning.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.018 | 0.006 |
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 source (direct Gemma or distilled Codex), 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".