The complexity of simulating local measurements on quantum systems
Bibliographic record
Abstract
An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity class PQMA[log] , and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is PQMA[log] -complete. In this paper, we continue the study of PQMA[log] , obtaining the following lower and upper bounds.Lower bounds (hardness results): - The PQMA[log] -completeness result of [Ambainis, CCC 2014] requires O(logn) -local observables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5 -local Hamiltonians is PQMA[log] -complete, resolving an open question of Ambainis.- We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly PQMA[log] -complete. - We identify a flaw in [Ambainis, CCC 2014] regarding a PUQMA[log] -hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a ``query validation'' technique, we build on [Ambainis, CCC 2014] to obtain PUQMA[log] -hardness for estimating spectral gaps under polynomial-time Turing reductions. Upper bounds (containment in complexity classes): - PQMA[log] is thought of as ``slightly harder'' than QMA. We justify this formally by exploiting the hierarchical voting technique of [Beigel, Hemachandra, Wechsung, SCT 1989] to show PQMA[log]⊆PP . This improves the containment QMA⊆PP [Kitaev, Watrous, STOC 2000]. This work contributes a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves a promise problem. This is particularly relevant for quantum complexity theory, where most natural classes such as BQP and QMA are defined as promise classes.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.043 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.004 | 0.006 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.011 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".