Stabilizer codes for Heisenberg-limited many-body Hamiltonian estimation
Bibliographic record
Abstract
Estimating many-body Hamiltonians has wide applications in quantum technology. By allowing coherent evolution of quantum systems and entanglement across multiple probes, the precision of estimating a fully connected <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-body interaction can scale up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mi>t</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> is the number of probes and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math> is the probing time. However, the optimal scaling may no longer be achievable under quantum noise, and it is important to apply quantum error correction in order to recover this limit. In this work, we study the performance of stabilizer quantum error correcting codes in estimating many-body Hamiltonians under noise. When estimating a fully connected <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi><mml:mi>Z</mml:mi><mml:mi>Z</mml:mi></mml:math> interaction under single-qubit noise, we showcase three families of stabilizer codes – thin surface codes, quantum Reed–Muller codes and Shor codes – that achieve the scalings of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mi>t</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>t</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mi>t</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>, respectively, all of which are optimal with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math>. We further discuss the relation between stabilizer structure and the scaling with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>, and identify several no-go theorems. For instance, we find codes with constant-weight stabilizer generators can at most achieve the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> scaling, while the optimal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> scaling is achievable if and only if the code bears a repetition code substructure, like in Shor code.
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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".