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Bibliographic record
Abstract
Quantum signal processing (QSP) and quantum singular value transformation (QSVT) currently stand as the most efficient techniques for implementing functions of block-encoded matrices, a central task that lies at the heart of most prominent quantum algorithms. However, current QSP approaches face several challenges, such as the restrictions imposed on the family of achievable polynomials and the difficulty of calculating the required phase angles for specific transformations. In this paper, we present a generalized quantum signal processing (GQSP) approach, employing general SU(2) rotations as our signal-processing operators, rather than relying solely on rotations in a single basis. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mrow><a:mo stretchy="false">|</a:mo></a:mrow><a:mi>P</a:mi><a:mrow><a:mo stretchy="false">|</a:mo></a:mrow><a:mo>≤</a:mo><a:mn>1</a:mn></a:math>, a restriction necessary due to the unitary nature of quantum computation. Furthermore, GQSP provides a straightforward recursive formula for determining the rotation angles needed to construct the polynomials in cases where <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><f:mi>P</f:mi></f:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><i:mi>Q</i:mi></i:math> are known. In cases where only <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><l:mi>P</l:mi></l:math> is known, we provide an efficient optimization algorithm capable of identifying in under a minute of GPU time, a corresponding <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><o:mi>Q</o:mi></o:math> for polynomials of degree on the order of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><r:msup><r:mn>10</r:mn><r:mn>7</r:mn></r:msup></r:math>. We further illustrate GQSP simplifies QSP-based strategies for Hamiltonian simulation, offer an optimal solution to the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><u:mi>ϵ</u:mi></u:math>-approximate fractional query problem that requires <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><x:mrow><x:mi mathvariant="script">O</x:mi></x:mrow><x:mrow><x:mo>(</x:mo><x:mo stretchy="false">(</x:mo><x:mn>1</x:mn><x:mo>/</x:mo><x:mi>δ</x:mi><x:mo stretchy="false">)</x:mo><x:mo>+</x:mo><x:mi>log</x:mi><x:mo></x:mo><x:mo stretchy="false">(</x:mo><x:mn>1</x:mn><x:mo>/</x:mo><x:mi>ϵ</x:mi><x:mo stretchy="false">)</x:mo><x:mo>)</x:mo></x:mrow></x:math> queries to perform where <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><fb:mrow><fb:mi mathvariant="script">O</fb:mi></fb:mrow><fb:mo stretchy="false">(</fb:mo><fb:mn>1</fb:mn><fb:mo>/</fb:mo><fb:mi>δ</fb:mi><fb:mo stretchy="false">)</fb:mo></fb:math> is a proved lower bound, and introduces novel approaches for implementing bosonic operators. Moreover, we propose a novel framework for the implementation of normal matrices, demonstrating its applicability through synthesis of diagonal matrices, as well as the development of a new algorithm for convolution through synthesis of circulant matrices using only <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><lb:mrow><lb:mi mathvariant="script">O</lb:mi></lb:mrow><lb:mo stretchy="false">(</lb:mo><lb:mi>d</lb:mi><lb:mi>log</lb:mi><lb:mo></lb:mo><lb:mi>N</lb:mi><lb:mo>+</lb:mo><lb:msup><lb:mi>log</lb:mi><lb:mn>2</lb:mn></lb:msup><lb:mo></lb:mo><lb:mi>N</lb:mi><lb:mo stretchy="false">)</lb:mo></lb:math> 1 and 2-qubit gates for a filter of lengths <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><rb:mi>d</rb:mi></rb:math>. Published by the American Physical Society 2024
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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 | 0.001 | 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 it