Triply Efficient Shadow Tomography
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Bibliographic record
Abstract
We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mi>k</a:mi></a:math>-body fermionic or <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:mi>k</d:mi></d:math>-local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mi>k</g:mi></g:math>-local Pauli operators with constant <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><j:mi>k</j:mi></j:math>. However, we prove that for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><m:mi>k</m:mi></m:math>-body fermionic observables and the full set of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><p:mi>n</p:mi></p:math>-qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.
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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 | 0.000 | 0.000 |
| Open science | 0.000 | 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