Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems
Bibliographic record
Abstract
Recently Chen and Gao \cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow></mml:math>, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow></mml:math> by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables.Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mi>q</mml:mi></mml:msub></mml:math> improving on subsequent work by Chen, Gao and Yuan \citeChenGao2018. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020QuantumCouponCollector}.
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".