Sample-optimal classical shadows for pure states
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Bibliographic record
Abstract
We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C1;</mml:mi></mml:math> in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">T</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>O</mml:mi><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> for any Hermitian observable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi></mml:math> to within additive error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03F5;</mml:mi></mml:math> provided <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">T</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>O</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>&#x2264;</mml:mo><mml:mi>B</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">&#x2016;</mml:mo><mml:mi>O</mml:mi><mml:mo fence="false" stretchy="false">&#x2016;</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math>. Our main result applies to the joint measurement setting, where we show <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi mathvariant="normal">&#x0398;</mml:mi><mml:mo stretchy="false">&#x007E;</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msqrt><mml:mi>B</mml:mi></mml:msqrt><mml:msup><mml:mi>&#x03F5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>&#x03F5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> samples of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C1;</mml:mi></mml:math> are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C1;</mml:mi></mml:math> can be compressed for observable estimation. In the independent measurement setting, we show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msqrt><mml:mi>B</mml:mi><mml:mi>d</mml:mi></mml:msqrt><mml:msup><mml:mi>&#x03F5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>&#x03F5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator.
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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 | 0.000 |
| 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